July 31st - Chris
There's a basic link between (1) systems of linear ODEs, (2) objects called "local systems", and (3) representations of fundamental groups. I'll begin by explaining how the 2nd and 3rd arise when you start from the ODE viewpoint. Then I'll talk about a particular geometric situation which gives rise to all three; the geometry often allows one to infer facts about the three types of objects and, equally importantly, one can often learn something about the geometry by studying one or more of the three objects.
Sources:
- "Period Mappings and Period Domains" by Carlson, Muller-Stach, and Peters
- "Differential Galois Theory" by van der Put and Singer, available at http://www4.ncsu.edu/~singer/ms_papers.html
July 25th - Laura
I'll talk about counting integer points on some cubic curves. The ingredients: basic properties of elliptic curves, a little linear algebra, and the Diophantine Approximation Theorem(s). Some applications: the gold standard for measuring the interestingness of your taxicab, as well as more general facts about Diophantine cubic equations in two variables.
Source: J. Tate and J. Silverman, Rational Points on Elliptic Curves, ch. 5
July 17th - Alden
I'll describe an algorithm to find the convex hull of the integer points contained in a rational polyhedron (the problem illustrated on the cover of the first source below), and maybe some other things related to this. Check out this page for a program which implements the algorithm.
Sources:
- Theory of Linear and Integer Programming by Alexander Schrijver.
- The Math 583 at University of Washington website/course notes by Rekha Thomas
July 10 - Brian
I will discuss the one to one correspondence between sequences of numbers in the unit disk and nontrivial probability measures on the unit circle. This will lead to an overview of some recent theorems relating properties of measures to the properties of the corresponding sequence. A special emphasis will be put on Baxter's Theorem, Szego's Theorem, and the role played by orthogonal polynomials.
Recommended Sources:
- Orthogonal Polynomials on the Unit Circle Part I: Classical Theory by Barry Simon (2005).
- Orthogonal Polynomials on the Unit Circle Part II: Spectral Theory by Barry Simon (2005).
July 3 - Robin
Measurable Cardinals.
From the source: "Set Theory: The Third Millennium Edition, Revised and Expanded" by Thomas Jech.
June 36 - Paul
Lie groups and stuff.
June 19 - Alden
Riemannian manifolds, metrics, connections, the Riemannian connection, curvature tensors, etc.
Some good books are:
- Riemannian Manifolds: An Introduction to Curvature, by John M. Lee
- Riemannian Geometry, by Manfredo Perdigao do Carmo (Author), Francis Flaherty (Translator)
- Riemannian Geometry, by Peter Petersen