Topology, Geometry, and Physics Seminar

Room 6417
Grad Center CUNY
365 Fifth Ave

Fall 2019

Dec 18

10:00-11:30
TBA
Introduction to Quantum Field Theory

12:00-1:30
TBA
TBA


2:00-3:30
TBA
TBA

Dec 11

10:00-11:30
TBA
Introduction to Quantum Field Theory

12:00-1:30
James Austin Myer
A1 Homotopy Theory


2:00-3:30
TBA
TBA

Dec 4

10:00-11:30
TBA
Introduction to Quantum Field Theory

12:00-1:30
Jeffrey Kroll
An introduction to flat (unitary) connections and parallel translation in smooth vector bundles

This is an introductory talk on connections and parallel translation in a smooth vector bundle (equipped with a bundle metric). We will begin with the definition of a connection compatible with a bundle metric, show how a flat connection allows one to define twisted de Rham cohomology with values in the bundle, and understand tangents to the space of flat connections. Then we will pivot to the geometric story wherein a connection allows one to parallel translate vectors in the bundle along paths in the base. I will cover some essential properties of parallel translation and give an explicit (local) formula for computing parallel translation.

2:00-3:30
Jeffrey Kroll
The Hamiltonian of trace of holonomy on the moduli space of flat unitary connections on a Kahler manifold
Given a smooth hermitian vector bundle over a closed Kahler manifold, we define the moduli space of flat unitary connections – that is, the moduli space of flat connections that are compatible with the bundle metric – and show that it has a symplectic structure. The difficulty is in establishing non-degeneracy. To do this, we use the fact that to each flat unitary connection there is a unique holomorphic structure on the bundle such that the connection becomes a Chern connection. This observation allows us to use a twisted version of the Hard Lefschetz Theorem which says that multiplication by the Kahler class induces isomorphisms on complimentary dimensional de Rham cohomology groups with values in the bundle. Next, for a fixed loop, we consider the function on the moduli space given by trace of holonomy around the loop, compute its the Hamiltonian vector field, and show that it can be represented by a twisted homology class constructed from the holonomy of the loop.

Nov 27

10:00-11:30
TBA
Introduction to Quantum Field Theory

12:00-1:30
James Austin Myer
The Lefschetz Fixed Point Formula and its Role in the Weil Conjectures


2:00-3:30
TBA
TBA

Nov 20

10:00-11:30
Raymond Puzio
Review of Quantum Field Theory --- Regularization and Renormalization

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.

This lecture will review and summarize some of the material from the last few lectures and illustrate abstract concepts with concrete examples. Specifically, it will go over the following topics:

  • Simplified model to explain renormalization and running coupling constant.

  • A few comments on and examples of smooth versus sharp cutoffs and parametrices.

  • Computing the four-point function of theory to second order, noting where and how the expressions diverge, exhibiting the running coupling and renormalization transform.


12:00-1:30
Jiahao Hu
Topological resolution of singularities

Steenrod asked the following question: can all singular homology classes be represented by manifolds, namely are they push-forward of fundamental classes of oriented smooth manifolds? Thom answered this question both positively and negativel: yes, for mod 2 and rational homology, but in generally no for integral homology. He observed there are infinitely many topological obstructions to “resolving the singularities” of a integral homology class, and constructed an example where those obstructions do not vanish.

However, Hironaka later showed, using heavy machinery from algebraic geometry, that all complex algebraic varieties admit resolutions. The topological consequence of that is, all those obstructions discovered by Thom must vanish on algebraic homology classes of a complex algebraic variety, which is quite surprising.

In this talk, I will try to explain what are those topological obstructions, and (if time permitted) show why those obstructions vanish for low-dimensional complex algebraic varieties without referring to Hironaka’s theorem.

2:00-3:30
Andrew Obus
Prismatic cohomology and perfectoid spaces

The theory of perfectoid spaces and perfectoid rings has taken the arithmetic geometry world by storm over the last decade. In the first half of the talk, I will give a brief introduction to the theory. In the second half, I will discuss another perspective on perfectoid spaces based on the theory of prisms, and show why perfectoid spaces are important when relating prismatic cohomology to étale cohomology. The first half of the talk is self-contained, but the second half will depend on the previous week's lecture.

Nov 13

10:00-11:30
Raymond Puzio
Review of Quantum Field Theory --- Diagrammatic Expansion

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.

This lecture will review and summarize some of the material from the last few lectures and illustrate abstract concepts with concrete examples. Specifically, it will go over the following topics:

  • The derivation of the perturbation series for the generating function. In particular, the derivation of the formula by way of Fourier analysis.

  • Feynman parameterization of propagator and its interpreation as a path integral.

  • Combining the above two topics, express the integral over field configurations as a sum of path integrals over graphs. Physical intuition for this as interacting particles and toy example in zero-dimensional spacetime.


12:00-1:30
Felix Wierstra
Introduction to deformation theory
In this talk I will explain how Lie algebras are used in deformation theory. The goal of deformation theory is to study deformations of algebraic objects. It turns out that all the information of such a deformation problem can be encoded in a differential graded Lie algebra. In this talk I explain how this is done and give some examples.

2:00-3:30
Andrew Obus
Prisms and prismatic cohomology

We define prisms and construct two important complexes of sheaves arising from a prism. The cohomology of these complexes relates a host of cohomology theories including étale cohomology, crystalline cohomology, and de Rham cohomology.

Nov 6

10:00-11:30
Jin-Cheng Guu & Ying Hong Tham
Renormalization in QFT

We introduced a generating function W(P,I) that helps compute path
integrals last time. As P varies, W behaves well, and we will use this
fact to define renormalization group flow in Costello's formulation.

12:00-1:30
Rob Thompson
Some applications of spectral sequences

We will go through several concrete applications of spectral sequences. You don't need to have attended any previous lectures on the construction or derivation of spectral sequences. If you have that's great but if not you can just treat a spectral sequence as a "black box" or a "machine" which takes a certain input, digests it, and produces a certain output. That's a perfectly reasonable way to start learning about spectral sequences. For each of the examples listed I will start by summarizing what the input and output is.

1) Given a smooth n-dimensional manifold and a good covering by open sets one can define something called the Cech cohomology of the cover. Using the differential forms on the manifold one can define something called the de Rham cohomology of the manifold. In fact, these are isomorphic. The proof we will give will be a standard argument using the spectral sequence of a double complex - a technique used throughout mathematics.

2) Serre spectral - we wlll give several Serre spectral sequence calculations for various examples of fiber bundles and fibrations. We'll compute the cohomology of some Eilenberg MacLane space - these are spaces that have only one non-zero homotopy group - and use this to deduce some homotopy groups of spheres.

3) We will use the Atiyah-Hirzebruch-Serre spectral sequence to compute the topological K-theory of complex projective spaces.

2:00-3:30
Andrew Obus
Algebraic detection of torsion in singular cohomology
If X is a compact complex manifold, then it is a classical result that the de Rham cohomology of X is isomorphic to the singular cohomology of X with complex coefficients. Unfortunately, de Rham cohomology is not a fine enough invariant to account for torsion in the singular cohomology of X with integral coefficients. We will describe an example of this involving an Enriques surface.

However, if X is an algebraic variety defined over the rational numbers, p-torsion classes in the integral singular cohomology of X can be related to the algebraic de Rham cohomology of the "reduction mod p" of X. We will carefully discuss the reduction mod p process, and foreshadow the theory of "prismatic cohomology" of Bhatt and Scholze that provides the link between singular cohomology and algebraic de Rham cohomology, via étale cohomology.

Oct 30

10:00-11:30
Jin-Cheng Guu
Feynman diagrams
We will continue our journey to the definition of Feynman diagrams and see how it helps compute path integrals of interest. This also allows us to define the renormalization group flow with concise notations. If time permits, we will discuss the geometric interpretation of the Feynman diagrams.

12:00-1:30
Rob Thompson
Bousfield's resolution model categories

In Bousfield and Kan's 1972 book "the yellow monster" (real title: Homotopy Limits, Completions, and Localizations) they construct for any commutative ring R with unit, an R-completion functor from spaces to spaces, and develop many properties of this object, including a spectral sequence which is designed to compute it. Thinking of this completion as being associated to singular homology with coefficients in R, in 2001 Martin Bendersky and I studied a generalization of this construction to complex topological K-theory. A few years later, Bendersky and John Hunton studied a further generalization of this construction to the homology theory represented by an arbitrary ring spectrum.

Around this time Bousfield published a paper titled "Cosimplicial resolutions and homotopy spectral sequences in model categories", which is what I will talk about. Given a suitable model category C, and a suitable collection of group objects in Ho(C), called a class of injective models, Bousfield constructs a model category structure on cC, the category of cosimplicial objects over C. From there one can defined derived functors, completions, and a homotopy spectral sequence, and prove a number of theorems. In addition to vastly generalizing completion with respect to a ring spectrum discussed above, this 'resolution model category structure' is a generalization of the 'E_2 model category structure' of Dwyer-Kan-Stover, which has applications to a number of realizability problems in homotopy theory.

For example Bousfield's resolution model category structure plays a key role in the work of Goerss and Hopkins on the realizability of E-infinity structures.

2:00-3:30
Felix Wierstra
Iterated suspensions and the little disks operad

In this talk I will describe the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every n-fold suspension is a coalgebra over the little n-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an n-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an n-fold suspension to be an (n+1)-fold suspension, i.e. if this cooperation is non-zero then the space is not an (n+1)-fold suspension. This is joint work with José Manuel Moreno-Fernández.

Oct 23

10:00-11:30
Jin-Cheng Guu
Feynman diagrams
As we have seen in the past few weeks, we ought to compute an integral over an infinite dimensional space, on which there's no well defined measure. To make sense of it, we expand the weight in terms of $h$, and hope to calculate the coefficients. To do this, we will need the techniques of Feynman's diagrams. In this talk, I will define Feynman's diagrams and a relating generating function, and compute them in some basic cases. If time permits, I will scratch on why this is helpful for our calculations of interest.

12:00-1:30
Felix Wierstra
Cochains and rational homotopy type
In this talk I will show that two simply-connected spaces of finite type are rationally equivalent if and only if their singular cochains considered as associative algebras can be connected to each other by a zig-zag of quasi-isomorphisms of associative algebras. This result is a consequence of the more general statement that two commutative algebras can be connected by a zig-zag of quasi-isomorphisms of commutative algebras if and only if they can be connected by a zig-zag of quasi-isomorphisms of associative algebras. This is joint work with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.

2:00-3:30
Rob Thompson
Bousfield's resolution model categories

In Bousfield and Kan's 1972 book "the yellow monster" (real title: Homotopy Limits, Completions, and Localizations) they construct for any commutative ring R with unit, an R-completion functor from spaces to spaces, and develop many properties of this object, including a spectral sequence which is designed to compute it. Thinking of this completion as being associated to singular homology with coefficients in R, in 2001 Martin Bendersky and I studied a generalization of this construction to complex topological K-theory. A few years later, Bendersky and John Hunton studied a further generalization of this construction to the homology theory represented by an arbitrary ring spectrum.

Around this time Bousfield published a paper titled "Cosimplicial resolutions and homotopy spectral sequences in model categories", which is what I will talk about. Given a suitable model category C, and a suitable collection of group objects in Ho(C), called a class of injective models, Bousfield constructs a model category structure on cC, the category of cosimplicial objects over C. From there one can defined derived functors, completions, and a homotopy spectral sequence, and prove a number of theorems. In addition to vastly generalizing completion with respect to a ring spectrum discussed above, this 'resolution model category structure' is a generalization of the 'E_2 model category structure' of Dwyer-Kan-Stover, which has applications to a number of realizability problems in homotopy theory.

For example Bousfield's resolution model category structure plays a key role in the work of Goerss and Hopkins on the realizability of E-infinity structures.

Oct 16 is a Monday schedule

Oct 9 No classes at CUNY

Oct 2

10:00-11:30
Raymond Puzio
Introduction to Quantum Field Theory
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.

This lecture will wrap up the introductory overview before proceeding to a more detailed study of the material. We will begin by reviewing the basic facts and definitions of quantum field theory. Then we will discuss the renormalization semigroup and introduce Wilson's point of view. We conclude with Costello's definition of perturbative quantum field theory and see how it gives a precise formulation of what one calculates in field theory independent of the heuristic derivation.

12:00-2:30
Christoph Dorn
A general framework for invariants: factorisation homology
This is going to be a very introductory and self-contained talk on the hot topic of factorisation homology. A core idea underlying the construction of factorisation homology is the basic observation that many mathematical entities are constructed from “simpler” local models (such as manifolds being locally constructed from Euclidean space). Inputting algebraic/categorical data for the local models can then be used to “glue” this data together into an invariant for a given entity. Many classical and more recent invariants can be recovered in the language of factorisation homology. We will focus our attention on the case of framed manifolds and E_n algebras in symmetric monoidal infinity-categories. The latter concepts (along with other useful ideas such as Kan extensions) will be introduced in the talk. If time permits we might also discuss the relation to topological field theories and/or generalisations to "vari-framed stratifications” as recently introduced by Ayala-Francis.

3:00-5:00 (Math Lounge)
Short Talks
TBA

Sept 25

10:00-11:30
Raymond Puzio
Introduction to Quantum Field Theory
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.  In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory."  This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.

In this session, we will focus on renormalization.  After reviewing the basics of the subject and the perturbation series, we will discuss regularization in terms of cut-off, Feynman parameter, and parametices.  Then we will introduce renormalization in terms of running coupling constants and exhibit the renormalization flow on the space of theories.

12:00-1:30
Micah Miller
The Reedy Model category structure on diagram categories
If C is a Reedy category and M is a model category, the category of C-diagrams in M can be given a model category structure called the Reedy model category structure. Furthermore, if M is a simplicial model category, the diagram category can be given a simplicial model category structure. In this talk, we will give the definitions of a Reedy category and Reedy model category, and look at the example of the category of cosimplicial simplicial sets.

2:00-3:30
Kris Klosin
Modular forms and applications to arithmetic
Modular forms have featured prominently in some of the milestone number-theoretic results and conjectures of the last decades. I will discuss their definition and basic arithmetic properties as well as their relation with other objects interesting from the number-theoretic point of view, e.g., class groups, elliptic curves and, if time permits, abelian surfaces.  The familiarity with any of these objects is not necessary.

Sept 18

10:00-11:30
Raymond Puzio
Introduction to Quantum Field Theory
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.

In Feynman's "sum over history" formulation of quantum mechanics, physical quantities are expressed as functional integrals. By a stationary phase argument, these integrals may be expanded as power series in powers of Planck's constant. The leading term of this series is the classical limit and subsequent terms are quantum corrections.

However, when one attempt to compute this series for non-trivial field theories, all the correction terms diverge! This is the famous "ultraviolet catastrophe". To solve this problem, Feynman and Dyson introduced the techniques of regularization and renormalization. By regularizing, we mean modifying the integrand so as to produce a finite, well-defined answer. There are many ways of doing this, but we will only consider regularization techniques based on parametrices which include, as special cases, energy cut-off and Feynman parameter regularizations. Renormalization is the process whereby we adjust the parameters in the action so as to produce a finite answer in the limit where the cut-off is sent to infinity. We will describe this process in terms of running couplings and renormalization flow.

12:00-1:30
Rob Thompson
An introduction to Spectral sequences with an application to differential topology
This talk will be an introduction to spectral  sequences, starting at the beginning and assuming no previous experience with them.  We’ll start with the general formulation in terms of an exact couple, and quickly specialize to the example of the Serre Spectral Sequence for DeRham cohomology, following the development in Bott and Tu.

2:00-3:30
Martin Bendersky
The Bousfield Kan spectral sequence
There is an unstable version of the Adams spectral sequence. For a reasonable space, X the E_2 term is described in terms of the homology of X. The spectral sequence converges to the homotopy groups of the completion which is many cases gives information about the homotopy of X. This is one of the main reasons Bousfield and Kan introduced the completion functor. There are natural ways to extend this construction. One is to generalized homology theories. For example the BK spectral sequence based on complex cobordism is the unstable Novikov Spectral Sequence. I will say a few words about the UNSS. Another generalization is to a relative version of the completion. That is to say for a fibration E-->B one can give a cosimplicial construction of the fibrewise R-completions. The "one" who do this are actually two - Bousfield and Kan. Unfortunately the E_2 term of the resulting spectral sequence cannot be described in terms of the homology groups of E and B. There is a modification (due to Dwyer, Miller and Neisendorfer) which does lead to a spectral sequence with a homologically identifiable E_2 term which in many cases converges to the homotopy of the space of sections of the BK fibrewise completion. I will sketch some of these ideas.

Sept 11

10:00-11:30
Raymond Puzio
Introduction to Quantum Field Theory
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we
are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session. After a review of basics, we will introduce the Feynman path integral approach to quantum field theory. We will start with the basics about observables and expectation values, then move on to the topics of regularization, Lorentzian vs. Euclidean signature, and renormalization. We will look at the specifics of Feynman parameterization, parametrices, and renormalization flow. This material corresonds to sections 2-4 of chapter 1 of Costello's book.

12:00-1:30
Collective Participation
Working with students on some fundamental concepts
We get together to talk math even more informally than the morning and afternoon sessions. One possible topic to discuss is invariants of flat bundles and the Gelfand-Fuchs cohomology. Other possible topics are Lie algebra homology of infinite by infinite matrices and cyclic homology. Feel free to bring your lunch.

2:00-3:30
Martin Bendersky
The Bousfield-Kan completion
In Bousfield and Kan's yellow monster the notion of the completion of a simplicial set with respect to a ring is introduced. Depending on the ring, it is related to the localization or p-adic completion of the geometric realization of the simplicial set. The R-completion is a functor from simplicial sets to simplicial sets. The process involves first constructing a cosimplicial simplicial set associated to X. The geometric realization of this cosimplicial simplicial set is the BK completion. Of course this abstract has no content. I will try to explain what these words mean in my talk.

Sept 4

10:00-11:30
Raymond Puzio
Introduction to Quantum Field Theory and Costello’s Programme
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere. In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are studying Costello's book "Renormalization and Effective Field Theory." This presentation does not presuppose any background in physics or quantum theory and each session is designed to be accessible to anyone with a general background in mathematics and comprehensible as a unit independently of other session.
We will start with an introductory overview. Firstly, we will begin with a little historical and scientific sketch to set the context. Then we will introduce the intuitions behind the basic concepts and techniques of the subject --- fields, observables, actions, Lorentzian vs. Euclidean signature, expectation values, path integration, regularization, and renormalization. We will end by going over the outline of Costello's book, relating his programme to our informal overview, and setting out the plan for the next few sessions.

12:00-1:30
Collective Participation
Working with students on some fundamental concepts
We get together to talk math even more informally than the morning and afternoon sessions. One possible topic to discuss is invariants of flat bundles and the Gelfand-Fuchs cohomology. Other possible topics are Lie algebra homology of infinite by infinite matrices and cyclic homology. Feel free to bring your lunch.

2:00-3:30
Mahmoud Zeinalian
An introductory discussion about smooth and holomorphic vector bundles and their characteristic classes.
One can present a vector bundle in terms of its transition functions {g_ij} which satisfy 1) g_ii=id, 2) g_ij=g_ji, and 3) g_kj g_ji=g_ki. These conditions correspond perfectly with the reflexive, symmetric, and transitive criteria for an equivalence relation. Because of this, a vector bundle has a total space. This total space can be used to construct an invariant for holomorphic vector bundles called the Atiyah Class.

The Atiyah class gives rise to characteristic class invariants that live in the Hodge cohomology. There are, however, more general situations where the transition functions do not fit together as perfectly and there is a discrepancy between g_kj g_ji and g_ki. This discrepancy can be analyzed additively or multiplicatively. Trivializing the additive discrepancy g_kj g_ji - g_ki gives rise to the concept of the Toledo-Tong complexes.

These are flexible objects which behave like bundles, but they do not have a total space in the naive sense. There is a however a sophisticated notion of a total space for these too. Toledo and Tong showed that even the Atiyah class makes sense and that these objects have characteristic classes. In this talk, I will give an introduction to this topic aimed at graduate students.

One can also trivialize the multiplicative discrepancy g_kj g_ji / g_ki and reach at the concept of a non-abelian gerbe. Aside from the particular case of twisted vector bundles, characteristic classes of non-abelian gerbes are not understood and defining them is an active area of research. We leave this for a future discussion.

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Spring 2019

May 8

10:00-11:30
Raymond Puzio
Renormalizability of the scalar field with quartic interaction in four dimensions
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.  In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

We will begin by recalling the basic ideas of perturbative quantum field theory and renormalization.  Then we will study the Wilsonian renormalization of a scalar field theory in momentum variables.  As Zee puts it, this can be thought of as “the trick of doing an integral a little bit at a time.” --- i.e. seeing how the partition function depends upon cut-off by only integrating over those Fourier modes of the field whose mode vectors lie in a spherical shell.  The result of this partial integration is that the change in the partition function can be expressed as the sum of a so-called geometrical and dynamical terms.  By ignoring the dynamical term, it is easy to solve for the renormalization flow and see that there are only three relevant terms.  To show that the scalar field with quartic interaction is renormalizable, we need to verify that this is still the case when one includes the dynamical term.  This was done by Polchinski in the attached papers and we will go through the main outline of his proof.  This proof begins by deriving a differential equation for the dependance of the effective Lagrangian on the cut-off.  This equation can be split into a piece corresponding to the relevant terms and a remainder.  By placing bounds on this remainder, one shows that taking the dynamical term into account does not qualitatively change how the renormalization flow behaves and hence the theory is perturbatively renormalizable.

This material corresponds to chapter 4 of Costello and to chapter VI.8 of Zee. Also see Renormalization and Effective Lagrangians by Polchinski.

2:00-3:30
Andrew Obus
Bhatt and Scholze's theory of prismatic cohomology.  
If X is a compact complex manifold, then it is a classical result that the de Rham cohomology of X is isomorphic to the singular cohomology of X with complex coefficients.  Unfortunately, de Rham cohomology is not a fine enough invariant to account for torsion in the singular cohomology of X with integral coefficients. However, if X is furthermore an algebraic variety defined over the rational numbers, p-torsion classes in the integral singular cohomology of X can be related to the algebraic de Rham cohomology of the "reduction mod p" of X.  The state-of-the-art approach to relating these cohomology classes is called "prismatic cohomology", and is due to Bhatt and Scholze.  We will attempt to give an accessible overview of the theory of prismatic cohomology while minimizing technicalities to the extent possible.  

May 1

10:00-11:30
Raymond Puzio
The Renormalization Group in Three Steps
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.  In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.
We will begin by recalling the basic ideas of perturbative quantum field theory, regularization and divergences.  Then we will present some ideas and techniques for understanding and dealing with divergences in a roughly historical order.  We start by introducing running coupling constants and solving for these in terms of observable quantities.  This leads to what is known as the "renormalized perturbation series".  Next, following Gell-Mann and Low, we will obtain an equation which describes how the coupling constants run from the condition that observable quantities should not depend upon the cut-off.  This gives us the first formulation of the renormalization (semi)group as the flow of the coupling constants.  Finally, following Wilson, we will look at how a theory at some scale of the cut-off can be expressed as an effective theory at a lower value of the cut-off by integrating over the degrees of freedom between those two values of the cut-off.  This effective theory will generally include more interaction terms than the original theory.  By repeating this operation, we obtain a semigroup which acts on the space of possible theories --- this is Wilson's reformulation of the renormaliztion group.  When this flow has a fixed point, we may, by linearizing about this fixed point, recover the earlier notions of renormalization flow and running couplings but now within the context of a more general theory.
This material corresponds to chapter 4, section 1 of Costello and to chapters III.1-3 and VI.8 of Zee.

2:00-3:30
Corbett Redden
Model structures on simplicial presheaves and the Freed-Hopkins Theorem
I will discuss the 2013 Bull. Amer. Math. Soc. paper "Chern-Weil forms and abstract homotopy theory" by Dan Freed and Mike Hopkins. The paper shows how a natural geometric problem can be reframed in the language of homotopy theory via simplicial presheaves. I will give the precise statement of their theorem and, in keeping with the paper's spirit, give an overview of model structures on simplicial presheaves. If time permits, I will explain details of the proof.

April 17

10:00-11:30
Raymond Puzio
Divergences and Perturbative Renormalization
Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.  In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

We will begin by recalling the basic ideas of perturbative quantum field theory and Feynman diagrams.  Examining the terms of the perturbation series, we note that they diverge in the limit of large cut-off and that we can decompose the terms as the sum of a finite piece and a singular part.  One can then ask given a field theory in a certain dimension, how singular the terms of the perturbation series can be.  As a first attempt at answering this question, we will introduce a heuristic power counting argument and find out that, for some theories, the degree of divergence is bounded for all orders in the series but for, other theories, the degree of divergence increases unboundedly.  We call the former class of theories perurbatively renormalizable and the latter class perturbatively unrenormalizable.  In the former case, we can obtain finite values for all observable quantities by making a finite number of parameters appearing in the action depend upon the cut-off.  To compute this systematically, we introduce counterterms.  To understand this conceptually, we will introduce the notion of renormalization flow on the space of theories.
This material corresponds to chapter 2, sections 5, 9, 10 of Costello and to part III of Zee.

2:00-3:30
Rob Thompson
Homology of structured spectra
In this talk I will survey, in an expository fashion, some results on the generalized homology of infinite loops spaces (their suspension spectra really) and other structured spectra.  I will start by looking at ordinary homology, then K-theory, then Morava K-theory and Morava E-theory. 

April 10

10:00-11:30
Raymond Puzio
QFT --- Expansions and Singular Parts

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.  In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

We will begin by recalling the basic ideas of perturbative quantum field theory and Feynman diagrams.  Expanding the field in a Fourier series, we will derive the Feynman rules in momentum space.  We will consider the perturbative expansion of the logarithm of the generating function and the semiclassical approximation and show how these correspond to connected diagrams and loop expansion.  Looking at the values of particular terms, we will note how we can seperate them into a divergent piece and a piece which stays finite as we take the value of the cut-off to infinity.  More systematic treatment of these observations will lead to the topics of power counting, counterterms, and renormalizability.

This material corresponds to chapter 2, sections 5, 9, 10 of Costello and to chapter I.7 and part III of Zee.

2:00-3:30
Martin Bendersky
Hopf Invarinat One

I will start by discussing various problems in algebra and geometry.  Namely the question of which Euclidian spaces admit the structure of a normed algebra - generalizing the complex numbers.  Which spheres have the maximal number of linearly independent vector fields - generalizing the fact that the circle has a trivial tangent bundle and which spheres have a "group" structure generalizing the fact that the product of unit complex numbers is a complex number. There is an integral invariant in homotopy theory that must be 1 if any of these problems have a positive solution.  The homotopy theory problem was solved by F. Adams in 1958.  I will outline the proof (with much detail omitted).

April 3

10:00-11:30
Raymond Puzio
Ultraviolet divergences and perturbative renormalization

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.

In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

After reviewing basic facts about perturbative quantum field theory, we will explicitly write down the first few terms in the perturbation series for the case of a scalar field theory with quartic interaction.  These terms will be integrals over the space-time manifold of products of Green's functions of the Laplacian (known in physics as "propagators").  We will note that beyond the first term, these integrals diverge at short distances --- this is the "ultraviolet catastrophe" of quantum physics.  To recover from this catastrophe, Feynman and Dyson introduced the notions of regularization and renormalization.  We will illustrate these notions in the context of our example calculations, see how they allow one to extract a sensible finite answer from a seemingly ill-behaved series and explain the deeper meaning of these techniques. This material corresponds to chapter 2, section 5 of Costello.

2:00-3:30
Alex Perry
Hochschild cohomology and group actions
 

I will discuss the behavior of Hochschild cohomology under the operation of taking group invariants for a finite group action. The motivation for this is a series of examples of fractional Calabi-Yau categories that occur inside the derived categories of smooth projective varieties. 

March 27


10:00-11:30
Raymond Puzio
An introduction to perturbation theory

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems. In order to make these developments accessible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

This Wednesday, we will continue our study of perturbation theory, which is a technique used for making computations in interacting field theories.  In order to make the session self-contained, we will begin with a review of the fundamentals of the subject --- observables, expectation values, path integrals, free fields.  By using the fact that the Fourier transform interchanges multiplication and differentiation, we will express the generating function for an interacting field in terms of formal differential operators. Expanding this expression, we will obtain Wick's theorem, which describes terms in the perturbation series in terms of Wick contractions of the interaction term in the Lagrangian.  Feynman diagrams provide a way of visualizing these terms by interpreting the contractions as edges of graphs.  This material corresponds to Chapter 2, section 3 of Costello.  For background material and an alternative discussion of the material, on may want to consult Zee.

Costello, Kevin. Renormalization and effective field theory. No. 170. American Mathematical Soc., 2011.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&type=pdf [citeseerx.ist.psu.edu]

Zee, A. (2003). Quantum field theory in a nutshell. Princeton, N.J: Princeton University Press.
https://www.worldcat.org/title/quantum-field-theory-in-a-nutshell/oclc/50479292 [worldcat.org]

2:00-3:30
Jingyu Zhao
From mirror symmetry for Fano manifolds to string topology 

Mirror symmetry for Fano symplectic manifolds relates enumerative Gromov-Witten invariants with certain period integrals of its Landau-Ginzburg mirror. In this talk, we outline a geometric proof of this relationship using Floer homology. If time allows, we will also discuss a further relationship with string topology operations.

March 20

10:00-11:30
Cheyne Miller
(joint w/ M. Miller, T. Tradler, and MZ)
An explicit description of the totalization of certain cosimplicial simplicial sets in the context of the theory of vector bundles with connection

In this talk, we consider functors from the opposite category of complex manifolds to the category of simplicial sets which we refer to as simplicial prestacks.  The two examples of interest in this talk will be (i) HVB, which assigns to each manifold, M , the nerve of the category of holomorphic vector bundles, with connections, where the morphisms in this category do not respect the connection and (ii) Perf, which assigns to each manifold, M, the maximal Kan subcomplex of the dg-nerve of the dg-category of finite chain complexes of vector bundles over M, with connection, where again the morphisms in this dg-category do not respect the connections.  For both prestacks (i) and (ii), we can consider the Cech-nerve of an open cover, of some M as a simplicial manifold.  By evaluating either prestack on this simplicial manifold, we obtain a cosimplicial simplicial set.  Applying the totalization, in lieu of the homotopy limit thanks to certain qualities of our setting which we will not go into, to these cosimplicial simplicial sets gives respective simplicial sets.  We will also consider the simplicial prestack HVB applied to the simplicial manifold, [M/G], whose n-simplices form the manifold: M x G x...x G.  The main goal of this talk will be to explicitly describe the simplicial sets: (a) Tot(HVB(NU)), (b) Tot(HVB(M/G)), and (c) Tot(Perf(NU)), while avoiding all distractions presented by the category-theoretic concerns which actually arise when working with these examples.

2:00-3:30
Cheyne Miller (joint w/ M. Miller, T. Tradler, and MZ)
Recovering Chern-Simons forms from the totalization of a map of prestacks

As a continuation of the previous talk, we focus on the simplicial prestacks, HVB, described above, and Omega, the simplicial prestack whose n-simplices for a given manifold are polynomials of degree-shifted DeRham forms on the manifold. Next, we explicitly describe a map between these prestacks, CS.  Evaluating this map on the nerve of a cover, and then applying totalization to the map, we obtain a map of simplicial sets.  The main purpose of this talk is to remark upon how this map, restricted to the 0-cells, recovers the Chern character as described by Atiyah and to explore what further invariants are present in the map as applied to higher k-cells.  With any remaining time, we will also see what happens to this map when we evaluate it on the simplicial manifold [M/G] and how this relates to the theory of G-equivariant vector bundles. 

March 11 and 13th

Mon Mar 11 (Dennis Sullivan’s class)
11:00-1:00
Dennis Sullivan
Background material for the afternoon talk below.

Mon Mar 11 (Einstein Chair Seminar)
2:00-3:30 + tea at 3:30
Dennis Sullivan (joint work with Alice Kwon)
Application of Perelman Theorem: All Closed Three Manifolds Have Schottky Presentations 

From the eight lie groups appearing in the Perelman Theorem, one explicitly constructs five lie group actions on the three sphere (only one is pesky). These generate a group G5 action on the three sphere where each element acts by a transformation which is real analytic outside a finite set.
For each closed oriented three manifold M one constructs a discrete subgroup of G5 which has a dense domain of discontinuity U on S3 with complement a totally disconnected limit set L so that U mod the discrete group is M. This is the Schottky presentation of the title. For a given M all Schottky presentations are topologically conjugate.
The construction depends on finitely many parameters that resemble usual Teichmuller spaces and a new analogue. The core of the construction is a G5 structure on M, namely a cover of M by coordinates in S3 whose transitions  are performed by elements from G5 that generate the Schottky presentation.

 Wed Mar 13
10:00-11:30
Sheldon Joyner
Period polynomials and relations on the Grothendieck-Teichmuller group

 The Grothendieck-Teichmuller group GT was defined by Drinfel’d by considering perturbations of the structure on the quasi-triangular quasi-Hopf algebras he introduced. This group admits a simple presentation and has been realized as the group of automorphisms of the Teichmuller tower of (fundamental groupoids of) moduli spaces of genus 0 surfaces with n marked points. At the same time, it is known that the absolute Galois group of the rationals, an exceptionally complicated object of great interest to number theorists, embeds into GT. In the first half of the talk I will introduce GT and discuss connections between the relations on this group and the geometry of the moduli spaces of genus 0 curves with 4 or 5 marked points respectively.
By recent work of Brown and Hain, this geometric perspective is also linked to the well-known Eichler-Shimura-Manin isomorphism, which provides a means of computing the dimension of a given space of cusp forms by identifying them with an explicit Euclidean space. This mapping can be effected using the so-called period polynomials. As Brown has observed, one of the symmetry properties satisfied by these period polynomials is reminiscent of the hexagonal relation on the Grothendieck-Teichmuller group. In fact the Lie algebra version of the hexagonal relation satisfied by associators is identical to this period polynomial symmetry property. In the second half of the talk I will discuss the geometric basis for this relationship.

Wed Mar 13
2:00-3:30
Christoph Dorn
On how to continue the sequence "Categories, Functors, Natural transformations, Modifications, ... "

Categories, functors, and natural transformations have proven themselves to be powerful and foundational tools in the study of many topics in modern mathematics. For instance general cohomology theories are certain functors on (homotopy) categories of spaces, and characteristic classes can be regarded as natural transformations to such cohomology theories. If we allow our categories to have higher structure, that is, allow them to take the form of a weak n-category, then we can also ask about "transformations (of transformations...) of natural transformations". The questions of what these transformations are is intimately linked with the geometric structure of the (n+1)-category of all n-categories. In this introductory talk we will only briefly mention this geometric story and then focus on a powerful combinatorial tool to classify the geometry: Polygraphs and their bi-closed monoidal structure, called the Crans-Gray tensor product. The goal will be to give an simplified version of higher transformations in the so-called lax and oplax case. If time permits, I will sketch how to obtain the full "pseudo-natural" case from this.
PS: Our polygraphs have nothing to do with those employed by the FBI or CIA and no algebraic structures will be subjected to any harm.

March 6

10:00-11:30
Rob Thompson
Splittings of configuration spaces and iterated loop spaces


Abstract:  In 1978 F. Cohen, P. May, and L. Taylor published a paper with a very elegant proof of a theorem which gives a stable splitting of iterated loop spaces. This theorem generalized the Milnor-James splitting, the Kahn splitting of QX, and the Snaith splitting. These splittings are fundamental to many basic constructions in homotopy theory. I will sketch the beautiful Cohen-May-Taylor construction and their theorem and discuss some examples. The prerequisites for the talk are minimal. 

2:00-3:30
Rob Thompson
Power operations in Morava E-theory


Returning to Rezk’s notes on Elliptic Cohomology, a crucial ingredient of this work is the use of multiplicative operations in generalized homology theories that are represented by structured ring spectra, e.g. Morava E-theory.  I will discuss some of these ideas, including a survey of some results about K-theory of infinite loop spaces, and Morava K-theory of infinite loop spaces. I will start slow by discussing older material on H-infinity ring spectra, then eventually graduate to the E-infinity case. 

Feb 27

10:00-11:30
Raymond Puzio
Renormaliztion and Effective Field Theory

Quantum Field Theory (QFT) was originally devised in order to study high energy particle physics but has since found applications elsewhere.  In topology, it has turned out to be relevant to such topics as multiplicative genera, knot invariants, and index theorems.

In order to make these developments accesible to a wider audience, we are holding a study group based upon Costello's book "Renormalization and Effective Field Theory".  This group does not presuppose any background in physics or quantum theory and should be accessible to anyone with a general background in mathematics.

We will begin with a brief review of expectation values and free fields. Then we will turn our attention to the topic of perturbation theory, which is a technique for studying interacting theories by expressing their expectation values as a series of correction terms to the corresponding values for free field theories.  This technique may be viewed as a generalization to functional integrals of the steepest descent approach to computing asymptotics of ordinary finite-dimensional integrals and hence we will begin by illustrating the technique with finite-dimensional integrals.

The remainder of the lecture will be devoted to methods for efficiently generating the perturbation series.  These methods are based upon applying formal functions of differential operators to suitable arguments.  Thus, we will obtain Wick's theorem, which describes terms in the perturbation series in terms of Wick contractions of the interaction term in the Lagrangian.  Then, by introducing a geometric interpretation of these contractions, we will obtain Feynman diagrams.

Costello, Kevin. Renormalization and effective field theory. No. 170. American Mathematical Soc., 2011.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&type=pdf [citeseerx.ist.psu.edu]

2:00-3:30
Rob Thompson
Power operations in Morava E-theory


Returning to Rezk’s notes on Elliptic Cohomology, a crucial ingredient of this work is the use of multiplicative operations in generalized homology theories that are represented by structured ring spectra, e.g. Morava E-theory.  I will discuss some of these ideas, including a survey of some results about K-theory of infinite loop spaces, and Morava K-theory of infinite loop spaces. I will start slow by discussing older material on H-infinity ring spectra, then eventually graduate to the E-infinity case. 

Feb 20

10:00-11:30
Raymond Puzio
Introduction to Perturbation Theory

This week, we will begin our study of quantum field theory with the topic of perturbation theory, which is about computing quantities as a series of corrections to a simplified approximation.  In our case, the quantities of interest are expectation values of observables, the simplified approximation is a free field theory, and the series of corrections account for interactions.
The material to be covered is found in sections 3 of chapter 2 of Costello's book.  Specifically, the main subtopics are as follows:

  • Review of expectation values and correlation functions.

  • Gaussian integrals and their generating function.

  • Differential and integral calculus over graded commutative algebras.

  • Asymptotic expansion of integrals.

  • Tensor contraction and formal differentiation.

  • Feynman diagrams.

The discussion will be self-contained and does not presume any prior acquaintance with quantum theory

2:00-3:45
Martin Bendersky

A survey of chromatic homotopy theory: A tale of two constructions

In the 60's and early 70's Adams, Smith and Toda constructed self maps on CW complexes which realize certain BP_* modules. These self maps were detected by (what we now call) Morava K(1), K(2) and K(3). These complexes and maps gave rise to infinite families in the stable homotopy groups of spheres. The Adams map gives rise to the alpha family which are the elements of order p in the image of the J-homormphism. For p>3 the family detected by K(2) gives rise to the beta family. The elements constructed in the homotopy groups of spheres by the third family are called the gamma family (I think you can detect a pattern here). As with the alpha family, the beta classes were shown to be an infinite non-zero family in the homotopy groups of spheres. The gamma family was much more subtle. Toda and his group proved that the first element, cleverly called gamma_1 was zero. At the same time Raph Zahler in his thesis (directed by Aranus Liulevicius – recently deceased) showed that gamma_1 was not zero. This led to an article in the NY times titled “A contradiction in mathematics”. The Toda group admitted defeat and Zahler with E. Thomas proved that infinitely many gammas were not zero. Zahler was the first to use the work of Quillen to make computations in the Adams Novikov spectral sequence. Unfortunately his methods were to compute the Adams Novikov spectral sequence using BP cohomology. This was in spite of Frank Adams' work pointing out the advantages of using homology to understand the E_2 term of a generalized Adams spectral sequences.

At the same time Jack Morava understood that Quillen's formal group law approach to BP allowed him to use powerful ideas from algebraic geometry and group cohomology to relate the E_2 term of the Adams Novikov spectral sequence for the BP_* modules of Adams, Smith and Toda (and algebraic generalizations) to the group cohomology of the stabilizer group of the formal group laws associated to generalizations of K-theory. Morava's contribution and its generalizations have been the topic of many talks in this seminar. Miller, Ravenel and Wilson realized that Morava's work can be used to detect the entire gamma family. The connection is via the chromatic spectral sequence. In their game changing paper – Periodic phenomena in the Adams Novikov Spectral Sequence they input the cohomology of the stabilizer group into a (Bockstein) spectral sequence which converges to the E_1 term of a spectral sequence (called the chromatic spectral sequence ) which converges to the E_2 term of the Adams Novikov spectral sequence (which converges to the stable p-local homotopy groups of the spheres). This is clearly the paper for mathematicians who are turned on by spectral sequences. They succeed in giving a complete calculation of the 2-line, generalizing the beta family (in E_2) and have enough information about the 3 line to show that the entire gamma family is non zero – the main goal of the paper (Steve Wilson told me that he bet a lobster dinner that they would not detect the entire gamma family).

I will give an outline of the ideas in the Miller Ravenel Wilson paper and try to connect it with previous lectures.

Lightning Talks 4:00-5:30 Math Lounge

James MyerAn Introduction to the Problem of Resolution of Singularities
We all have an idea of what a singularity is. For example, we think of a singularity on a curve defined by a polynomial equation f(x,y) = 0 as a point where the partial derivatives fx,fy and the function f simultaneously vanish. Our natural inclination is to want to fix a singularity. Wouldn’t it be nice if we could find a curve very much like our original curve, except nonsingular? We refer to such a curve as a resolution of the singularity. There is a beautiful geometric interpretation of a resolution of a plane curve singularity described by an operation called blowing up. The terminology should elicit thoughts of inflating a bubble with a straw at a point of the curve, rather than an explosion, as described by David Eisenbud in his book Commutative Algebra with a View Toward Algebraic Geometry.

Perhaps a singularity occurs for another reason though? An important philosophy in algebraic geometry is that an affine algebraic variety and its underlying ring are effectively the same thing. In fact, there is an equivalence of categories between the algebraic world of reduced, finitely generated algebras over algebraically closed fields and the geometric world of affine algebraic varieties. Fascinatingly, a purely algebraic property known as normality of a ring exten- sion offers a purely algebraic explanation of a plane curve singularity and nor- malizing the ring corresponding to the variety serves to resolve the singularity immediately.

The normalization of a ring is a highly computable object, and can in fact be computed via Macaulay2, open source web-based software available to everyone for free! So, we might think then that the problem of resolution of singularities is solved by this purely algebraic and highly computable normalization operation. Think again! The issue is that normal and nonsingular are not quite the same when we move to higher dimensions. It is a happy coincidence that normal and nonsingular are the same in dimension 1, i.e. for curves. That normalization is not sufficient to resolve singularities leads to the beautiful and historically rich story of resolution of singularities pioneered by the classical Italian school and Oscar Zariski, still unresolved for algebraic varieties over fields of positive characteristic for dimensions greater than 3. And despite the existence of a solution in other cases, there are still a number of open problems.

My plan is to introduce the problem of resolution of singularities from the naive point of view, and attempt to convince the audience that the problem is more subtle and nuanced than it might first appear with several examples. 

Joseph DiCapua
On Modules over Certain Group Rings

This talk is on the structure of a certain group ring Λ and the structure of finitely generated Λ-modules. Λ is defined to be the inverse limit limZp[Γn] where Γn is the p-part of a certain Galois group. We go into some detail on the construction of Λ and give an example of a Λ-module by taking limits of p- parts of class groups of a tower of field extensions. We then state some of the lemmata necessary for proving the structure theorem for finitely generated Λ-modules. 

Laura Lopez
Bi-interpretability of a free monoid with arithmetic and applications

We proved bi-interpretability of arithmetic $\mathbb{N} = \langle N, +,\cdot, 0, 1\rangle$ and the weak second order theory of $\mathbb{N}$ with the free monoid $\mathbb{M}_X$ of finite rank greater than 1 and with the partially commutative monoid without the center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable.  Moreover, any recursively enumerable language in the alphabet $X$ is definable in $\mathbb{M}_X$.   Primitive elements, and, therefore, free bases are  definable in the free monoid.   It has the so-called QFA property, namely there is a sentence $\phi$ such that every finitely generated monoid satisfying $\phi$ is isomorphic to $\mathbb{M}_X$. The same is true for a partially commutative monoid without center. We also prove that there is no  quantifier elimination in the theory of  any structure that is bi-interpretable with $\mathbb N$ to any boolean combination of formulas from $\Pi_n$ or $\Sigma_n$.

Alice Kwon
Augmented knots in the Thickend Torus

I will be talking about augmented links in the thickened torus and angle structures on triangulations on their link complements. We can prove if link complements whose decomposition into tetrahedra have an angle structure the link is hyperbolic. In this talk I will show that semi regular biperiodic alternating links with augmentations are hyperbolic.

Kieran O’Reilly
What Are the Markoff Numbers?

The Markoff numbers encode information about how well rational numbers can approximate real numbers. More precisely, the Markoff numbers are in one to one correspondence with equivalence classes of ”badly approximable” real numbers. In this talk, we will explore in what sense real numbers can be ”badly approximable”, how the Markoff numbers help with understanding this and the surprising connections this all has with hyperbolic geometry.

Corey Switzer
What Is the Continuum Hypothesis?

The continuum hypothesis (CH) is a statement about the size of the real line. Specifically it states that there is a linear ordering 􏰁< of R so that each initial segment {y | y 􏰁< x} is countable. Thus it says that R is “almost” countable. CH is the typical example of a question that cannot be decided by the axioms of set theory. In 7 minutes, I will explain more about what CH says and aim to give at least one concrete application, in this case to measure theory.

Peter Thompson
Commuting planar polynomial vector fields for conservative Newton systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field in the plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. Let f be an element of K[x], where K is a field of characteristic zero, and let d be the K-derivation on K[x, y] that corresponds to the differential equation x′′ = f in a standard way. This is called a conservative Newton system, as it describes the position of a particle under the influence of a conservative force. Let also H be the Hamiltonian polynomial for d, that is H = y^2 − 2int(f), where int(f) denotes the antiderivative of f with respect to x with zero constant term. It is known that the set of all polynomial K-derivations that commute with d forms a K[H]-module M_d. We show that, for every such d, the module M_d is of rank 1 if and only if the degree of f is at least 2.

Feb 13

10:00-11:30 
Manuel Rivera
Categorical and algebraic constructions related to path spaces

I will explain the relationship between the following three functors:

(1) the functor P that associates to any space X the topological category P(X) whose objects are the points of the space and morphisms space P(X)(a,b) is the space of paths from a to b.
(2) the cobar functor from the category of coaugmented differential graded coalgebras to the category of augmented differential graded algebras.
(3) the rigidification functor from simplicial sets to simplicial categories.

Note that for any pointed space (X,b), P(X)(b,b) is the based loop space of X at b, a classical construction in homotopy theory. Functor (2) is based on an algebraic construction originally introduced by Frank Adams, which, together with the bar construction establishes an important duality between coalgebras and algebras. Functor (3) was originally introduced by Cordier and then used by Lurie in order to compare different models for infinity categories (quasi-categories and simplicial categories). Functor (3) is the left adjoint of the homotopy coherent nerve functor and such adjunction establishes a Quillen equivalence between the A. Joyal’s model category of simplicial sets and J. Begner’s model category of simplicial categories.  The key to relate these three functors is to introduce a cubical version of (3). 

As a consequence, I will deduce an extension of a classical result of Adams that relates the cobar construction and the based loop space of a simply connected space to the case of non-simply connected spaces. I will also use these results to explain the sense in which the algebraic structure of the singular chains on a space determines the fundamental group functorially.


2:00-4:00 
André Joyal 
Goodwillie's calculus of functors and higher topos theory.

We develop an approach to Goodwillie’s Calculus of Functors using the generalised BM theorem. Central to our method is the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality. We show that the pushout product of a P(m)-equivalence with a P(n)-equivalence is a P(m+n+1)-equivalence. We then prove a BM theorem for the Goodwillie tower of functors. We rederive some foundational theorems in the subject, such as delooping of homogeneous functors.

Feb 11


11:00-12:45 (Dennis Sullivan’s Class)
André Joyal
Duality in (higher) topos theory.

We claim that topos theory is best understood from a dual algebraic point of view. We shall use the term *logos* for the notion of topos dualized. A logos is a ring-like structure and the theory of logoi has many things in common with the theory of commutative rings. The free logos on one generator Set[X] can be described explicitly. The 2-category of topoi is defined to be the opposite of that of logoi. The topos-logos duality is closely related to the locale-frame duality, and to the duality between affine schemes and commutative rings. Similar observations apply to higher topoi and higher logoi. Rezk's descent principle plays a central role in the theory of higher logoi. We shall sketch a proof of Rezk's descent principle in the category of spaces (=infinity groupoids). We shall sketch the connection between higher topos theory and Homotopy Type Theory.

2:00-3:30 (Einstein Chair Seminar)
André Joyal
A generalised Blakers-Massey theorem in higher toposes.


We prove a generalization of the classical connectivity theorem of Blakers-Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system (L, R) in which the left class is stable by base change. We rederive the classical result, as well as some recent generalization. Our proof of BM theorem uses Rezk's  descent principle and it is inspired by the proof discovered in Homotopy Type Theory.

Feb 6

10:00-11:30
Collective participation

Initial meeting for a reading group on Kevin Costello’s "Renormalization and Effective Field Theory". Please join to listen in or take part.


2:00-4:00
Martin Bendersky

A survey of chromatic homotopy theory: A tale of two constructions

In the 60's and early 70's Adams, Smith and Toda constructed self maps on CW complexes which realize certain BP_* modules. These self maps were detected by (what we now call) Morava K(1), K(2) and K(3). These complexes and maps gave rise to infinite families in the stable homotopy groups of spheres. The Adams map gives rise to the alpha family which are the elements of order p in the image of the J-homormphism. For p>3 the family detected by K(2) gives rise to the beta family. The elements constructed in the homotopy groups of spheres by the third family are called the gamma family (I think you can detect a pattern here). As with the alpha family, the beta classes were shown to be an infinite non-zero family in the homotopy groups of spheres. The gamma family was much more subtle. Toda and his group proved that the first element, cleverly called gamma_1 was zero. At the same time Raph Zahler in his thesis (directed by Aranus Liulevicius – recently deceased) showed that gamma_1 was not zero. This led to an article in the NY times titled “A contradiction in mathematics”. The Toda group admitted defeat and Zahler with E. Thomas proved that infinitely many gammas were not zero. Zahler was the first to use the work of Quillen to make computations in the Adams Novikov spectral sequence. Unfortunately his methods were to compute the Adams Novikov spectral sequence using BP cohomology. This was in spite of Frank Adams' work pointing out the advantages of using homology to understand the E_2 term of a generalized Adams spectral sequences.

At the same time Jack Morava understood that Quillen's formal group law approach to BP allowed him to use powerful ideas from algebraic geometry and group cohomology to relate the E_2 term of the Adams Novikov spectral sequence for the BP_* modules of Adams, Smith and Toda (and algebraic generalizations) to the group cohomology of the stabilizer group of the formal group laws associated to generalizations of K-theory. Morava's contribution and its generalizations have been the topic of many talks in this seminar. Miller, Ravenel and Wilson realized that Morava's work can be used to detect the entire gamma family. The connection is via the chromatic spectral sequence. In their game changing paper – Periodic phenomena in the Adams Novikov Spectral Sequence they input the cohomology of the stabilizer group into a (Bockstein) spectral sequence which converges to the E_1 term of a spectral sequence (called the chromatic spectral sequence ) which converges to the E_2 term of the Adams Novikov spectral sequence (which converges to the stable p-local homotopy groups of the spheres). This is clearly the paper for mathematicians who are turned on by spectral sequences. They succeed in giving a complete calculation of the 2-line, generalizing the beta family (in E_2) and have enough information about the 3 line to show that the entire gamma family is non zero – the main goal of the paper (Steve Wilson told me that he bet a lobster dinner that they would not detect the entire gamma family).

I will give an outline of the ideas in the Miller Ravenel Wilson paper and try to connect it with previous lectures.    

Jan 30

Topology, Geometry, and Physics Seminar
10:00-11:30 (slow intro) & 2:00-4:00 (self-contained lecture)
Mahmoud Zeinalian
Chern-character and Chern-Simons forms in terms of the transition function

Let FVB be the simplicial presheaf that assigns to a smooth manifold the nerve of the category whose objects are smooth vector bundles with flat connection and morphisms are bundle isomorphisms that ignore the connections. 

Let Omega be the simplicial presheaf that assigns to a smooth manifold a simplicial abelian group, remembered only as a simplicial set, obtained by applying the Dold-Kan functor to the non-positively graded cochain complex obtained by tensoring the non-negatively graded deRham cochain complex with the polynomials in one variable u of degree -2 and then quotienting it by the sub complex of elements in positive degrees. 

Said simply, this is the simplicial set whose k-simplicies are decorations of all i-dimensional faces of the standard k-simplex with sequences of forms, all even for i even, and all odd for i odd, in such a way that the exterior d of what is sitting on an i-dimensional face is the summation of all those forms sitting on its (i-1)-dimensional faces of that face.

We construct a map of simplicial presheaves CS: FVB —> Omega, as follows. 

In simplicial degree 0, we assign to a flat connection the decoration of the standard 0-simplex by the sum of the (dim of the fibre)+0+0+…, where the j-th zero means the zero 2j-form.

In simplicial degree 1, we assign to a bundle isomorphism g: (E, D0) —> (E, D1) that ignores the flat connections D0 and D1, the decoration of the standard 1-simplex obtained by the trace of the odd powers (g^-1dg)^{2s-1} u^s with appropriate coefficients (i.e. the odd Chern character). Here, dg represents the derivative of g by pre and post composing operators D0 and D1 in the domain and the range, 

In simplicial degree 2, we assign to a pair of compossible morphisms (E, D0) —> (E, D1) —> (E, D2), not respecting the connection, labelling of all cells of the standard 2-simplex by a combinatorial formulae, in terms of combinations of the left as well as the right invariant Maurer-Cartan forms together, which can be thought of as the higher analogues of the odd Chern character living on GxG. Note the odd Chern Simons form lives on G. There are interesting higher forms living on GxGxG and all the higher Cartesian products giving rise to a closed form in the deRham forms on the stack [pt/G] represented by the simplicial manifold pt, G, GxG, GxGxG, ...

Application: Given a manifold M with a cover U we can apply the simplicial presheaf FVB to the Cech nerve of the cover U, which is a simplicial smooth manifold, to obtain a cosimplicial simplicial set. The totalization of this cosimplicial simplicial set is a simplicial set that has good geometric meaning: its vertices are the vector bundles on M together with a choice of flat connection on each open set of U. The edges are bundle isomorphisms not respecting the locally chosen flat connections, etc … Further applications are to equivariant theories and more generally bundles on simplicial manifolds. 

Similarly we can evaluate the simplicial presheaf Omega to the nerve of U and pass to the totalization to obtain a space (simplicial set) whose points are the closed elements of the Cech-DeRham bicomplex, the edges are the cohomological witnesses to two such closed elements representing the same Cech-deRham class, and so on and so forth with elements witnessing how a sum of the lower witnesses is trivialized.

A natural question is what this induced simplicial set map obtained through totalization is? The answer is that on the vertices this map gives a “combinatorial” formula for the Chern Character of the bundle, in terms of the transition functions of the bundle, in the Cech-deRham complex. Over the 1-simplices we obtain a formula for the Chern-Simons invariant of a bundle isomorphism, with respect to the domain and range connections, in the Cech complex, in terms of the transition functions of the bundle, interpolating between the previous formulae for the Chern Characters, and so on and so forth moving up the higher cells of the space.

Note that totalization does something very interesting. Before totalization, the map on the vertices was very trivial ie just assigning the dimension of the bundle (=the Chern character of the flat connection) However, after totalization the map on the vertices become the Chern character of the non flat bundle obtained by gluing bunch of flat bundles together via maps that did not respect flatness. 

Remark 1) There is an infinity homotopy coherent version of all of this, where vector bundles are replaced by derived families whose clutching functions are only fit together up to a system of coherent homotopies (TT resolutions), to which the above story is applied. Here, I have avoided discussing this because there is already new phenomena observed in the fully strict case.

Remark 2) Aside from the conceptual clarity of the above pre-totalization CS natural transformation which leads to the O’Brian-Toledo-Tong (OTT) Chern character formulae, this simplicial presheaf point of view leads to a map of simplicial sets whose value on the vertices reproduces the OTT Chern character formulae. The value on the edges and higher dimensional simplicies of the totalization are new invariants that should be thought of as an infinite hierarchy of Chern-Simons type relative invariants for the Toledo-Tong twisted resolutions. These can be used in the study of secondary and higher invariants for coherent sheaves on complex manifolds.

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Fall 2018

Dec 19

10:00-11:30
Micah Miller
The Bousfield-Kan map and computing homotopy limits

Let M be a simplicial model category.  The Bousfield-Kan map induces a map from the totalization of a cosimplicial object in M to its homotopy limit.  Under certain conditions, this map is a weak equivalence.  Since the totalization is easier to compute, this is a useful tool for calculating the homotopy limit. We give a discussion of the Reedy model structure on a model category M, and state when the Bousfield-Kan map is a weak equivalence.

2:00-4:00
Micah Miller
The space of twisted complexes as a totalization of a cosimplicial simplicial set

Given a simplicial manifold X, we can apply the stack Perf of perfect complexes of locally free sheaves, to get a cosimplicial simplicial set.  The vertices of this totalization can be identifies as the the Toledo-Tong twisted sheaves. If time permits, we will discuss a Chern character (on the vertices) and a Chern Simons form (on the edges), and generalizations to the higher n-simplices in this totalization. 

Dec 12

No K-theory seminar today. Note the following talk at Stonybrook Dec 12th and Dec 13th.

2:30-3:30
Math Tower P-131
Manuel Rivera, University of Miami
A new point in topology

The homotopy theory of geometric spaces can be recast into the language of infinite dimensional topological groups determined by function spaces of closed based loops in the spaces. The zeroth homology of the function space can be naturally identified with the group algebra of the fundamental group of the geometric space. This algebra has a compatible coproduct determined by the diagonal map on the natural basis defining an algebraic structure known as a bialgebra. Through a new perspective we now understand how this bialgebra and its characterizing prolongation to higher dimensions at the level of homological chains on the function space can be defined in satisfactory and complete generality directly from the algebraic structure of the singular chains of the geometric space. The algebraic construction that does this was introduced sixty years ago for simply connected spaces and remarkably is understood only now to work for all geometric spaces. The new idea beyond technique is to combine a duality theory for algebraic structures and the infinite homotopical symmetry of chain approximations to the perfectly symmetrical diagonal map on a topological space with the algebraic construction from the past. 

2:30-3:30
Math Tower P-131
Bhargav Bhatt, University of Michigan
Etale cohomology of affinoid spaces

This talk has two distinct but related parts. First, I will discuss a new Grothendieck topology (the arc topology) on the category of schemes and its usefulness in addressing some foundational questions in etale cohomology (including excision as well as new proofs of the Fujiwara-Gabber theorem and some results of Huber). Secondly, I will explain how to prove the analog of the Artin vanishing theorem in rigid analytic geometry. (Joint work with Akhil Mathew.)

Thursday December 13, 2018
4:00-5:00
Math Tower P-131
Bhargav Bhatt, University of Michigan
Interpolating p-adic cohomology theories

Integration of differential forms against cycles on a complex manifold helps relate de Rham cohomology to singular cohomology, which forms the beginning of Hodge theory. The analogous story for p-adic manifolds, which is the subject of p-adic Hodge theory, is richer due to a wider variety of available cohomology theories (de Rham, etale, crystalline, and more) and torsion phenomena. In this talk, I will give a bird's eye view of this picture, guided by the recently discovered notion of prismatic cohomology that provides some cohesion to the story. (Based on joint work with Morrow and Scholze as well as work in progress with Scholze.)

Dec 5

10:00-11:30
Corbett Redden
Equivariant de Rham theory

This talk will serve as a basic (pun intended) introduction to equivariant cohomology (Borel) and will focus on the complex of equivariant differential forms (Weil or Cartan model).  Most likely, I will sneak in some discussion about stacks, as this perspective will be used in the afternoon talk.

2:00-4:00
Corbett Redden
Equivariant connections on (higher) bundles

Suppose a compact Lie group G acts on a smooth manifold M. I will explain how the language of stacks (on the site of smooth manifolds) provides a natural framework for encoding and classifying certain G-equivariant structures on M. For example, there is a natural refinement of the equivariant Chern-Weil homomorphism that lives in an equivariant extension of differential cohomology. Furthermore, in joint work with Byungdo Park (GC alumnus) we prove that equivariant gerbe connections are classified by equivariant differential cohomology in degree 3. I will not assume that audience members are comfortable with the words in this abstract. I will try to explain everything in terms of natural geometric constructions on principal bundles.

Nov 28

10:00-11:30
No Talk

2:00-4:00
Rob Thompson
K(n)-local spectra, the Morava Stabilizer group, and Morava E-theory

In this talk, I will survey some facts about K(n)-local spectra and how the Adams spectral sequence can be modified to converge to the K(n)-localization of a spectrum.  Combined with Morava’s change of ring theorem, this leads to a homotopy fixed point spectrum of Devinatz and Hopkins.  This material is foundational to Rezk’s approach to elliptic cohomology.

Nov 21

10:00-11:30
Martin Bendersky
Cannibalistic Characteristic Classes

We will define the cannibalistic characteristic classes and its relation to the e-invariant. We will calculate the Adams’s e-inviariant for the image of the homotopy of U under the J-homomorphism.

2:00-4:00
Martin Bendersky
Adams’s e-invariant and the p-adic K-theory

We will relate the Adams conjecture to the Lubin-Tate theories.  In particular, Lubin-Tate  (or Morava) E_1 is p-adic K-theory. We will describe how the localization of the sphere spectrum with respect to E_1 is the fibre of a self map of the p-adic completion of BU. 

Nov 14

10:00-11:30
Alexander Milivojevic

Setting up the Frölicher spectral sequence

The sections of the exterior bundle of the complexified cotangent bundle of a complex manifold naturally form a bigraded commutative algebra on which the de Rham differential splits into two anti-commuting bigraded pieces (one "holomorphic" and one "anti-holomorphic"). Filtering this algebra in a natural way yields a spectral sequence whose first page is the Dolbeault cohomology and whose last page is isomorphic to the (complexified) de Rham cohomology. We will discuss Frölicher's original treatment of this spectral sequence as presented in his 1955 paper "Relations between the cohomology groups of Dolbeault and topological invariants". The existence of this spectral sequence implies relations between the Betti numbers of a smooth manifold and the dimensions of the Dolbeault cohomology groups of any complex structure it may come equipped with.

2:00-4:00
Alexander Milivojevic
Calculations with the Frölicher spectral sequence

The differentials on each page of the Frölicher spectral sequence can be explicitly described, and one can wonder whether these differentials uniformly vanish from some page onwards on a given complex manifold. If the manifold is compact and Kähler, or a compact complex curve or surface, then this phenomenon happens on the first page. We will look at some examples of non-Kähler compact complex threefolds for which the spectral sequence has non-trivial differentials past the first page. These examples will come from the easy-to-work-with family of nilmanifolds, and we will obtain explicit descriptions of each page of their spectral sequences. 

Nov 7

10:00-11:30
Micah Miller
A primer of homotopy colimits

We go over the definition of a homotopy limit of a diagram in a simplicial model category M and some elementary computations.  When we have a cosimplicial object in M, we can define its totalization.  The Bausfield-Kan map is a map from the totalization to the homotopy limit.  Under certain conditions, this map is a natural weak equivalence.  Since the totalization is easier to compute than the homotopy limit, this gives us another way to calculate homotopy limits.

2:00-4:00
Dennis Sullivan
Complex K-theory, manifolds and condensed matter physics


Nov 5 (Special Mon)

11:00-12:00
Dennis Sullivan [in Sullivan’s class]
Manifold with singularities construction of complex K-homology [connective version] and relation to Conner-Floyd theorem

2:00-3:30
Dennis Sullivan [Einstein Chair Seminar]
How complex K-theory classifies manifolds at odd primes

Oct 31

10:00-11:30 & 2:00-4:00
Martin Bendersky

On the J-Homomorphism

 I will describe Adams' work on ImJ.  First I will review the basic material. Specifically, I will restate the definition of the J homomorphism and the Adams conjecture.  I will outline the proof of the calculation of ImJ.  At some point I would like to show how the Adams conjecture connects with Morava E theory.   

Oct 24

10:00-11:30
Christoph Dorn
Directed triangulations
In last week’s talk we introduced the classifying category of posetal maps, and a category of posets modelling intervals with singularities. After recalling these concepts, we will build a category of singular n-cubes in 2 different ways (one using double categories, one using towers of bundles) and show that they are equal. Singular n-cubes provide “directed" triangulations of submanifolds of the cube. We will find to natural relations on cubes A,B, namely, sub-cubes (A < B) and quotient-cubes (A -> B). We will prove that the latter satisfies the Church-Rosser property. This translates into the powerful statement that any two directed triangulations of the same submanifolds of the cube have a mutual coarsening (not a mutual refinement, as stated e.g. in the disproven Hauptvermutung [1]).

[1] https://en.wikipedia.org/wiki/Hauptvermutung


2:00-4:00
Christoph Dorn
Presented associative n-categories
We will build a “type theory” for higher dimensional categories, which allows us to define a category by its generators. As an example we will give generators for the infinity-category Bord of cobordisms. To our knowledge, this is the first (claimed) presentation of the extended cobordism category [2]. Using this category, and recalling the generalised Thom-Pontryagin construction from last week, we translate CW complexes into associative n-categories. As a consequence of this translation, we illustrate the fully algorithmic nature of both listing and verifying maps into these CW complex.

[2] https://ncatlab.org/nlab/show/extended+cobordism

Oct 17

10:00-11:30
Christoph Dorn
Getting rid of the continuum: Posetal models for (interesting) manifolds

We will study towers of posetal Grothendieck fibrations (which can be described in completely elementary, foundations-independent terms), and examine some of their many surprising properties. These towers can be understood as a combinatorial description of certain stratified manifolds (called manifold diagrams) if one introduces “local consistency” conditions.  As an application we will rediscover (n+1)-framed n-cobordisms from a combinatorial perspective.

2:00-3:45
Christoph Dorn
Using manifolds to define (and understand!) general higher categories
Manifold diagrams allow us to give definitions of higher categories, and moreover, distinguish different flavours of them. We will build combinatorial notions of semistrict and fully weak higher categories, and see how coherence data in the latter arises as a combination of two now cleanly separated concepts: Semi-strict homotopies and the theory of dualisibility (which we will describe in terms of cobordisms). To motivate that these new notions of categories are “correct”, we will connect manifolds diagrams to the more common cellular approach to higher categories by a generalised Pontryagin construction.

Oct 10

10:00-11:30
Jeffrey Kroll
Symplectic structure on moduli space of flat connections

We will introduce the moduli space of flat connections on a fixed principal G-bundle whose points correspond to gauge equivalence classes of flat connections. Our interest is when the base is a Lefschetz manifold, which is a symplectic manifold for which wedging with powers of the symplectic form induces isomorphisms between complimentary dimension cohomology groups. If G has both an orthogonal structure and a class function, the moduli space of flat connections on a principal G-bundle with a Lefschetz manifold base has a symplectic structure.

2:00-4:00
Jeffrey Kroll
On the Poisson bracket of traces of holonomy

The moduli space of flat connections on a suitable principal bundle over a Lefschetz manifold is symplectic, and therefore one can investigate Hamiltonian vector fields associated to functions on the moduli space. We will define, and compute the Hamiltonian vector field associated to, the class of functions given by trace of holonomy around closed curves in the base and then investigate whether or not these functions are closed under the Poisson bracket. While this remains an open problem in general, we will see that traces of holonomy form a sub-Lie algebra of the Poisson algebra of functions on the moduli space when the base is a product of closed oriented surfaces.

 

Oct 3

10:00-11:30
Mahmoud Zeinalian
On the homotopy coherent nerve of cubical categories and Adams’s cobar construction

I will give an introductory talk on the homotopy coherent nerve of cubical categories and how it leads to a model of chains on the based loop space of a connected space through Adams’s cobar construction. This is on a joint work with Manuel Rivera who will speak in our seminar on Oct 24th.


2:00-4:00
Martin Bendersky
On the J-homomorphism
I will define the J-homorphism.   Frank Adams in  a mathematical tour-d-force computed the image of the stable J homorphism in the stable homotopy groups of spheres (actually with some ambiguity which was called the Adams Conjecture which was solved by Dennis and Quillen).  I will outline Adams' computations  (and the Adams conjecture).  I will make use of K theory (which I will define).  I will describe the role of the Bernoulli numbers.   I hope to describe the unstable image of J (actually the desuspension of the stable image of J).  Perhaps if there we are left with nothing to do and lots of time to do it in (apologies to Mae West) I will describe how the image of J appears in the stable Adams- Novikov spectral sequence. 

 

Sept 26

10:00-11:30
No Lecture

2:00-4:00
Mahmoud Zeinalian
From 2D Hyperbolic Geometry to the Loday-Quillen-Tsygan Theorem

The space of hyperbolic structures on an oriented smooth surface has a rich geometry. The study of the Hamiltonian flow of various energy functions associated with a collection of closed curves has led to a host of algebraic structures: first, on the linear span of free homotopy classes of non trivial closed curves (Wolpert-Godman-Turaev Lie bi-algebra) and then, more generally, on the equivariant chains on the free loop spaces of higher dimensional manifolds under the umbrella of String Topology (Chas-Sullivan + Others). 
While it is understood that the most natural way to organize the relevant algebraic data would be through topological field theories whereby chains on the moduli space of Riemann surfaces with input and output boundaries label operations, various aspects of these structures are not well understood. 
In this talk, I will provide some historical context and report on a recent joint work with Manuel Rivera on algebraically modeling the chains on the based and free loop spaces. One tangible result is that the long held simply-connected assumptions in various theorems, such as Adams’s 1956 cobar construction, can be safely removed once a cofibrant version of chains is utilized. This makes the chains on the free loop space of a non-simply connected space algebraically accessible. Towards the end of the talk I will report on a related ongoing joint work with Owen Gwilliam and Gregory Ginot based on the celebrated Loday-Quillen-Tsygan theorem which calculates the Lie algebra cohomology of the matrix algebras of very large size. This suggests a remedy to the problem that the trace of holonomy map from the vector space of homotopy classes of closed curves to the Poisson algebra of function on the moduli space of flat connection not being one-to-one.

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Spring 2018

Wed Mar 18
Morning Session 10:00-11:30
Gregroy Ginot
The circle action on Hochschild Homology and HKR Theorem

In this talk we will present a modern, infinity categorical approach to classical Hochschild homology as an object with a circle action in various contexts, including topological Hochschild homology and explain modern proof of the celebrated Hochschild-Kostant-Rosenberg theorem in this line after the work of Toen-Vezzosi, Hoyos, Antonio-Petit-Porta.  
In particular, we will explain the relationhip between André-Quillen Homology and Hochschild homology. Further, we will recall the definition of negative cyclic homology in this context and explain its relationship with K-theory in characteristic zero, which is a fundamental result of Goodwilie. 

Afternoon Session 2:00-3:45
Gregory Ginot
Nikolaus-Scholze definition of Topological Cyclic Homology

This talk is dedicated to the modern construction of the topological cyclic homology functor, which is a better approximation of K-theory then negative cyclic homology in non-zero characteristic. To do this, we will recall the Tate construction, give Scholze-Nikolaus infinity categorical definition of cyclotomic spectra, and their approach to topological cyclic homology.

Wed Apr 11
No lecture. Friday Schedule at CUNY.

Wed Apr 4
No lecture. Spring recess.

Wed Mar 28
Morning Session 10:00-11:30
Martin Bendersky
On p-adic Adams operation
I will review the notion of a special lambda algebra.  Atiyah and Tall describe the free special lambda ring generated by a single element and the ring of operations on the collection of special lambda rings.  We are heading to the construction of Adams operations indexed by the p-adics.

Afternoon Session 2:00-3:45
Rob Thompson
Operads and Obstructions
This session has two goals. Firstly, to define operads and give some examples. We will be interested in algebras over operads. Roughly, in the category of spectra, an algebra over an A-infinity operad is what's known as an A-infinity ring spectrum. 
Secondly, we will look at Bousfield's 'resolution model category construction',  which gives a way of constructing a spectral sequence for computing obstructions to getting algebra structures over an operad and maps between operads. 
This is all just one piece of the Hopkins-Miller theorem, which in turn is part of the foundation of Rezk's work on power operations in Morava E-theory, and in particular elliptic cohomology.

Wed Mar 21
No lectures due to snow
 

Wed Mar 14 (morning and afternoon meetings)
Companion meeting to the K-theory seminar (morning)
Martin Bendersky

10:00-11:30 AM
On p-adic Adams operation
I will review the notion of a special lambda algebra.  Atiyah and Tall describe the free special lambda ring generated by a single element and the ring of operations on the collection of special lambda rings.  We are heading to the construction of Adams operations indexed by the p-adics.

K-theory Seminar (afternoon)
Rob Thompson
2:00-3:45 PM

Power operations in Morava E-theory

One of the main goals of Rezk's Bonn notes is to give a homotopy theoretic explanation of some of the properties of the Witten genus.  The homotopy theory required is formidable.   I plan to give two expository lectures on several pieces of this background material. Specifically I will discuss some aspects of the Hopkins-Miller-Goerss theorem which states that Morava E-theory admits an A_{\infty}-structure (Hopkins-Miller) and in fact an E_{\infty)-structure (Hopkins-Goerss).  Some of the components of the required obstruction theory are:  Simplicial Operads,  Hochschild homology in the A_{\infty} case, Andre-Quillen homology in the E_{\infty} case, and Dwyer-Kan-Stover resolution model categories (as formulated by Bousfield), and I will summarize some of this.   Furthermore this all has to take place in a nice point-set model category of spectra.  There are a number of options for this - Lewis-May-Steinberger,  Elmendorff-Kriz-May-Mandell, May-Mandell, Hovey-Shipley-Smith, etc.   I will not actually present any of these in detail, as each one would take a significant amount of time, but rather just describe axiomatically what features of a category of spectra are required for the Hopkins-Miller theorem.

The Hopkins-Miller-Goerss theorem enables the construction of power operations. Power operations in elliptic cohomology are the basis for Rezk's analysis of the Witten genus.   The talks will be expository and introductory, as I am just learning a lot of this stuff myself for the first time.

Wed Mar 7
No lectures due to snow
 

Wed Feb 28 (morning and afternoon meetings)
Companion meeting to the K-theory seminar

Mahmoud Zeinalian
10:00-11:30

An Informal discussion on genera
I will review some of the basic material Corbett covered in his talk two weeks ago.

K-theory Seminar
Corbett Redden
2:00-3:45 PM
Obstructions from degree 4 characteristic classes
The Witten genus can be constructed (heuristically or mathematically) in several different ways.  In each of these constructions, there is an obstruction related to the degree 4 characteristic class p_1(M). Namely, a cohomology class must vanish in order for the construction to exist or have desired properties, and the choice of a trivialization is often referred to as a string structure.  I will explain the following specific instances in the talk.
- When defining the Witten genus via characteristic classes, the Witten genus of a spin manifold is a modular form if p_1(M)=0 [HBJ].
- For a spin manifold with the choice of a string structure, there is an element in the cohomology theory TMF that refines the Witten genus [Hop].
- Heuristically, the Witten genus is the S^1-equivariant index of the (not mathematically defined) Dirac operator on the free loop space LM.  In attempting to describe such an operator, one must construct a spin structure on LM [CP, Wit].
- Heuristically, the Witten genus is the partition function of a certain 2-dimensional nonlinear sigma model.  The degree 4 class arises here as an anomaly that must be canceled [Wit, AS, Bun].
Similarly, in Costello's "geometric construction of the Witten genus," there is an obstruction to quantization that arises from the second Chern class of a compact complex manifold [Cos].  Most likely, I will not discuss this example.
References:
[AS] Alvarez--Singer. "Beyond the elliptic genus."
[Bun] Bunke. "String structures and trivialisations of a Pfaffian line bundle."
[Cos] Costello. "A geometric construction of the Witten genus I" and "... II."
[CP] Coquereaux--Pilch. "String structures on loop bundles."
[HBJ] Hirzebruch--Berger--Jung. "Manifolds and modular forms."
[Hop] Hopkins. "Algebraic topology and modular forms."
[Wit] Witten. "The index of the Dirac operator in loop space" and "Index of Dirac operators."

Wed Feb 21 (morning and afternoon meetings)
Martin Bendersky

Introduction to the Hopf Invariant One problem
9:45-11:45 AM
I will discuss Hopf invariant 1.  On the geometric side I will talk about how an n-dimensional normed algebra => a division algebra => H-space structure on a sphere => an element of Hopf Invariant One. Also I will draw a picture to show that a trivial tangent bundle on a sphere => an H-space structure on the sphere.  So proving that there are no elements of Hopf Invariant One implies non of these happen in dimension n.  I will use Adam operations to determine when an element of Hopf Invariant One can exist.

Lambda rings and the J-homomorphism
2:00-3:45 PM
I will briefly mention what is necessary to extend the operations to \psi^k on K(X; Z_p) (p-adic K-theory) with k a p-adic number. We will then look at lambda-rings in a more general setting, following Atiyah and Tall (who follow Grothendieck).  They introduce special lambda rings (which are equivalent to lambda rings where the induced Adam's operations satisfy the above relations).  They use special lambda rings to prove a theorem about the J-equivalent classes of representations.  I will only have time to state their theorem.  Future lectures (by others) will apply lambda ring technology to define operations on elliptic cohomology.

Wed Feb 14
Martin Bendersky

Lambda rings and the J-homomorphism

I will talk about Lambda rings and induced operations.  The most famous induced operations are the Adam's operations, \psi^k.  They enjoy the property that they are ring homomorphisms and \psi^k( \psi^ t) = \psi^{kt}.  We will prove these relations for KU and allude to the fact that they are true for KO (the proof involves representation theory).   The Adam's operations are defined on K^0(X).  It is possible to extend \psi^k  to K^*(X) [1/k] (i.e. stable operations). 

Wed Feb 7
Corbett Redden

Brief introduction to elliptic genera

This talk will be a brief introduction to elliptic genera, and it will most likely be review of material covered in the early Fall 2017 talks.  I will begin by defining complex-oriented cohomology theories and looking at the two special cases of complex cobordism and K-theory.  We will see that a complex orientation of a cohomology theory naturally leads to a formal group law. Furthermore, Quillen’s theorem states that the universal complex-oriented theory (complex cobordism) encodes the universal formal group law. This implies that complex genera, or homomorphisms from the complex cobordism ring to a ring R, are equivalent to formal group laws over R. The group structure on an elliptic curve naturally leads to the notion of an elliptic genus. Finally, we use the Landweber exact functor theorem to produce an elliptic cohomology theory whose formal group law is given by the universal elliptic genus.

References:
Ochanine. "What is an elliptic genus?”
Landweber. Introductory chapter to "Elliptic curves and modular forms in algebraic topology."
Hirzebruch--Berger--Jung. "Manifolds and modular forms."
Redden. "Elliptic genera and elliptic cohomology" chapter from TMF book.

Wed Jan 31
Jeffrey Kroll

Derived Moduli Space of Flat Bundles

An affine derived stack is a representable functor from commutative differential graded algebras to simplicial sets. We will define the derived stack RBG, which is the derived analog of the classifying space BG of a group, and give an explicit construction for GL(r) due to Kapranov. The derived stack RBG is used to define the derived moduli space of flat principal G-bundles. We will see that the tangent complex of the derived moduli space at a flat vector bundle is weakly equivalent to the 1-shifted and truncated complex of cochains with values in the endomorphism bundle.

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Fall 2017

Wed Dec 6
Micah Miller

The Atiyah class for coherent sheave

The Atiyah class of a holomorphic vector bundle E → X is the obstruction to the existence of a holomorphic connection. The construction of this class goes as follows. Associate to E a short exacts equence of the form
0→E⊗Ω1 →P(E)→E→0. A splitting of this sequence is a holomorphic connection. From homological algebra, we know that short exact sequences can be identified with Ext1(X; E, E⊗Ω1) and a split short exact sequence is identified with the zero element in Ext1(X; E, E ⊗ Ω1). The Atiyah class is the negative of the class representing the short exact sequence in Ext1(X; E, E ⊗Ω1). A representative of the Atiyah class can be given in terms of the transition functions of E. The Chern character can then be defined by exponentiating the Atiyah class and taking the trace.

The definition of the Atiyah class extends readily to a coherent sheaf F on X. O’brian, Toledo, and Tong construct a representative for the Atiyah class in the twisted complex C•(U;Hom•(E, E⊗Ω1)). This complex can be thought of as filling the role of the Cech complex associated to a projective resolution for F, which may not exist for F. O’brian, Toledo, and Tong construct a trace map on twisted complex, and the Chern character for a coherent sheaf is then defined by exponentiating its Atiyah class and applying the trace map.

Wed Nov 29
Rob Thompson

The  Lubin-Tate Theorem and Morava E-theory

The goal is to construct Morava E-theory, a generalized homology theory which is at the heart of contemporary homotopy theory and its applications.
First we'll discuss the Lubin-Tate theorem.  For the statement of the theorem I will follow the elegant exposition in Rezk's paper on the Hopkins-Miller theorem.     The Lubin-Tate theorem is concerned with the following:  Given a field k of char p, and an FGL Gamma over k, consider deformations (i.e. lifts) of Gamma to complete local rings B with maps k -> B/m_B.   Lubin-Tate identifies the universal such deformation.  Also the group of automorphisms of Gamma acts on the groupoid of deformations and this leads to the Morava stabilizer group. 
We will describe the structure of the universal deformation ring  and sketch a proof of a portion of the theorem.
In order to construct Morava E-theory, we have to establish that the Lubin-Tate FGL is Landweber exact, and therefore gives a generalized homology theory and a representing spectrum.  I believe there is an easy, elegant proof of this using ideas of Andy Baker. Along the way we will relate Morava E-theory to a menagerie of other useful homology theories like Morava K-theory, and Johnson-Wilson theory.

Wed Nov 22
Rob Thompson

Formal group laws, the Lubin-Tate theorem and Morava E-theory

We will talk about Formal Groups Laws over rings in characteristic p > 0, and define the height of a FGL.  I'll define the Honda FGL and prove that over an algebraically closed field, any height n FGL is isomorphic to the Honda FGL. The affine groupoid scheme of p-typical FGL's over an F_p-algebra is filtered by height, but the schemes in the filtration are not affine schemes, they are stacks. We will mention the fact that an Elliptic curves gives a FGL which is  height 1 or 2.

Then we'll get to the Lubin-Tate theorem.  For the statement of the theorem I will follow the elegant exposition in Rezk's paper on the Hopkins-Miller theorem. The Lubin-Tate theorem is concerned with the following:  Given a field k of char p, and an FGL Gamma over k, consider deformations (i.e. lifts) of Gamma to complete local rings B such with maps k -> B/m_B.   Lubin-Tate identifies the universal such deformation.  Also the group of automorphisms of Gamma acts on the groupoid of deformations and this leads to the Morava stabilizer group. 
We will describe the structure of the universal deformation ring in important special cases, and say something about a sketch of the proof.

This sets the stage for the construction of Morava E-theory.  The Lubin-Tate FGL is Landweber exact, and therefore gives a generalized homology theory and a representing spectrum.  This homology theory, known as Morava E-theory, is fundamental to contemporary homotopy theory. We will relate it to other well known homology theories like Morava K-theory and elliptic cohomology.  Finally we'll say something about the construction of power operations in Morava E-theory.

Wed Nov 15
Rob Thompson
Formal group laws, the Lubin-Tate theorem and Morava E-theory, I

I plan to talk about basic material that makes up a portion of the prerequisites to Rezk's notes "Elliptic cohomology and Elliptic curves".   To begin, I would like to go over some basic facts about formal group laws.  I will define a p-typical FGL over an arbitrary p-local ring and prove Cartier's theorem which states that any FGL over a p-local ring is canonically isomorphic to a p-typical one.  I will point out the simple observation that while (L,LB), the Hopf algebroid which corepresents FGL's and their strict isomorphisms, gives a split groupoid scheme,   (V,VT), the Hopf algebroid corepresenting p-typical FGLs and their strict isomorphisms, is NOT split.   (This idea will play a role later when we introduce Morava E-theory, which IS split.)

Then we'll look at FGL's over rings of characteristic p, and I'll define the height of a FGL.  I'll define the Honda FGL and prove that over an algebraically closed field, any height n FGL is isomorphic to the Honda FGL.     The affine groupoid scheme of p-typical FGL's over an F_p-algebra is filtered by height, but the schemes in the filtration are not affine schemes, they are stacks.

At this point it's probably worth discussing the FGLs that come from elliptic curves - they are either height 1 or 2, with the height 2 ones coming from super singular elliptic curves. Unfortunately I am not that knowledgeable about elliptic curves, so I won't attempt to actually lecture on this, it's just something that we ought to discuss because it's obviously relevant.

Then we'll get to the main event of the first part  - the Lubin-Tate theorem.  For the statement of the theorem I will follow the elegant exposition in Rezk's paper on the Hopkins-Miller theorem.   For the proof, which I will only sketch a portion of, I'll probably follow along the lines of Andy Baker's proof.    The Lubin-Tate theorem is concerned with the following:  Given a field k of char p, and an FGL Gamma over k, consider deformations (lifts) of Gamma to complete local rings B such with maps k -> B/m_B.   Lubin-Tate identifies the universal such deformation.   The group of automorphisms of Gamma acts on the groupoid of deformations and this leads to the Morava stabilizer group.  

This all sets the stage for the second part, which will be the construction of Morava E-theory.  We'll show that it's a Landweber exact homology theory and relate it to other well known homology theories like Morava K-theory and elliptic cohomology.   Then we'll say something about the construction of power operations in Morava E-theory.  Also, the fact that Morava E-theory is an A_infinity ring spectrum is important so we can say something about the statement of that.  The proof of that is hard so that would have to be for another seminar sometime.

Wed Nov 8
Mahmoud Zeinalian

BD structure of the Chevalley Eilenberg cochains of Lie algebras with invariant co-inner products

-Flat connection on the jets of section of a bundle (the holomorphic case)
-A description of the bundle of first jets as a twisted direct sum.
-Atiyah class as the obstruction to having a holomorphic connection
-A theorem of Kapranov regarding the graded Lie algebra structure of the sheaf cohomology of the shifted tangent bundle
-Action of the above Lie algebra on the sheaf cohomology of holomorphic bundles
-Differential operators on a manifold
-Left D-modules and flat connection

Next week and the week after Rob Thompson will talk about Morava E-theory. The week after Micah Miller will talk about the Toledo Tong twisting cochains, the Atiyah class, and the Chern Character via the Atiyah class.

Wed Nov 1
Mahmoud Zeinalian
Duality between D-modules and dg modules over the deRham forms

-curved L-infty algebras and their Chevalley Eilenberg (CE) cochain complex
-examples of curved dg-Lie alegbras
-bundle of infinite jets and its flat connection in the smooth and holomorphic settings

-interpreting the flat connection on the bundle of infinite jets of functions as a curved L-infty algebra over the differential ground ring of differential forms \Omega. the underlying  \Omega-module on which the bracket and higher brackets live is the differential forms with values in the tangent bundle. The Lie algebra structure and the higher brackets are not related to the Lie bracket of vector fields!

-the cohomology of the above CE cochain complex is concentrated in degree 0 and equals the \Omega-module of smooth functions. note that algebra of smooth functions received a map from \Omega since the algebra of functions can be identified with the space of forms modulo the dg-ideal of forms in positive degrees).

-BV structure on the CE complex of L-infty algebras with an invariant degree 3 non-degenerate parings

-BD algebras

-differential operators on a manifold

-left invariant differential operators on a Lie group and the universal enveloping algebra

-lie-rienhart pairs aka lie algebroids

-chevalelly-eilenberg cochains of lie-reinhardt pairs

-going from left modules over the differential operators to dg-modules over the differential forms

-construction of a BV operator on the poly-vector fields from a right D-module structure of functions

-connection between the following concepts:

1) local volume form up to a constant (i.e. flat connection on top exterior power of tangent bundle)
2) BV operator on polyvector fields compatible with the nijenhuis bracket
3) right D-module structure on functions

 

Wed Oct 25
Sreekar Shastry
Integral models of modular curves, II

Given a formal group over a perfect field, consider its universal deformation. Morava E-theory is associated to this universal deformation; Theorem 2.9 page 15 of Rezk’s lecture notes explains how this association is functorial. An important example arises from the universal deformation of the formal group arising from the p-infinity torsion of a supersingular elliptic curve in characteristic p.

We’ll pick up from where we left off on 10/18, and complete the explanation of how to go from the group scheme GL_1 (also known as G_m) to the formal group F(X,Y) = X+Y+XY.

After this we’ll see the picture of the (coarse) modular curve with the universal elliptic curve sitting over it. This will give a geometric idea of how we obtain height two formal groups from supsersingular curves, and their universal deformations, and where these objects live relative to familiar objects such as (upper half plane)/SL_2(Z).

Wed Oct 18
Sreekar Shastry
Integral models of modular curves

 I am going to survey some aspects of the geometry of modular curves, with the goal of being able to draw the pictures behind the results invoked in sections 2 and 3 of Rezk’s Bonn lectures. As elaborate as the subject is, we will see that this picture can mostly be reduced to explicitly given rings and Hopf algebras. 

(The scheme theoretic prerequisites will be kept to a minimum, when working with affine schemes we are reduced immediately to rings, and elliptic curves over a base scheme can be thought of as bundles of tori)

The topics covered:

Overview of the theory of (finite) group schemes and Hopf algebras
Overview of formal group schemes and how they arise from group schemes
Overview of p-divisible groups (inductive systems of finite group schemes satisfying some conditions)
Statement of the connected etale sequence
Rough definition of elliptic curves over an arbitrary base scheme
A discussion of why the moduli functor classifying isomorphism classes of elliptic curves is not representable
A definition of the basic moduli problems in terms of Drinfeld level structures, which is where an understanding of finite group schemes becomes indispensable.
Time permitting, I'll state the Serre-Tate theorem which relates the deformation theory of p-divisible groups and the deformation theory of elliptic curves. This is the central result which will allow us to understand the local geometry on modular curves.

Two introductory survey papers:

Finite flat group schemes by John Tate
Fermat’s Last Theorem by Darmon, Diamond, Taylor

A reference book:
Arithmetic Moduli of Elliptic Curves by Nick Katz and Barry Mazur

 

Wed Oct 11
Martin Bendersky
Conner Floyed and Landweber Exactness (part II)

 Using MU* or MSO_* genera to a ring R  induces a functor from CW complexes to R modules.  Sometimes these are cohomology theories.

In order to give a general criterion for the functor to be a cohomology theory (due to Peter Landweber) I will review some facts about formal group laws.  I will give the first example of a genus inducing a cohomology theory due to Conner and Floyd - Here R = Z[u,u^{-1}] and the resulting cohomology theory is K-theory (actually Conner and Floyd's construction gave  Z/2 graded K-theory.  I will use the Landweber exact functor theorem to construct Z-graded K theory).  

There are various cohomology theories called Elliptic cohomology.  The non-periodic theory (with coefficients in Z[1/2][delta, epsilon]) is constructed by the Baas-Sullivan method of killing cobordism classes by introducing singularities.   Periodic Elliptic theory  (where the discriminant, epsilon(delta^2-epsilon)^2 is inverted) is shown to be a cohomology theory using Landweber exactness.

  Then a quick introduction to K-theory. K^0(X) is the ring of virtual bundles over X.   Bott periodicity extends this to a cohomology theory (K^{-n}(X), n \geq 0 is readily defined, Bott periodicity defines the functors for n <0).   A brief description of stable cohomology operations (i.e. operations which commute with the suspension operator - which for K-theory is the Bott isomorphism) and unstable operations.  Introduction to the Adams operations (which are not stable).  An application to Hopf invariant 1 (e.g. division algebras, H-space spheres and parallelizable spheres are in low dimensions).  A quick introduction to extending Adams operations to the p-adics.

 

Wed Oct 4
Mahmoud Zeinalian
BD algebras and L-infinity spaces

       Over the past few weeks, we have covered some basic facts about the genera and their relation to the characteristic classes. Martin discussed when an oriented genus MSO(pt)—>R, or a complex genus MU(pt)—>R, can be obtained from evaluating a map of multiplicative cohomology theories MU(X) —> R(X) on a point. To find a multiplicative cohomology theory X—>R(X) whose value at a point R(pt) equals R, a natural choice is to tensor the appropriate bordism theory, say X—> MU(X), with R, over the ring MU(pt). This natural candidate is not always a cohomology theory. A sufficient condition for it to be a cohomology theory is for the ring R to be flat as a MU(pt)-module, but this condition is so strong that it is in practice rarely satisfied.

Luckily, there is a weaker condition, still sufficient, that is met in several noteworthy cases. This phenomenon was first observed by Conner and Floyd who discovered the complex K theory, X—>KU(X), is obtained in this way from the Todd genus MU(pt)—>KU(pt) . The condition was later made abstract into what is now referred to as the Landweber exactness. Martin explained last time that the elliptic genus satisfies this condition and therefore gives rise to a cohomology theory called the Elliptic Cohomology.

It is remarkable that some aspects of this beautiful theory were later unveiled by the physicists through their calculations.

There are other instances in recent times when physicists have gained deep insight into abstract mathematical constructions. Mathematicians have been trying to learn from these advancements by way of making the physicists’ constructions mathematically meaningful and also by way of inspiration from the physicists’ pictures.

Aside from specific useful constructions, sometimes the physics point of view has helped mathematicians to reorganize several theories they already knew deeply in ways that new insights were gained. For instance, the theory of moduli spaces of surfaces was always intensely studied by mathematicians. When these objects naturally appeared in physics of the path integrals, they appeared in a more holistic way which shed light on the algebraic structures expressed in terms of surfaces composing with one another via gluing. This gave a boost to the theory of algebraic operations and gave new insights into higher category theory, as well as invariants of knots and manifolds via topological field theories.

Mathematicians have tried to understand what is behind all this. For example, Kevin Costello and Owen Gwilliam have written several books explaining some of the physics concepts behind these discoveries in the more familiar language of homotopical algebras. Reading these books makes one realize the physicists’ constructions are very similar to techniques of homotopical algebras as in Stasheff’s A-infinity world and Sullivan and Quillen’s rational homotopy theory that are familiar to us.

Costello, applied these constructions to show how the Witten genus can be constructed as the partition function (a projective volume element) of a theory (a sigma model). Also see Grady-Gwilliam paper on the A-hat genus.

One goal of our seminar is to understand this approach to the Witten genus. It appears, aside from some Ansätze here and there, everything is based on first principals and logic and therefore within the grasp of working mathematicians. 

This Wednesday I will discuss some of the rudiments as follows.
-flat connection on the top exterior tangent bundle and projective volume element
-divergence operators, BV, and BD algebras (ask in-house expert John Terrila about the difference an h-bar makes)
-examples of BV algebras and differential BV algebras coming from symplectic geometry
-an example coming from the CY geometry giving rise to genus zero B-model of mirror symmetry
-one more example coming from Poisson manifolds
-relations between projective volume forms, divergence operators, right D-module structure of the structure sheaf
-the infinite jet bundle and its flat connection
-the Spencer complex
-L-infinity algebras and curved L-infinity algebras
-L-infinity spaces and the L-infinity space associated to smooth and complex manifolds
-Chevalley-Eilenberg complex of a curved L-infinity algebra
-Koszul duality


Wed Sept 27
Martin Bendersky
Conner Floyed and Landweber Exactness

 Using MU* or MSO_* genera to a ring R  induces a functor from CW complexes to R modules.  Sometimes these are cohomology theories.

In order to give a general criterion for the functor to be a cohomology theory (due to Peter Landweber) I will review some facts about formal group laws.  I will give the first example of a genus inducing a cohomology theory due to Conner and Floyd - Here R = Z[u,u^{-1}] and the resulting cohomology theory is K-theory (actually Conner and Floyd's construction gave  Z/2 graded K-theory.  I will use the Landweber exact functor theorem to construct Z-graded K theory).  

There are various cohomology theories called Elliptic cohomology.  The non-periodic theory (with coefficients in Z[1/2][delta, epsilon]) is constructed by the Baas-Sullivan method of killing cobordism classes by introducing singularities.   Periodic Elliptic theory  (where the discriminant, epsilon(delta^2-epsilon)^2 is inverted) is shown to be a cohomology theory using Landweber exactness.

  Then a quick introduction to K-theory. K^0(X) is the ring of virtual bundles over X.   Bott periodicity extends this to a cohomology theory (K^{-n}(X), n \geq 0 is readily defined, Bott periodicity defines the functors for n <0).   A brief description of stable cohomology operations (i.e. operations which commute with the suspension operator - which for K-theory is the Bott isomorphism) and unstable operations.  Introduction to the Adams operations (which are not stable).  An application to Hopf invariant 1 (e.g. division algebras, H-space spheres and parallelizable spheres are in low dimensions).  A quick introduction to extending Adams operations to the p-adics.

 

Wed Sept 20
No seminar duo to CUNY holiday schedule
 

Wed Sept 13
Martin Bendersky
Witten Genus and Elliptic Cohomology II

As an example, he will discuss how K-theory can be obtained from the complex cobordism theory using the Todd genus. I am attaching Conner and Floyd’s original work on this to this email. Martin will discuss how, in a similar fashion, one can construct the elliptic cohomology. 

If you missed the previous 2 lectures, you can read the writeup below for the past two weeks and come to the seminar with your questions and comments. 

We strive to make each lecture as self contained as possible so that everyone can participate. 

 

Wed Sept 6
Martin Bendersky

Witten Genus and Elliptic Cohomology I

 

In our last lecture, I gave a description of what an elliptic genus was. This week Martin Bendersky will speak about the closely related topic of elliptic cohomology. We will keep the lecture as self-contained as possible so that it would be easier to follow if you missed the previous lecture.

Recall that an oriented genus, or simply a genus, is a ring map from the oriented cobordism ring to an algebra R over the rational numbers (see the writeup from last week below). Since the oriented cobordism ring over the rationals is the polynomial algebra over the complex projective spaces of real dimensions 4, 8, 12, etc., the information of the genus can be fully stored in a formal power series called the logarithm series of the genus. 

Among the genera, the ones with a strict multiplicative property with respect to bundles of compact spin manifolds with compact connected structure group are nicely characterized. More precisely, suppose \phi is a genus whose value on the total space of bundle of closed spin manifolds with compact connected structure group is the product of the \phi of the base times \phi of the fibre. These genera are exactly the ones whose logarithm series are given by the Jacobi elliptic integrals. For their immediate connection to elliptic integrals, these genera are named the elliptic genera. 

The signature and the A-hat genera are examples of elliptic genera, even though they are rather degenerate cases in the following sense. The composition inverse of an elliptic genus is generically a doubly periodic meromorphic function on the complex plane or a meromorphic function on a complex elliptic curve. For the signature and A-hat genus, the quartic polynomial defining the elliptic integral has repeated roots and the corresponding inverse functions are respectively u=tanh(x) and u=2sinh(x/2) which are only singly periodic meromorphic functions on the complex plane. Singly periodic functions can be thought of as a doubly periodic function where the length of one of the period vectors is infinite.

It is more natural to consider all elliptic genera together as a single genus with values in the ring of modular forms for the level 2 congruence subgroup of modular group PSL(2, Z) marked by those 2x2 invertible matrices that are congruent to the identity matrix mod 2.

This week, Martin Bendersky will give a description of the Witten Genus.  He will recall all the necessary definitions so that anyone who missed the last lecture can still follow his talk. In particular, he will define modular forms and discuss their relation to the Witten genus.

One can try to use a given genus to construct a homology and cohomology theory from the oriented bordism theory. The idea is that a genus \phi: \Omega^SO —> R makes R into a module over \Omega^SO. Given a space X, one can tensor, over the oriented bordism ring \Omega^SO, the oriented bordism homology group of X with R. When R is a flat module, tensoring with R is an exact functor and as a result this new assignment of groups to spaces defines a homology theory. Unfortunately, R is rarely flat. Fortunately, Conner and Floyd discovered that for certain genera, miraculously, one obtains homology and cohomology theories even though technically the flatness assumptions which would have been sufficient fail. The inherent weaker flatness conditions of the examples were made abstract by Peter Landweber into conditions now referred to in the literature as the Landweber exactness conditions. For instance, the complex K-theory KU can be obtained from a Landweber exact genus as Martin will explain.

Martin will also discuss Quillen's theorem that the complex bordism Omega^U is the Lazard ring. Ring homomorphisms out of the Lazard ring are in one to one correspondence with formal group laws. In fact, this is the defining property of the Lazard ring. 

Also, Scott will use some of the time to expand on his comments from last week which in and of themselves can expand into a future lecture (Scott?). These will probably fill our time for this week. 

Beyond these, Martin plans to discuss the following material. 

Aside from their additive and multiplicative structures, many multiplicative cohomology theories enjoy more sophisticated operations. For example, ordinary cohomology theories with coefficients in Z/p, have the Steenrod operations. Complex K-theory has power operations and the Adams operations. 

After defining the Adams operations, Martin will explain how to use them to characterize those spheres that admit an H-space structure. He also discusses which spheres are parallelizable as well as how to classify finite dimensional division algebras over the reals.  

This will lay the background for studying cohomology operations in elliptic cohomology in the future. Hope to see you this Wednesday. 

Wed Aug 30
Mahmoud Zeinalian

This Wednesday’s seminar at 2PM in the Math Lounge will be about Elliptic Genera. 

The signature of closed oriented manifolds of dimensions 4, 8, … is defined as follows. The cup product composed with the evaluation on the fundamental class gives rise to a symmetric bilinear form over the reals on the middle cohomology.  Such a form diagonalizes over the reals with a number of +1s and -1s on the diagonal. The signature is defined to be the number of +1s minus the number of -1s. The signature of a disjoint union and a cartesian product are respectively the sum and the product of the signatures. 

Even though this invariant can be defined for topological manifolds or even Poincare duality spaces of dimension 4k, we will focus on smooth manifolds for now because some of the upcoming concepts are either not defined for more general spaces, or their definitions will require further nontrivial considerations. For example, the Pontryagin classes which are defined readily for smooth manifolds can also be defined for triangulated topological manifolds but not as easily. We will postpone such subtleties to a later time. 

Perhaps the most important property of the signature beyond additivity and multiplicativity is its invariance under cobordism, or equivalently, that the signature of the boundary of a compact and oriented manifold of dimension 4k+1 is zero. The signature of a 4k-dimensional sphere, which is the boundary of a ball, is obviously zero because the middle dimensional cohomology is zero. The signature of the unit sphere bundle of a real vector bundle is zero because it is the boundary of the unit disc bundle. The signature of the projectivization of an even dimensional complex vector bundle is zero because these manifolds also bound (why? see hint below). Note that it is possible to have manifolds with zero signature that do not bound. In fact, an oriented closed manifold bounds if and only if all its Stiefel-Whitney and Pontryagin numbers are zero.

Let Omega^SO denote the ring of cobordism classes of smooth closed oriented manifolds under the disjoint union and cartesian product. The structure of this ring is complicated. However, if one tensors this ring with the rationals Q this ring becomes isomorphic to the free polynomial ring on the cobrodism classes of complex projective spaces of dimension 4, 8, …

In fact, this ring modulo the torsions is isomorphic to the polynomial ring over the integers. However, the generators now are not the familiar complex projective spaces but rather certain hyper surfaces M4, M8, … discovered by Milnor. In fact, Omega^SO \otimes Z[1/2] = Z[1/2][M4, M8, …].

By definition, an oriented genus, or simply a genus, is an algebra map from Omega^SO to a Q-algebra R. In this sense, signature is a Q-valued genus which happens to have an additional integrality property. Genera with similar additional integrality properties for closed oriented manifolds, or manifolds that carry further geometric structures beyond an orientation, are of great importance. 

For this and other reasons, one can define a host of cobordism rings that take into account various geometric structures. For instance, Omega^Spin is the ring of cobordism class of manifolds where the structure group of their stable normal bundles is given a lift from the special orthogonal group SO to its double cover Spin. Similarly, Omega^MU is defined as the cobordism ring of manifolds whose stable normal bundles are endowed with an almost complex structure.

Understanding the structure of these rings is also very interesting and important. Some cobordism rings such as Omega^Spin are to this day not fully understood. One amazing result of Milnor is that Omega^MU is isomorphic to Z[M2, M4, …], which after tensoring with Q happens to be isomorphic to Q[CP^1, CP^2, CP^3, ...]. 

It is interesting that CP^1, CP^3, CP^5, etc bound smooth oriented manifolds but the normal bundles of those manifolds do not admit almost complex structures no matter how many copies of the trivial line we add to them. To see CP^2k+1 bounds an oriented manifold, note that CP2k+1 which is the space of lines in C^2(k+1) has a natural map to HP^(k+1) by sending a complex line to the quaternionic line it generates. The fibre of this map is a 2-sphere and the structure group is given by rotations. Therefore, one can fill in the fibre 2-spheres with 3-balls in such a way that they glue nicely together to give a manifold whose boundary is CP^{2n+1}. Use this as the hint for the earlier claim. 

Going back to oriented genera Omega^SO —>R, we may ask if there exist other genera aside from the signature. Since R is a Q-algebra and Omega^SO \otimes Q is a polynomial ring over CP2, CP4, etc, one can arbitrarily assign R-values to the generators to obtain a ring map. An interesting question is how to relate such genera to other known cobordism invariants of manifolds such as the Pontriagin numbers. For instance, the signature of an oriented 4-maniold is 1/3 of the first Pontryagin number i.e. the integral of the first Pontryagin class over its fundamental class. 

Can we express an arbitrary genus in terms of the Pointriagin numbers? The answer to this question which culminated in the work of Hirzebruch is yes. We will discuss how to do this in our seminar. 

Given a genus \Phi: Omega^SO —>R, we encode the information of the values attained by the generators in a formal power series called logarithm or the log series of the genus defined as follows. 

log_\phi(u)=u+\phi(CP^2)/3 u^3, \phi(CP^4)/5 u^5 etc. Thus, the log series associated to the signature becomes u+1/3 u^3 +1/5 u^5 … which can be rewritten as the integral from 0 to u of 1+x^2+x^4+… with respect to x. The integrand is of course just 1/(1-x^2) which can again be rewritten as 1/sqrt(1-2(1)x^2 + (1)x^4). The reason I am rewriting the log series in seemingly increasingly artificial forms is that the genera whose log series have expressions of the form integral 0 to u of 1/sqrt(1-2(\delta)x^2 + (\epsilon)x^4) happen to have a characterizing rigidity property. 

These genera which for their obvious connection to elliptic integrals above are named the elliptic genera, are precisely singled out as those genera that possess the following stronger multiplicative property: their value on the total space of a smooth fibre bundle compact and connected structure group with spin manifold fibres is the product of the genus of the base and the genus of the fibre. 

So for any choice of delta and epsilon one obtains a genus with values in the complex numbers. Note that, as it was the case with the signature, the genus associated to a particular choice of delta and epsilon may have its image in a smaller subring of R. 

It is however more natural not to specialize to particular values of delta and epsilon. Delta and epsilon are actually best interpreted as modular forms. More precisely, one can show there exist two holomorphic functions delta and epsilon on the upper half plane in such a way that for any given tau in the upper half plane, delta(tau) and epsilon(tau) determine a curve inside CP^2 given by the equation y^2=1/sqrt(1-2(\delta(tau))x^2 + (\epsilon(tau))x^4). This is the Jacobi presentation of an elliptic curve i.e. a smooth complex curve of genus 1. Furthermore, for a fixed \tau, this elliptic curve is isomorphic to the torus obtained by quotienting the complex plane C by the lattice Z+Z\tau. The actual isomorphism is given by two doubly periodic functions x=s(z) and y=s’(z)=derivative of s. Here, s is the composition inverse of the integral 0 to u of 1/sqrt(1-2(\delta)x^2 + (\epsilon)x^4) i.e the inverse of the log series of the elliptic genus.

One word of caution is that the case of signature was rather degenerate because the polynomial 1-2x^2+x^4=(1-x^2)^2 has repeated roots. In this case the log series u+1/3u^3 +… equals tanh^-1(u) whose inverse s(u)=tanh(u) is only singly periodic. This is because in this case we have a singular elliptic curve. 

Elliptic genera satisfy a certain "flatness” condition, referred to as Landweber exactness, similar to the condition discovered by Conner and Floyd for the Todd genus. This additional flatness property of the elliptic genus leads to a construction of a cohomology theory called Elliptic Cohomology. 

Let me stop here now and we will further discuss this interesting crossroad of classical ideas in complex analysis and number theory as well as modern homotopy theory slowly over the next few lectures. If you have not heard about these things, this writeup which is really meant for you and I to organize our thoughts, may be uninviting. But, we don’t have to go through this material fast and can take our time parsing the concepts. 

Some excellent references for this material are as follows:

1) Serge Ochanine’s Notices "What is an elliptic genus”, which is only 2 pages and beautifully written

2) My LIU colleague Corbett Redden’s account “Elliptic Cohomology: A Historic Overview”

3) Graeme Segal’s 1987 Bourbaki Seminar notes which fueled a lot of the activities in the past three decades

4) Landweber’s intro to elliptic genera