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The University of Michigan Student Geometry/Topology Seminar |
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The Student Geometry/Topology seminar will be held 3-4 pm on Friday afternoons,
in 3096 East Hall.
To volunteer to speak on any topic you wish, contact Jose or Aaron
| January 5th | (not meeting this week) | |
| January 12th | Paul Johnson | An introduction to orbifolds (abstract) |
| January 19th | Jose Manuel Gomez-Guerra | An introduction to operads (abstract) |
| January 26th | Aaron Magid | The Margulis Lemma and the Structure of a Hyperbolic n-Manifold (abstract) |
February 2nd |
Kyle Hofmann | The Lefschetz Hyperplane Theorem (abstract) |
| February 9th | Marshall Williams | Gromov Hausdorff Convergence: an Introduction(abstract) |
| February 16th | Daniel Kneezel | H-spaces and (Co)homology (abstract) |
| February 23rd | (not meeting this week due to Spring Break) | |
March 2nd |
(not meeting this week due to Spring Break) | |
| March 9th | Kyle Ormsby | Classifying Spaces and the Milnor Construction (abstract) |
| March 16th | Jose Manuel Gomez-Guerra | Introduction to Simplicial Spaces |
| March 23rd | Nina White | The Moment Map |
| March 30th | (not meeting) | not meeting this week due to the seminar at MSU on Saturday this weekend |
| March 31st | Michigan State Seminar | |
| April 6th | Johanna Mangahas | Teichmuller Geometry (abstract) |
| April 13th | (not meeting) | Michigan is hosting a workshop on "Aspects of Teichmueller Theory" April 12-14 that we encourage you to attend. Register here. |
| April 20-22 | (not meeting) | Graduate Student Topology Conference at the University of Chicago |
Click here
to see the student seminar webpage from Fall 2006.
Click here
to see the student seminar webpage from Winter 2006.
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.
An introduction to orbifolds
An introduction to Operads
The idea of this talk is to give an intoduction of Operads. Basically I
will talk about Stasheff's work on A_{\infty} spaces. From that, one can see
the motivation behind operads. If time allows, I will explain how these
are used in modern algebraic topology.
The Margulis Lemma and the Structure of a Hyperbolic n-Manifold
The Margulis lemma gives sufficient conditions for when a discrete subgroup of a Lie group is nilpotent. No knowledge of nilpotency is required as nilpotent subgroups of isometries of hyperbolic n-space are virtually abelian! While the consequences of the Margulis lemma apply to hyperbolic n-manifolds, we will concentrate on dimensions 2 and 3 during this talk. I will begin with an introduction to hyperbolic space, isometries of hyperbolic space, and hyperbolic manifolds. Then we will outline how to prove the Margulis lemma. Finally, we explain the importance to hyperbolic geometry by discussing the thick-thin decomposition. A nice application of this is that for any dimension n, there exists a lower bound for the volumes of all hyperbolic n-manifolds.
The Lefschetz Hyperplane Theorem
Let X be a smooth compact complex manifold of complex dimension d. Invariants such as \pi_1 and H_1 describe the low-dimensional topology of X. If d is large, it is possible to cut away some of X while still retaining all the information about the homotopy and homology groups. The Lefschetz Hyperplane Theorem is one way to make this precise: Assume that X can be embedded inside some projective space CP^N and that H is a hyperplane inside CP^N. The inclusion of X \cap H into X induces restriction maps in homology, cohomology, and homotopy, and the theorem says that these maps are isomorphisms except in the top two degrees.
We will explain a proof of the Lefschetz Hyperplane Theorem using Morse theory.
Along the way, we will show the surprising result that if X is a smooth complex
manifold in C^N of complex dimension d, then X has the homotopy type of a
CW-complex of real dimension d. As time permits, we will discuss generalizations
and refinements of the Lefschetz Hyperplane Theorem.
Gromov Hausdorff Convergence: an Introduction
Oftentimes, in geomerty and analysis, its useful to be able to take limits of sequences of metric spaces. I'll introduce the notion of Gromov-Hausdorff Convergence, and discuss some basic properties that make it fun to work with, including a nice compactness result. Then, I'll survey some applications of this compactness to geometry, geometric group theory, and analysis of general metric spaces. One particularly interesting application to group theory, due to Gromov, is the fact that groups have polynomial growth if and only if they are virtually nilpotent. I'll discuss this and other results in the second half of the talk.
H-spaces and (Co)homology
The notion of an H-space, short for Hopf space, broadly generalizes that of topological group. In this talk I would like to give a definition of H-space, present several examples and non-examples (some familiar spaces, some less familiar), and briefly survey how the H-space structure of a space X manifests itself in homological computations.
Classifying Spaces and the Milnor Construction
After developing the theory of fiber bundles and defining the classifying space BG of a topological group G, I will present the Milnor construction of BG as an infinite join of G. Along the way, I will present several examples to motivate and solidify the concept of a classifying space, and, if time permits, I will comment on the connection between the Milnor and bar constructions of BG. Classifying spaces are widely applied in mathematics ( e.g., in Lie groups, Chern-Weil theory, cobordism, group cohomology, and intuitionistic logic), and while the speaker makes no claim to understand any of these applications, the talk should be accessible to all those who might be interested.
Teichmuller Geometry
I hope to give a helpful introduction for graduate students interested in the Teichmuller workshop next week. We'll start from scratch but go beyond what Aaron and I presented in seminars last semester. We'll see why the hyperbolic and conformal viewpoints are interchangeable, what a geodesic in the Teichmuller metric means as a family of "almost Euclidean" surfaces, and maybe even get to a nice example.