**K-Theory Seminar and its Companion Meeting**

Room 6417

Grad Center CUNY

365 Fifth Ave

Room 6417

Grad Center CUNY

365 Fifth Ave

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**Fall 2018**

**Dec 19**

**10:00-11:30**

**Micah Miller**

**Toledo-Tong resolutions of coherent sheaves**

**2:00-4:00**

**Micah Miller**

**Chern character of infinity vector bundles**

**Dec 12**

**10:00-11:30**

**TBA**

**TBA**

**2:00-4:00**

**Alex Perry**

**Hochschild cohomology and group actions**

**Dec 5**

**10:00-11:30**

**Corbett Redden**

**On twisted equivariant K-theory and the Verlinde algebra**

**2:00-4:00**

**Corbett Redden**

**On twisted equivariant K-theory and the Verlinde algebra**

**Nov 28**

**10:00-11:30**

**TBA**

**TBA**

**2:00-4:00**

**Rob Thompson**

**Complex bordism of infinite loop spaces**

**Nov 21**

**10:00-11:30**

**Martin Bendersky**

**Adams e-invariant**

**2:00-4:00**

**Martin Bendersky**

**Adams e-invariant**

**Nov 14**

**10:00-11:30**

Alexander Milivojevic

**Setting up the Frölicher spectral sequence**

Alexander Milivojevic

## 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

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

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

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**

Martin Bendersky

## 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]).

Directed triangulations

## [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.

Christoph Dorn

Presented associative n-categories

## [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.

Getting rid of the continuum: Posetal models for (interesting) manifolds

**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.

Christoph Dorn

Using manifolds to define (and understand!) general higher categories

**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

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*

*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*

*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.*

*No lecture. Friday Schedule at CUNY.*

**Wed Apr 4**

*No lecture. Spring recess.*

*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.

*Morning Session 10:00-11:30*

*Martin Bendersky*

*On p-adic Adams operation*

*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.

*Afternoon Session 2:00-3:45*

*Rob Thompson*

*Operads and Obstructions*

**Wed Mar 21**

*No lectures due to snow*

*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.

*Companion meeting to the K-theory seminar (morning)*

Martin Bendersky

Martin Bendersky

*10:00-11:30 AM*

*On p-adic Adams operation*

*K-theory Seminar (afternoon)*

*Rob Thompson*

2:00-3:45 PM

*Power operations in Morava E-theory*

*K-theory Seminar (afternoon)*

*Rob Thompson*

2:00-3:45 PM

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*

*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

*Companion meeting to the K-theory seminar*

*Mahmoud Zeinalian*

10:00-11:30

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.

Martin Bendersky

**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 **

Martin Bendersky

## 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**

Corbett Redden

## 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**

Jeffrey Kroll

## 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 **

Micah Miller

## 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**

Rob Thompson

## 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**

Rob Thompson

## 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

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**

Mahmoud Zeinalian

## -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

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

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

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)

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

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

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**

Martin Bendersky