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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.
For information, or to volunteer to speak on any topic you wish, contact Felipe (felipear AT umich DOT edu) or Dave (constand AT umich DOT edu)
Click here
to see the student seminar webpage for Winter 2008.
Click here
to see the student seminar webpage from Winter 2007.
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.
Introduction to geometric group theory
Stallings' Theorem
Stallings' Theorem is a foundational result in geometric group theory. It relates the geometry of the Cayley graph of a group to the algebraic structure of the group.
The fundamental group of a compact semisimple Lie group is finite
The fundamental group of a compact semisimple Lie group is finite.
An introduction to locally symmetric spaces
I'll prove the equivalence of two notions of locally symmetric space.
Quasi-isometries of symmetric spaces
In this talk I'll survey a result by Kleiner and Leeb stating that a quasi-isometry of a higher rank symmetric space is a finite distance from an isometry. The talk will be a very loose sketch of the proof, with a focus on the many diverse tools from analysis, geometry, topology and even logic that go into proving it. Besides quasi-isometries, we'll see bi-Lipschitz maps, asymptotic cones, ultrafilters, buildings, local homology groups etc.
Groupoids: Abstract nonsense, or fundamental?
In a chapter of "Towards a Philosophy of Real Mathematics", David Corfield examines the mathematical debate over groupoids. Though they have many uses throughout mathematics, they are often rather abstract (stacks, noncommutative geometry), and so groupoids are often waved away as being unnecessary - aren't groups good enough? We will spend some time examining the debate, and examine how it plays out in a concrete example: the fundamental groupoid.
Cocycles and cohomology in dynamics
Many problems in dynamics boil down to solving a cohomological equation. So many, in fact, that cocycles have become a subject of study in their own right. I will define cocycles and cohomology from the point of view of dynamics, and discuss some problems where these notions arise. I will also outline some well-known and basic results concerning cocycles.
Isoperimetric functions and the Dehn function
Among planar curves of fixed length, it is known that the circle encloses the maximum area (this is the isoperimetric problem). There is an aspect of the word problem for finitely-presented groups which turns out to be equivalent to the isoperimetric problem on the Cayley complex of the group. I will define the Dehn function, which measures maximal enclosed area on a Cayley complex, and discuss known results about its rate of growth.
Introduction to Currents
In this talk I will introduce k-dimensional currents, which generalize the notion of a k-dimensional submanifold in Euclidean space. One of the main motivations for this generalization is the need, particularly in variational problems, for completeness and compactness properties not satisfied by spaces of submanifolds. I will introduce two versions of this theory. In the first part of the talk, I will give a brief overview of the theory of currents in Euclidean spaces described by Federer and Fleming in 1960. In the second half, I will describe a recent generalization of the theory to arbitrary metric spaces, introduced by Ambrosio and Kirchheim in 2000.
Introduction to Stable Homotopy of Spheres I
This is the first talk in a two-part series. Stable homotopy -- the "nice" part of the homotopy groups of spheres -- has inspired much of the work in algebraic topology in the past six decades. While its definition is simple, it follows few simple patterns and is one of the most mysterious objects in modern mathematics. In this talk, we cover the definition of higher homotopy groups, the Freudenthal suspension theorem, the definition of the stable homotopy groups, and Serre's finiteness result.
Introduction to Stable Homotopy of Spheres II
This is the second talk in a two-part series. Building off of the definitions and results of last week's talk, we construct some of the well-known structure of the stable homotopy groups of spheres. While full descriptions of these groups are well beyond the state-of-the-art in algebraic topology, J.F. Adams specified some of their nicer properties in 1965. We construct the real J-homomorphism from the higher homotopy groups of the infinite orthogonal group to the higher homotopy groups of spheres. Using basic KO-theory (which we will develop in this seminar), we show that the image of J constitutes a direct summand of the n-th stable homotopy group of spheres in a manner that depends on the congruence class of n mod 8.