The University of Michigan Student Geometry/Topology Seminar
Winter 2006


The Student Geometry/Topology seminar will be held 3-4 pm on Friday afternoons, in 3096 East Hall.
If you would like to give a talk, just send an email to vavriche at umich dot edu.

Schedule of Talks

January 13th Kyle Hofmann Meromorphic functions on Riemann Surfaces (abstract)
January 20th Cagatay Kutluhan Intersection of surfaces in 4-manifolds (abstract)
January 27th Dave Constantine Marked length spectrum for nonpositively curved surfaces(abstract)
February 3rd Jose Gomez-Guerra Eilenberg-Maclane Spaces (abstract)
February 10th Aaron Magid Seifert Fibered Spaces (abstract) (notes)
February 17th Eric Zupunski 3-Manifold Groups and Group Splittings (abstract)
February 24th "Spring" break No meeting
March 3rd "Spring" break No meeting
March 10th Diane Vavrichek The Pure Mapping Class Group (abstract)
March 17th Marshall Williams Topological Groups: An introduction (abstract)
March 24th Giancarlo Urzua Compact Riemann surfaces with a Z/pZ action for p prime and p > genus, part I (abstract)
March 31st Ryan Kinser Introduction to Quivers
April 7th Giancarlo Urzua Compact Riemann surfaces with a Z/pZ action for p prime and p > genus, part II
April 14th Johanna Mangahas Loop Theorem (abstract)
April 21st Zhenyi Liu Dehn Filling: An Introduction (abstract)

Click here to see the student seminar webpage from Fall 2005.
Click here to see the student seminar webpage from Winter 2005.
Click here to see the student seminar webpage from Fall 2004.
Abstracts

January 13th, Kyle Hofmann, Meromorphic functions on Riemann Surfaces
An abstract Riemann surface is a one-dimensional complex manifold. Riemann surfaces arise naturally by analytically continuing functions on the complex plane and allow a precise formulation of the concept of a many-valued function. Riemann surfaces are often represented as branched coverings of the Riemann sphere CP^1. This representation makes a Riemann surface look like a covering space except at a finite number of points, called branch points. At the branch points, the projection map looks like z->z^n for some n (depending on the point). In fact, all Riemann surfaces can be represented as branched coverings of the sphere; equivalently, every Riemann surface has a meromorphic function. We will sketch a proof of this deep result. As time permits, we will also discuss the implication that all Riemann surfaces are complex algebraic varieties.

January 20th, Cagatay Kutluhan, Intersection of surfaces in 4-manifolds
The classification problem for 4-manifolds is much harder than that of 3-manifolds mainly because not every topological 4-manifold admits a smooth structure and there may be infinitely many non-diffeomorphic smooth structures on a given 4-manifold whenever one exists. In fact, for example R^4 admits uncountably many "exotic" structures on it. Moreover, the tools developed for investigating existence and uniqueness of smooth structures on manifolds of dimensions n>=5 proved useless in the case n=4. Although there was much progress in understanding the topology of manifolds of dimensions n\=4, remarkable progress in classification of 4-manifolds has been done only after 1980 starting with the work of Freedman. His resulting theory gave a complete classification of closed, simply connected 4-manifolds with respect to their "intersection forms", a unimodular integral bilinear form defined on the second homology of the manifold. In this talk, we will see how cup product in cohomology descends literally to intersection of embedded surfaces in 4-manifolds and defines a topological invariant of 4-manifolds, the intersection form. And I will try to present some of the results gotten in classifying 4-manifolds via their intersection forms. However, actual work on existence and classification of smooth structures on 4-manifolds is done in the last 20 years via Gauge Theory and Kirby Calculus which I will not have time to mention.
See http://www.mathunion.org/medals/Fields/1986/index.html#0x82496a1f_0x0005ea3c for an indication of how remarkable Freedman and Donaldson's contributions have been.

January 27th, Dave Constantine, Marked length spectrum for nonpositively curved surfaces
For any element of the fundamental group of a surface, find the shortest path on the surface representing it (or take an infimum if you need to) and associate its length to that element. The function that assigns these lengths is called the marked length spectrum. Often this doesn't tell us much (for example, if the surface is simply connected). But if the surface is closed and has nonpositive curvature the marked length spectrum determines it up to isometry. I think this is rather surprising, and therefore you are going to hear about it in the seminar. For the proof I'll introduce Morse correspondence of geodesics and geodesic currents.

Februrary 3rd, Jose Gomez-Guerra, Eilenberg-Maclane Spaces
An Eilenberg-Maclane space is a space for which all its homotopy groups but one vanish. These spaces are of great importance because they represent singular cohomology. This is is a very surprising fact because the Eilenberg-Maclane are defined entirely by homotopy groups, which on the gace of it have nothing to do with cohomology! and also because they give an "unexpected" relationship between homotopy and cohomology.
I'll existance and uniqueness of theses spaces and then we'll see how to get this connection.

Februrary 10th, Aaron Magid, Seifert Fibered Spaces
Much like the Pittsburgh Steelers going into the playoffs, Seifert fibered spaces are an underrated part of low-dimensional topology. These ubiquitous 3-manifolds play an important role in our understanding and classification of 3-manifolds. The evil step-mother to hyperbolic 3-manifolds, Seifert fibered spaces refuse to admit hyperbolic structures. However, through the JSJ decomposition, they play a crucial role in setting up Thurston's geometrization conjecture.

We will begin with some basic definitions, discuss the classification of these 3-manifolds, and prove a few of their interesting properties. Also, we will discuss examples including the Hopf fibration of the 3-sphere, other fibrations of the 3-sphere, the Poincare dodecahedral space, and lens spaces.

February 17th, Eric Zupunski, 3-Manifold Groups and Group Splittings
Any finitely presented group is the fundamental group of some 4-manifold. However, in 3 dimensions there are many more restrictions on what groups may appear as the fundamental group of a 3-manifold. There are also many connections between the topology and fundamental group of a 3-manifold. I will present a few results on splittings of 3-manifold groups that illustrate this connection.

March 10th, Diane Vavrichek, The Pure Mapping Class Group
The mapping class group of a compact orientable surface S is the group of orientation-preserving diffeomorphisms of S, modulo isotopy. This group plays an important role in the theory of surfaces, and of Teichmuller spaces. Dehn and Lickorish showed that these groups are finitely generated. We will present a simpler result: that the pure mapping class group is finitely generated, where the pure mapping class group of S is the group of orientation-preserving diffeomorphisms of S which fix the boundary of S, modulo isotopy. This result is also due to Dehn and Lickorish, with a simplified proof due to Birman.

March 17th, Marshall Williams, Topological Groups: An introduction
Topological groups, (groups endowed with a topology where multiplication and inversion are continuous), appear throughout mathematics, and are of interest to algebraists, geometers, and analysts. While there is a great deal of interest in the special case of Lie Groups, in this talk I will focus on the more general setting of locally compact, and especially compact, topological groups. For such groups, we can find an invariant measure (the Haar Measure) which will allow us to extend many results from finite groups to arbitrary topological groups - in particular, we will discuss the Peter-Weyl theorem, which will give us, as a corollary, a structure theorem for compact metrizable groups. I intend this talk to be fairly introductory in nature, though some familiarity with finite group representations will be helpful.

March 24th, Giancarlo Urzua, Compact Riemann surfaces with a Z/pZ action for some p > genus
I will be explaining a way of classifying these objects. This will use the analytic approach for Riemann surfaces (as in the book "Riemann surfaces" by H. Farkas and I. Kra) and some hyperbolic geomety along the way. I'll write down a few equations and also say something about the relation with moduli spaces of curves.

April 14th, Johanna Mangahas, Loop Theorem
One can say three-manifold topology begins with the Loop and Sphere Theorems, which give conditions for embedding disks and spheres into 3-manifolds. Our topic, the Loop Theorem, says that, given a map from the disk into a 3-manifold M such that the boundary of the disk maps to an essential curve on the boundary of M, one can find an embedding of the disk with the same property. A cleverly constructed "tower of covering spaces" appears in the proof, due originally to Papakyriakopoulos and modified later by others.

April 21st, Zhenyi Liu, Dehn Filling: An Introduction
Any closed, connected, orientable 3-manifold may be obtained from the 3-sphere by Dehn surgery on a link L in S^3, or equivalently, by a finite sequence of surgeries on knots in 3-manifolds. To study 3-manifolds, it is of interest to study the relationship between the global properties of a knot's exterior and the manifold obtained by surgery on the knot.

Basic concepts such as knot exteriors, Dehn fillings, slopes, etc. will be introduced. We'll also discuss examples of exceptional surgeries. As time permits, we'll talk about the Knot Complement Problem and Cabling Conjecture and some results on Dehn fillings on hyperbolic knots.
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