Mathematics & Computer Science

Seminars and Colloquia

Wednesday, October 14, 2015

04:15 pm - 05:00 pm

Topology Seminar: Tue Ly (Brandeis): "Diophantine approximation on number fields, homogenous dynamics and Schmidt game"

Abstract: In 1960's, Wolfgang Schmidt used his (\alpha, \beta)-game to prove that the set of badly approximable numbers has countable intersection property. In this talk, I will discuss about extending Schmidt's result to the set BA_K of vectors badly approximable by elements of a fixed number field K using the connection to homogeneous dynamics and recent developments of Schmidt's game. This strengthens a recent result by Anish Ghosh, Beverly Lytle and Manfred Einsiedler concerning the intersection of BA_K with curves. Joint work with Dmitry Kleinbock.

Thursday, October 15, 2015

04:15 pm - 05:15 pm

Math CS Colloquium, Alex Eskin (University of Chicago): "The SL(2,R) action on Moduli space"

Abstract: There is a natural action of the group SL(2,R) of 2x2 matrices on the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models. I will survey some recent developments in the area, and give some applications to the study of billiards in polygons and other problems.

Exley 121

Wednesday, October 21, 2015

04:15 pm - 05:00 pm

Topology Seminar: John Schmitt (Middlebury): "Two tools from the polynomial method toolkit"

Abstract: The polynomial method is an umbrella term that describes an evolving set of algebraic statements used to solve problems in arithmetic combinatorics, combinatorial geometry, graph theory and elsewhere by associating a set of objects with the zero set of a polynomial whose degree is somehow constrained. Algebraic statements about the zero set translate into statements about the set of objects of interest. We will examine two tools from the polynomial method toolkit, each of which generalizes the following, well-known fact: a one-variable polynomial over a field can have at most as many zeros as its degree. The first generalization which we will discuss is Alons Non-vanishing Corollary, a statement for a multivariate polynomial introduced in the 1990s that follows from his celebrated Combinatorial Nullstellensatz. The second generalization is the Alon-Furedi Theorem, a statement which gives a lower bound on the number of non-zeros of a multivariate polynomial over a Cartesian product. We give an application for each of these tools. For the first we show how to apply it to a combinatorial problem of the polymath Martin Gardner known as the minimum no-three-in-a-line problem. For the second we show how it quickly proves a number-theoretic result from the 1930s due to Ewald Warning, a statement which gives a lower bound on the number of common zeros of a polynomial system over a finite field.

Exley Science Center (Tower)