Topology and Geometry Seminar Archive

2007

2006

2005

2004

2003

Unlike trigonometric functions that are simply-periodic, elliptic functions are relatively well-behaved functions of a complex variable that have two independent periods. In the last 20 years or so, elliptic functions have come to play an important role in topology, through the notions of elliptic genus and elliptic cohomology. The talk, aimed mainly at the first and second year graduate students, is meant to be a gentle non-technical introduction to the subject. No prior knowledge of topology is required.

On September 23, we considered some basic notions: manifolds, differential forms, DeRham cohomology, looked in some detail at the index (signature) of an oriented manifold and discovered that the index can be viewed as a genus, i.e. as a ring homomorphism from the oriented cobordism ring into a number ring or field. We also looked at the particular role played by the complex projective spaces in this picture. The September 30 talk will explore these ideas further. In particular, we will see the interaction between the analytical properties of the functions involved (trigonometric, elliptic) and the geometry of the manifold, in particular the existence of circle actions on the manifold. Like the previous talk, this talk will be very informal. It aims at giving the audience some taste of the subject, rather than any precise knowledge of any particular result. Everybody is welcome.

The moduli space of pointed stable curves of genus g was introduced by Deligne, Mumford and Knudsen to give a natural compactification of Mumford's moduli space of nonsingular curves of genus g. A new construction of the genus zero case is due to Keel. Kontsevich and Manin develop a cohomological field theory which, together with Keel's presentation, imply the WDVV equations. Hence, it is natural to refer to the cohomology ring given by Keel's presentation as the WDVV ring. En route to understanding the combinatorics of the WDVV ring, I study the topology of a related ring, which I call the pre-WDVV ring. In this talk I will show how the pre-WDVV ring is directly related to the Whitehouse complex. Using discrete Morse theory, I give an elementary proof of the Cohen-Macaulayness of the pre-WDVV complex. Additionally, I describe recurrences for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.

Hopkins and Miller have introduced a generalized cohomology theory TMF*(X) called “elliptic cohomology” or “topological modular form theory”; the latter terminology is motivated by the fact that the coefficient ring TMF*(point) agrees with the ring of modular forms after tensoring with C. From a geometric point of view, TMF*(X) is the home of the “family Witten genus” of a family of string manifolds parametrized by a space X (i.e., a fiber bundle E --> X whose fibers are string manifolds). This is analogous to the “family Â-genus” of a family of spin manifolds parametrized by X, which lives in KO*(X) (the real K-theory of X). However, unlike KO*(X) which is build from vector bundles over X, there is so far no geometric definition of TMF*(X) in terms of suitable “elliptic objects”. According to an old idea of Segal, elliptic objects should be closely related to 2-dimensional conformal field theories. In these talks I will try to motivate this relationship and I will explain the analogous statement for K-theory. Then I will present a modification of Segal's elliptic objects in order to make “excision” possible. Conjecturally these “enriched elliptic objects” represent elements of TMF*(X). This is joint work with Peter Teichner.

In the previous talk, Hirzebruch’s Index Theorem and, more generally, Hirzebruch’s formalism were discussed. In this talk, I will show how the analytical properties of the generating functions (e.g. the hyperbolic tangent) are reflected in some topological-geometric properties of the manifold. Then I will introduce the elliptic genus and discuss its properties. No prior knowledge of elliptic functions will be assumed.

In this final talk of the series, I will sketch the proof of the vanishing of elliptic genera of projective bundles and then discuss the rigidity phenomena for Spin manifolds.

Ritter and Segal proved that the Burnside ring of finite G-sets maps onto the rational representation ring of G for G a finite p-group, making a connection between stable maps and K-theory. We extend their result to the double Burnside ring of finite G-sets with a free Z/p-action, motivating the question topologically and focusing on the algebraic solution.

Embed C (a copy of the n-1 dimensional sphere S(n-1)) in S(n). In 1960, singular homology had already shown that that this separates S(n) into two components, U and V, whose common boundary is C. Question: Are the closures of U and V in fact n-cells? For n = 2, the answer was already known to be yes (Schoenflies). Brown's paper examines the answer for n > 2.

We consider two classes of left R-modules, P and C, such that P is included in C. If the module M has a P-resolution and a C-resolution then for any module N and all non negative integers n we define generalized Tate cohomology modules E(M,N,C,P,n) and show that we get a long exact sequence connecting these modules and the modules E(M,N,C,n) and E(M,N,P,n). When C is the class of Gorenstein projective modules, P is the class of projective modules and when M has a complete resolution we show that the modules E(M,N,C,P,n) for n>0 are the usual Tate cohomology modules and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate cohomology E(M,N,R,n) can be computed using either a complete resolution of M or a complete injective resolution of N. And so, using our dual result, we obtain Avramov and Martsinkovsky's exact sequence under hypotheses different from theirs.

I will discuss circular symmetries of the real and complex Dirac operators and give a proof of the vanishing of the elliptic genus on odd spin S^1-manifolds.

We will look at some classical indecomposable continua and their description as inverse limits.

We will look at some classical indecomposable continua and their description as inverse limits.

A simplicial complex consists of a set {v} of vertices and a set {s} of finite nonempty subsets of {v} called simplices. A Euclidean ring is a commutative ring in which the division algorithm holds. We construct examples of contractible simplicial complexes on which groups of matrices with entries in Euclidean rings act with finite stabilizers. Then we show the relevance of these examples for studying various topics in group cohomology and algebraic K-theory.

Wall’s Finiteness Obstruction Theory arises from the question: “When is a finitely dominated CW-complex Y homotopy equivalent to a finite complex?” To answer this question, Wall algebraically defined an obstruction, sigma(Y), in the reduced projective class group of Y, and showed that Y is homotopy equivalent to a finite complex if and only if sigma(Y) = 0. Later, Ferry gave a geometric interpretation of Wall’s finiteness obstruction. The object of this talk is to use Ferry’s geometric interpretation to prove Wall’s theorem.

We discuss the definition of VA's and prove an important theorem about their structure.

We report on recent progress made towards solving an old standing conjecture of Quillen about the homology with finite coefficients of a group of matrices over algebraic integers.

While matroid theory dates back to Whitney, 1935, the fundamentals of oriented matroids were introduced in 1978 in two papers by Bland and Las Vergnas and Folkman and Lawrence. This talk will begin with a discussion of some different representations of oriented matroids, follow with a look at two major theorems, the Folkman-Lawrence Topological Representation Theorem and the Universality Theorem, and conclude with a look at some open problems and current research in the field.

The fundamental theorem of algebra states that the field C of complex numbers is algebraically closed. There will be a short introduction to the concept of Poincare group of a topological space after which I shall go on to prove the theorem by using this group. The talk is intended for graduate students.

We state Sard's Theorem, then we give an outline of the proof of the theorem. If time permits, we will state a few applications of the theorem. The talk is intended for graduate students.

An imaginary hotel in which there are infinitely many rooms, numbered 1, 2, 3, 4, ... Even if the hotel is full, one more guest can be fitted in by asking everybody to move to the next room up, which frees up room 1. In fact, by asking everyone to move up n rooms, you can free up the first n rooms, and accommodate any finite number of new guests. If a (countable) infinite number of new guests arrive, you can fit them in by asking the person in room 1 to move to 2; the person in 2 to move to 4, the person in 3 to move to 6, etc... This frees up all the odd numbered rooms. The Hilbert hotel is a good demonstration of some of the unexpected properties of infinite sets. These properties are the subject of this talk. The talk is intended for graduate students.

An introductory talk intended for graduate students.

This is part of a series of elementary lectures. Everybody is welcome.

This is part of a series of elementary lectures. Everybody is welcome.

This is part of a series of elementary lectures. Everybody is welcome.

Dissertation.

Dissertation.

In this talk, we will review the "higher" character theory of Hopkins, Kuhn, and Ravenel, which gives a description of the rational Morava E-theory of classifying spaces for finite groups. We will then describe a generalization of their result, and how it leads naturally to the idea of "higher equivariance".

We will give an overview of the classical approach to elliptic cohomology, leading up to the construction of the spectrum of topological modular forms (tmf) by Hopkins and Miller. We will then introduce the language of derived algebraic geometry, and explain how it can be used to give a new (and easier) construction of tmf.

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint". (Time permitting, some additional comments will be made on regular values.)

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

We discuss the point-set part of a paper by J.H.C. Whitehead.

This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".

This program is a spectacular interplay between number theory, geometry, representation theory and analysis. Mostly we will concentrate on the geometric version of the program, which comes down to complex geometry and representation theory of infinite dimensional Lie algebras. The first introductory talk will be part of a series of talks based broadly on some work of Beilinson, Drinfeld, and others. Some lectures will be contributed by the other participants. Everybody is welcomed to attend!

We will explain how a topologist may look at classfield and then focus on the Frobenius element and its corresponding adele.

We will recall what is the Frobenius element and continue its study. Then we will construct two dimensional Galois representations by using elliptic curves.

This is a talk for master degree.

Quandles, introduced by D. Joyce in 1982, are algebraic structures which model the Reidemeister moves in knot theory. These structures were discovered independently at the same time by S. Matveev under the name of distributive groupoids. Joyce associated a quandle to a knot, called knot quandle, and proved that it is a complete invariant of knots. Quandle cohomology was introduced by S. Carter et al. in early 2000 as a modification of rack cohomology theory of R. Fenn, C. Rourke, and B. Sanderson. We will give a survey of quandle cohomology and cocycle knot invariants, describe some of our recent joint work with S. Carter and M. Saito and conclude with some open problems.

This is a talk for master degree.

This is a talk for master degree.

This is part of a qualifying exam.

This is part of a qualifying exam.