Student Geometry/Topology Seminar

2010 Graduate Student Topology and Geometry Conference

It is almost here! The 2010 Graduate Student Topology Conference will be held here in Ann Arbor on April 10th and 11th. There will be great keynote lectures, open problem sessions, and 32 short talks given by graduate students from across the country.

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Schedule of Talks

Abstracts

Using the Bochner technique, Steve Kerckhoff and I recently proved the following theorem. Let M be a compact hyperbolic manifold with totally geodesic boundary. If M has dimension at least four, then the holonomy representation of M is infinitesimally rigid. This is an infinite volume analog of the Calabi-Weil rigidity theorem. I will explain some of the background and ideas used in the proof.

In this talk, I will state what is a classifying space and how it is used, and raise some interesting questions. After Milnor's construction of the classifying space for free G-action, the study on classifying space has been expanded to the study of classifying space for families of subgroups, especially for the family of compact subgroups, which is so called the classifying space for proper action. I will introduce several models for such classifying space, some of which will be shown explicitly in the case of Fuchsian group. For arithmetic subgroup, a cocompact model for classifying space is given by the Borel-Serre compactification. I will give sketch of a proof using the upper half-plane model.

Last time, I introduced the moduli space of curves and motivated the study of its cohomology ring (in particular the tautological classes). I concluded with three relations among intersection numbers of tautological classes which allow the computation all genus zero intersection numbers. In the present talk, I will outline how these relations may be obtained, and indicate how they are special cases of the Witten-Kontsevich theorem, which determines the analogous intersection numbers for any genus.

Witten's 1991 survey paper Two-Dimensional Gravity and Intersection Theory on Moduli Space provided the motivation for surprising and significant developments in Geometry and its interaction with Physics. I will present a selection of the arguments from this paper, which give an intuitive and illuminating sense of the techniques in the field. Time permitting, I will outline some of Witten's original conjectures, and the extent to which they have been resolved after two decades of intense study.

After defining the mapping class group of a closed surface and Dehn twists, we prove the claim for the torus and then outline the induction up to closed surfaces of higher genus. Along the way we highlight use of the group's action on a particular graph: the 1-skeleton of the complex of (nonseparating) curves, which we'll define.

Given an almost-invariant set of a group G, we can construct a cubing on which G acts. I will define "almost-invariant set" and "cubing", and give plenty of examples of this procedure. After exploring several basic properties of Sageev's cubing, we will give a generalization of Stallings' theorem that employs the cubing.

Last week, some topological properties of the p-adic numbers were discussed. Many of these properties carry over naturally when discussing matrix groups with p-adic entries. We'll give some motivation for studying p-adic groups and how they compare and differ from real Lie groups. If time permits, I will also mention the Bruhat-Tits building and its ubiquity in the study of p-adic representation theory.

Several older graduate students will give short, informal talks about their and their adviser's research. They will also be delivering insightful/amusing anecdotes and taking questions. First- and second-year students considering working in geometry/topology are especially encouraged to attend.

In this talk I will define the p-adic numbers and try to motivate their significance in topology and geometry. I will begin by construct the real and p-adic numbers. I will then discuss some interesting topological differences between these two fields. For example the real integers are unbounded in R but the p-adic integers are compact in Q_p. As there is a meaningful notion of convergence in Q_p, we can define analytic functions from Q_p^m to Q_pⁿ. As such we may construct analytic p-adic manifolds. I will discuss some important properties of these manifolds and mention how they can help us understand certain real manifolds, in particular Lie subgroups of GL(n, R).

In this talk we will discuss two combinatorial principles from set theory which have been used with great success to decide problems related to the topology of the real numbers. As an example we will consider the following problem, which remained unsolved for 50 years after it was first proposed by Suslin in 1920: Let S be a complete dense linear order without endpoints equipped with the order topology, and suppose that every collection of pair-wise disjoint open subsets of S is at most countable. Is S homeomorphic to R?

Coming Soon.

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:

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Useful Links

UM Math Seminars Page

Last Updated:

April 1, 2010

Past seminar pages:

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