Student Geometry/Topology Seminar

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)

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:

Schedule of Talks

Abstracts

My hope for this talk is discuss one instance where ideas from dynamical systems can be used to derive topological information. Expanding maps on manifolds are a class of maps that generate "chaotic" dynamics. I will derive some elementary topological constraints on spaces which admit expanding maps, leading up to a statement of a theorem of Shub and Gromov which classifies these maps up to topological conjugacy.

Intuitively, a spine of a 3-manifold M is a 2-polyhedron onto which M deformation retracts. If we impose certain restrictions on spines, they give an elegant combinatorial representation of 3-manifolds. In the first half of the talk, I will summarize the basic theory of spines. In the second half, I will present one application of this theory: the Turaev-Viro 3-manifold invariants. Along the way, I will mention some current research being done on these topics. This talk will be accessible to everyone and includes several colorful pictures.

This talk will be a survey on results related to semisimple Lie groups. In particular, I will discuss some ways in which these groups arise when one tries to solve geometric problems such as those related to superrigidity, strong rigidity, and the isospectral problem. We will see how the rich structure of semisimple Lie groups allows one to use tools from other areas of mathematics (such as measure theory and algebraic number theory) to answer these geometric questions.

A building can be thought of as a simplicial complex satisfying a few additional axioms. The concept was introduced by Jacques Tits to encode the algebraic structure of linear groups in geometric terms. As a unifying principle, buildings have found a wide variety of applications: the classification of groups (algebraic, Lie, or finite simple), Kac-Moody groups, combinatorial geometry, and rigidity phenomena in negatively curved spaces. In this talk, we will define buildings, consider some examples, and (time permitting) look at some applications.

In the context of dynamical systems, the topological entropy is a dynamical invariant which measures how "wild" the system is. When our system is given by a smooth map on a compact manifold then by a well known result of Yomdin we can estimate from below this quantity in terms of the spectral radius of the action on the cohomology. If we also assume that our manifold is Kahler and our map is holomorphic, then we can obtain the same bound from above by a famous argument given by Gromov. In this talk we will introduce the topological entropy and show some basic properties and examples (particularly from holomorphic dynamics). At the end we will sketch the beautiful proof of Gromov.

After classifying surfaces, a natural object to classify is homeomorphisms of surfaces. Given a topological space X, the mapping class group of X is the group of its self--homeomorphisms modulo those homeomorphisms isotopic to the identity. Given an orientable surface, elements of its mapping class group fall into an elegant trichotomy (much like the classification of isometries of H²). This classification is due to Nielsen and Thurston, with an alternate proof by Bers. In this talk I'll present the simple (i.e. linear) case of the mapping class group of the torus. Then I'll discuss the analogous classification for higher genus surfaces. Lastly, I'll present some applications of the mapping class group classification to the geometry of mapping tori.

To any compact, orientable surface one can associate a simplicial complex, C(S), called the curve complex that encodes information about the intersection pattern of simple closed curves on that surface. We will discuss some basic properties of the curve complex and an application to hyperbolic geometry.

On a bundle over a smooth manifold, we will define connections, their curvature, and their holonomy. I will try to say why they give the right setting for geometry. The level will be fairly elementary.

Group actions on trees is a fundamental tool in geometric group theory. First I will show how a splitting of a group corresponds to an action of the group on a tree. Next I will define the number of ends of a group. Finally I will present Stallings' theorem, an application of these two concepts. The only prerequisite for this talk is basic covering space theory.

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UM Math Seminars Page

Last Updated:

November 19, 2009

Past seminar pages:

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