Student Geometry/Topology Seminar

Schedule of Talks

Abstracts

An orbifold is a topological space which looks locally like R^n modulo the action of a finite group. They pop up in physics, algebraic geometry, and a bunch of other places. Because orbifolds are so similar to manifolds, many constructions and facts about smooth manifolds carry over to orbifolds. But in some ways orbifolds are radically different. This talk will introduce the theory of orbifolds and their relation to manifolds, with particular emphasis on ways in which the theory differs. No prerequisites are required beyond a basic understanding of differential topology.

A wall space is space with a collection of walls, each "wall" separating the space into two parts. Cube complexes are the right- angled analogue of simplicial complexes (think: glue together a bunch of Euclidean cubes together). Add a couple combinatorial restrictions to a cube complex, and it becomes CAT(0).

In this talk, I will show how, given a proper action of a group on any wall space, to construct a CAT(0) cube complex on which the same group acts properly. My talk will come equipped with lots of examples. The only prerequisite is that you know what a group action is.

Spectral sequences, while having earned a reputation for being difficult for the learner to work with, are a prevalent and powerful tool in many areas of mathematics. In this talk, I will state and attempt to motivate the definition of a spectral sequence. From there, I will turn to a particularly important spectral sequence in topology, namely the Serre spectral sequence associated to a fibration, and discuss its applications to computations of homology and homotopy groups.

The Cayley graph allows us to think of groups as geometric objects. As such, they have associated growth functions, telling us asymptotically the volume of balls with respect to radius. I'll discuss several important applications of these growth functions-- some in more detail (Milnor's theorems) and some in less detail (Gromov's theorem). This will be a very introductory talk, including all necessary definitions to understand the statements and proofs of results.

A symplectic toric manifold is a symplectic manifold M together with a hamiltonian torus action on M, where the dimension of the torus is half the dimension of M. In this talk, I'll define all the terms necessary to understand the above sentence before sharing some of the beautiful combinatorial properties of sympletic toric manifolds. Time permitting, I'll explain how sympletic toric manifolds arise naturally in the study of completely integrable systems.

A locally symmetric space (e.g. the genus 2 surface) is a quotient of symmetric spaces (e.g. hyperbolic 2-space) by the action of a discrete subgroup of a Lie group. When this discrete subgroup is arithmetic, we then may invoke a wide range of number theoretic tools to say something about this space's topology and geometry. In this introductory talk, I will illustrate this technique by studying the case of SL(2,R). In particular, I will describe arithmetic subgroups of SL(2,R) and use their arithmetic to determine whether the hyperbolic 2-manifold they define is compact. Time permitting, I will explain the general compactness criteria for semisimple Q-groups. This talk will be concrete with many examples.

This is the second part of series of talks on lattices. In this talk, I will introduce methods of constructions of non-arithmetic lattices. In the late 1960s, Vinberg introduced the groups generated by reflections on hyperplanes in Lobachevski space (i.e. hyperbolic space). With certain geometric conditions, this groups form crystallographic groups, that is, discrete lattices whose fundamental domains are polyhedrons. Considering such groups as subgroups of SO(n,1), Vinberg proposed a criterion for the arithmeticity of crystallographic groups. He also proved that there is a upper bound for n where such groups exists. To overcome this limitation, Gromov and Piatetski-Shapiro introduced the method of hybridation of arithmetic manifolds to construct non-arithmetic lattices for all dimension n. In the late 1970s, Mostow introduced the notion of complex reflections in 2-dimensional complex hyperbolic space, and showed the existence of non-arithmetic lattices in SU(2,1). Later, Deligne and Mostow constructed non-arithmetic lattices in SU(n,1) for n <= 3. No example of non-arithmetic lattices in SU(n,1) for n>3 is known.

The Laplacian, being the simplest elliptic operator, is well connected to many branches in Physics - diffusion, Wave propagation and quantum mechanics. In those examples, we often are concerned with finding its eigenvalues and eigenfunctions. The Laplacian can also be generalized to the case of a Riemannian manifold. In the talk I will discuss some very basic setups and the eigenvalue problems of the Laplacian (or the Laplace-Beltrami operator) in Riemannian manifolds, and investigate the properties of its eigenvalues and the connection to physics with examples. If time permits, I will talk about research in spectral geometry which is unrelated to physics but of independent mathematical interest.

I will discuss mirror symmetry, a phenomenon which comes from String Theory. Then I will outline a construction developed in FJR-theory, and how mirror symmetry shows up in the construction of the FJRW-ring. FJR-theory is a Gromov-Witten type theory developed in the last few years by Tyler Jarvis, Huijun Fan, and Yongbin Ruan to prove a conjecture made by Witten regarding the solutions to the so-called Witten equation.

About The Seminar

This seminar is organized by and run by University of Michigan graduate students. Past topics have come from many different areas of geometry and topology, such as:

• Twisted K-theory (algebraic topology)
• Morse theory (differential topology)
• Quasi-isometries of symmetric spaces (differential geometry)
• Growth of groups (geometric group theory)
• Fundamental group of a compact semisimple Lie group (Lie group theory)
• Mathematical physics (differential topology)

For more past topics, see the "past seminar" links on the sidebar. Talks can be a broad survey of a theory or a focused discussion of a single result (or anywhere in between... the space of possibilities is path connected).

For information, or to volunteer to speak on any topic you wish, contact:

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Last Updated:

September 10, 2010

Past seminar pages:

Winter 2010
Fall 2009
Winter 2009
Fall 2008
Winter 2008
Fall 2007
Winter 2007
Fall 2006
Winter 2006
Fall 2005
Winter 2005
Fall 2004