Based on joint work with Jacek Świątkowski (Wroclaw University).

A particularly useful class of spaces in geometric group theory is the one of (locally) CAT(-1), (and CAT(0)) spaces: metric spaces of negative (or nonpositive) curvature with no assumption of smoothness.

The main source of high dimensional spaces of that kind are combinatorial constructions involving cubical complexes. In 1987 Gromov realized that there is a beautifully simple combinatorial condition on a cubical complex, which is equivalent to the a priori noncombinatorial CAT(0) condition. The theory of CAT(0) cubical complexes is fairly well developed by now.

In recent work, Jacek Świątkowski and myself asked for the corresponding story for simplical rather than cubical complexes. What we got was somewhat surprising: we could not formulate a simple combinatorial condition equivalent to the CAT(0) property, but found a very simple condition--Simplicial NonPositive Curvature (SNPC)--which, without implying CAT(0), has many of its consequences. The condition also admits strengthenings which do imply CAT(0).

I plan to explain some background on CAT(0), tell you what SNPC is, how one works with it, and show some similarities (and dissimilarities) with nonpositively curved spaces we are familiar with.
 

Colloquium: The Moduli Space of Curves of Genus g
Algebraic curves of genus g are parametrized by an algebraic variety (more accurately a scheme or a stack) called M_g, the moduli space of curves of genus g. There has been intense interest in M_g from Physics and many areas of mathematics including algebraic geometry and topology. The moduli space is both interesting in its own right and because by studying it, one can learn about the individual curves it parametrizes. In this talk I'm going to give a survey of this beautiful space including mentioning an open problem, which if true when the genus is zero, then is true for all genera.

November 8th Grad Student Seminar, 4pm, THO 119: What is a Family of Algebraic Curves of Genus g?
In algebraic geometry one of the main objects of study are algebraic varieties. One dimensional algebraic varieties are first of all classified by an invariant called genus. I'm going to introduce families of curves of genus g and by giving lots of examples, motivate how one can learn about a particular curve by viewing it as a member of a family of curves.
 

Colloquium: Conformal Invariants, Q-curvature and Ricci Tensor
Based on the work of C. Fefferman and R. Graham in 1985, there have been systematic study of the conformal invariants and conformally covariant operators on manifolds of any dimensions.  A special case of such an operator is a fourth order linear elliptic operator with its leading symbol the bi-Laplace operator called the Paneitz operator.  In this talk, I will discuss study of this operator on four manifolds, its associated curvature function called the Q-curvature, connection of Q-curvature to the study of eigenvalues of the Ricci tensor and applications to some problems in geometry.  I will also discuss some open questions in the area.

November 22nd Grad Student Seminar, 4pm, THO 119: How can we tell it is the sphere?
Given a compact, closed manifold, how can we tell it is differomorphic to the standard sphere?  When the dimension of the manifold is two, i.e. when we are on a compact surface, we may apply the Gauss-Bonnet formula and decide from the sign of the integral of the Gaussian curvature over the surface to tell if the surface is a sphere, a torus or a hyperbolic ball.  When the dimension of the manifold is bigger than 2, there are many great efforts to seek analytic or curvature quantities to distinguish the sphere.  In this talk, I will describe some effort in this direction using analysis and PDE methods.  Exposing the fact that the conformal transformation group of the sphere is not compact, I will discuss a blow-up sequence of functions which are extremals of a Sobolev embedding inequality and relation of the embedding to the Yamabe problem.  I will also discuss efforts in characterizing the n-sphere via some n-th order conformal invariants.
 

Click here to view slides from this talk.
 

February 14th Grad Student Seminar, 4pm, THO 134: General Relativity, Overview and Prospects
We will first outline the geometrical definition of a spacetime of general relativity, together with its original physical motivations. Then we will explain what are the Einstein equations governing its dynamics and give the classical model of the most current use which has received confirmation under observation. Finally we will briefly survey some recent results and open problems in black holes and cosmology which make General Relativity the most remarkably successful and also puzzling mathematical and physical theory.
 

I will give an overview and in particular explain how a homotopy theoretic approach led to the recent proof by Madsen and Weiss of Mumford's conjecture on the rational stable cohomology of moduli space (obtained by letting the genus tend to infinity).

April 4th Grad Student Seminar, 4pm, THO 119: Groups, Categories, and Classifying Spaces
A good way to study a discrete group is to study spaces on which it acts. Much can be learned from the topology of its orbit space and the stabilizer subgroups.  Of particular interest are those spaces which admit a free group action and which are topologically trivial.  Their orbit spaces are 'classifying spaces' for covering spaces, as I will explain.  Vice-versa, it is easy to recover the group from its classifying space.

In his thesis in 1968, Graeme Segal introduced the notion of a classifying space for categories which has now become a basic construction in topology.  In part as a preparation for the colloquium talk I plan to discuss how extra structure on the category leads to extra structure on the associated classifying space, and how much the classifying space remembers of the category.
 

April 25th Grad Student Seminar, 4pm, THO 119: Whitney's Extension Theorem and Related Problems
H. Whitney studied in 1934 which functions $f:\,A\to\ {\mathbb R}$ can be extended from a subset $A\subset{\mathbb R}^n$ to ${\mathbb R}^n$ as a $C^m$ function. In particular, he showed that there is a continuous curve on the plane and a $C^1$-function $f$ with $f'(x)=0$ at each point of the curve, such that $f$ is not constant along the curve. That is, one can climb up to the top of a mountain along a curve in such a way that he moves all the time only horizontally!

We will study Whitney's extension theorem and related problems, old and new.
 

The talk will consist of two parts. The first one (hopefully short) will deal with the philosophical question of what makes a field a part of mathematics (Hint: a collection of rigorous definitions, theorems and proofs is not necessarily a field of mathematics). In the second part I will argue (surprise, surprise) that probability is indeed a field of mathematics. The concept of "probability" is now an abstract mathematical notion, often unrelated to any real world randomness. I will present a number of examples of purely technical uses of probability measures in mathematics.
 

A classical problem in topology is that of characterizing those finite groups that act fixed-point freely on a sphere. In this talk we will review these results and describe recent work towards extending this to (i) a product of two spheres and (ii) actions of discrete groups. The methods we use are a combination of techniques in topology and group theory.
 

June 1st Grad Student Seminar, 4pm, THO 119: Imaging and Inverse Problems
Radar imaging, seismic imaging, and medical imaging all require sophisticated mathematical processing to form an image. The mathematics involved often takes the form of an ``inverse problem'', in which one tries to deduce the coefficient in a differential equation from partial knowledge of the solution of the equation. This talk will discuss examples of various inverse problems, features of such problems, and some of the mathematics involved in addressing them.