K-Theory Seminar
Wed  2:00-4:00
Room 6417
Grad Center CUNY
365 Fifth Ave

 

Wed Apr 18
Micah Miller

Coherent duality and the Hirzebruch-Riemann-Roch theorem
TBA

Wed Apr 11
Gregory Ginot
An introduction to topological cyclic homology

TBA

Wed Apr 4
Gregory Ginot

An introduction to topological cyclic homology
TBA

Wed Mar 28
Cheyne Miller

Hodge to deRham degeneration
TBA

Wed Mar 21
Rob Thompson

TBA
TBA

Wed Mar 14
No Lecture

No Lecture

Wed Mar 7
Rob Thompson

TBA
TBA

Wed Feb 28
Corbett Redden

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 talks)
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.

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