| Date | Speaker | Title
Abstract
|
| Oct. 6 | Jeffrey Case | An Introduction to the Atiyah-Singer Index Theorem
The Atiyah-Singer index theorem relates the analytic and topological index of an elliptic operator. I will give a few examples to illustrate why such a relation should hold, and, hopefully, state the theorem in an understandable way.
|
| Oct. 13 | Robert Ream | Hirzebruch-Riemann-Roch
An important case (proved beforehand, of course) of the Atiyah-Singer index theorem is the Hirzebruch-Riemann-Roch theorem. In fact Atiyah and Singer's first proof of the index theorem was a generalization of Hirzebruch's case. I will discuss the theorem, specifically how it becomes the usually Riemann-Roch theorem in dimension 2.
|
| Oct. 20 | Charles Martin | Periodic Solutions to the KdV
The KdV is a well-known PDE of nonlinear type and has many interesting properties. In this talk we consider periodic solutions, and in doing so introduce a number of interesting structures and methods, such as inverse scattering and Riemann surfaces.
|
| Oct. 27 | Kevin Brighton | Clifford Algebras and Spin Structures
I'll give a basic introduction to Clifford algebras including their
construction and basic properties, such as their Z_2 grading and their
relation to the exterior algebra. I'll then introduce the groups
Spin(n) and Pin(n). Then, time permitting, I'll introduce the Steifel-Whitney
classes and define a spin structure on a manifold.
|
| Nov. 3 | Jeffrey Case | Spin Structures II
Continuing on Kevin's talk, I will introduce Clifford bundles and spin structures on manifolds. I will discuss principle bundles, the Steifel-Whitney classes, connections on spin bundles, and the Dirac operator, time permitting.
|
| Nov. 10 | Kevin Brighton | The Bochner Technique
I will talk about the Bochner technqiue, and hopefully the Lichnerowicz theorem in spin geometry.
|
| Nov. 17 | Robert Ream | Characteristic Classes
The Index of many interesting elliptic operators can be
expressed in terms of certain invariants, notably the signature and the
\hat A-genus. These invariants arise from certain multiplicative
sequences in the characteristic classes of the manifold.
I will introduce characteristic classes via invariant
polynomials of Lie Groups. I will then define multiplicative
sequences and discuss important examples. This talk should be
accessible to anyone who knows the definition of a manifold.
|
| Nov. 24 | John Mangual | Amoebas
Amoebas were introduced in the 1994 in a book by Gelfand, Kapranov and
Zelevinsky in relation to zeros of polynomials in several variables.
However, recently they have found use in real algebraic geometry and in
string theory. In this talk we will show an amoeba-based proof of
$\sum_{k=1}^\infty 1/k^2 = \pi^2/6$ using some second-year calculus and
Euclidean geometry. Then, we will look at amoebas of Laurent polynomials
and their relation to tropical geometry.
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