The Student Geometry/Topology seminar will be held 3-4 pm on Friday afternoons, in 3096 East Hall.

For information, or to volunteer to speak on any topic you wish, contact Nina (whitenj AT umich DOT edu) or Michelle (mishlie AT umich DOT edu)

Schedule of Talks

September 5th	organizational meeting
September 12th	Mark Krawitz	A Gentle Introduction To Moduli Spaces (abstract)
September 19th	Aaron Magid	The Geometry and Topology of Surfaces (abstract)
September 26th	Jeff Meyer	note! changed time: 4-5pm. An Introduction to Cohomology Operations (abstract)
October 3rd	Kyle Ormsby	The Steenrod Algebra: Squares in the Homotopy of Spheres (abstract)
October 10th	Daniel Kneezel	What Would You Do With A Steenrod Square? (abstract)
October 17th	not meeting
October 24th	Julian Rosen	"Exotic" Manifolds (abstract)
October 31st	Great Lakes Geometry Conference	Seminar not meeting.
November 7th	Michelle Lee & Nina White, 2 hour special, 3-5pm	Three Manifolds: Foliations and Topology (abstract)
November 14th	Paul Johnson	The Holonomy Groupoid (abstract)
November 21st	Ben Fehrman	Morse Theory (abstract)
November 28th		Thanksgiving No Meeting
December 5th	Various Speakers, Five 10-minutes talks	What kind of Geometry/Topology does (insert name here) do?
January 16th	Nina White	What is a Gromov Hyperbolic Space? (abstract)
January 23rd	David Constantine	Symmetric Spaces (abstract)
January 30th	Michelle Lee	Algebraic and Geometric Convergence (abstract)
February 6th	Johanna Mangahas	Growth of groups: a short survey with examples
February 13th	canceled
February 20th		Not meeting
February 27th	``Spring Break''	Not meeting
March 6th	Marshall Williams	TBA
March 13th	Cagatay Kutluhan	Another way of studying 3-manifolds (abstract)
March 20th	Becky Haoi	Geometrization Conjecture (abstract)

Abstracts

Becky Hoai, March 20

Geometrization Conjecture

The purpose of this talk is to provide a brief survey of Thurston's geometrization conjecture, which establishes a complete characterization of geometric structures on 3-manifolds. We will begin by discussing its 2-dimensional analogue, the uniformization theorem for surfaces. A vague notion of what curvature is and some knowledge of covering space theory will be assumed.

Cagatay Kutluhan, March 13

Another way of studying 3-manifolds

I will talk about some applications of Floer homology to the topology of 3-manifolds. I will start with the notion of Morse homology, which perhaps the audience is more familiar with, and then describe the common framework of Floer homologies by establishing analogy with Morse homology. In particular, I will focus on one type of Floer homology, so-called monopole Floer homology.

Michelle Lee, January 30

Algebraic and Geometric Convergence

We will discuss algebraic and geometric convergence of Kleinian groups.

David Constantine, January 23

Symmetric Spaces

In this talk I'll discuss what a symmetric space is, coming from three viewpoints - a basic metric geometry-flavored approach, a Riemannian geometry approach (curvature, parallel translation etc. show up), and a Lie groups approach. We'll keep things basic and focus on looking at some key properties of this important class of spaces via examples we all know and love. Then we'll use our new found wisdom to take a quick peek at one example we don't already know and see what we can say.

Nina White, January 16

What is a Gromov hyperbolic space?

Anyone can attend this, but it will be aimed at providing background needed for the next lecture in Juan Souto's class. I'll give a definition of Gromov hyperbolic space, and a few of its stunning properties. One example of a Gromov hyperbolic space is hyperbolic space itself, and many interesting and fundamental properties of hyperbolic space depend only on its courser, Gromov hyperbolic structure. This will also allow us to define a hyperbolic group.

Various Speakers, December 5

What kind of Geometry/Topology does (insert name here) do?

Several older graduate students will give short informal talks about their research and their advisors' research. First and second year students thinking about studying topology and/or geometry are especially encouraged to attend.

Ben Fehrman, November 21

Morse Theory

The tools of Morse theory provide an extraordinary insight into the global topology of smooth manifolds. After developing the theory, we will observe almost as afterthoughts that every smooth manifold has the homotopy type of a CW-complex. And, that the summed indices of a vector field with isolated zeros is necessarily the manifold's Euler-characteristic. This talk will emphasize the foundations of finite-dimensional Morse theory. That is, given a manifold and a generic smooth function we will determine the manifold's topology through a simple analysis of the function's critical points. The development is concrete. To emphasize this we will show that a manifold admitting a generic smooth function with exactly two critical points is necessarily homeomorphic to the sphere. Time-permitting, applications of these ideas to certain infinite-dimensional spaces or to the development of homology theories will be mentioned if only briefly.

Paul Johnson, November 14

The Holonomy Groupoid

We pick up the foliation theme of Nina and Michelle from another direction. Given a foliation, an obvious thing to consider is the space of leaves. We might have hoped for this space to have some nice structure, but in fact it can be horribly behaved: the space of a leaves of an irrational foliation on the torus is an uncountable set with the concrete topology. The holonomy groupoid is a way to study the space of leaves that keeps more structure. Most of the work will go into explaining and motivating the definition, but with luck we will explain why the Reeb foliation cannot be real analytic, and how this story leads to noncommutative geometry.

Michelle Lee and Nina White, November 7

Three Manifolds: Foliations and Topology

Julian Rosen, October 24

"Exotic" Manifolds

Topological manifolds are interesting, but it is often useful to have a differentiable structure. Pick your favorite topological manifold (mine is the projective plane). One can ask, "What kind of differentiable structure can I put on this manifold?" In the case of the projective plane, there is exactly one possible differentiable structure (up to diffeomorphism). For topological manifolds of higher dimension, however, the answer is not so simple. There are topological manifolds that do not admit ANY differentiable structure, as well as manifolds that admit inifinitely-many pairwise non-diffeomorphic differentiable structures. There is also an interesting class of manifolds admitting finitely-many different differentiable structures. The so-called exotic spheres are one example, and this will be the focus of this talk.

Daniel Kneezel, October 10

What Would You Do With A Steenrod Square?

In this third and final installment of the series on cohomology operations I will present some standard applications of Steenrod squares. We will first use the squares to give a (partial) solution to the "Hopf invariant one" problem, which will show that every division algebra over the reals must have real dimension a power of 2. The real numbers, complex numbers, quaternions, and octonions are familiar examples of real division algebras of dimensions 1, 2, 4, and 8, respectively. Time permitting, we will also consider the question of how many linearly independent tangent vector fields there can be on a sphere of given dimension by examining the action of the squares on the cohomology ring of an appropriate Stiefel manifold.

Kyle Ormsby, October 3

The Steenrod Algebra: Squares in the Homotopy of Spheres

The Steenrod algebra A* encodes the algebraic structure of the Steenrod squares, the stable mod 2 cohomology operations. It plays a special role in homotopy theory because the mod 2 cohomology ring of any topological space is a highly structured algebra over A*. I will investigate the construction and formal properties of the Steenrod squares and specify the algebraic structure of A*. If time permits, I will indicate how the homological algebra of A* approximates the stable homotopy of spheres. To mitigate the decidedly algebraic nature of this talk, I will draw several pictures.

Jeff Meyer, September 26

An Introduction to Cohomology Operations

Cohomology operations, as described by Mosher and Tangora, are a "technique for supplementing and enriching the algebraic structure of the cohomology ring." This talk will be the first of three on cohomology operations and their applications to homotopy theory. We will begin by introducing the notions of a cohomology ring and cohomology operations. Due to the representability of cohomology, these operations are fundamentally related to Eilenberg-MacLane spaces. We will conclude by exploring the correspondence between cohomology operations of a certain type and the cohomology of Eilenberg-MacLane spaces.

Aaron Magid, September 19

The Geometry and Topology of Surfaces

We will investigate the moduli space of a surface (a.k.a. curves if your reasoning is complex) through its universal cover, Teichmuller space. This talk will be introductory, beginning with several equivalent ways to define Teichmuller space. The mapping class group (the group of homotopy classes of homeomorphisms of a surface) acts on Teichmuller space, giving the moduli space as the quotient. We will discuss (but not necessarily prove) some facts about the topology and geometry of Teichmuller space, and how one can use the mapping class group action to understand surface homeomorphisms.

Marc Krawitz, September 12

A Gentle Introduction To Moduli Spaces

I will describe the Moduli Space of Curves and its Universal family in analogy with more familiar objects like Grassmannians (e.g. Projective Spaces) and their tautological bundles. The required background will be kept to a minimum.