Title:  Galois theory over the p-adic numbers

Abstract:  The inverse Galois problem asks whether every finite group appears as the Galois group of some finite Galois extension of the rational numbers. One can ask the same question for other fields. In this talk, we will discuss Galois theory over the p-adic numbers. We will see that ramification groups provide a useful tool for analyzing the structure of these Galois groups. Using ramification groups, we will find limitations on the finite groups which can occur as Galois groups over the p-adic numbers.  A review of the necessary facts about the p-adic numbers will be included. This talk is meant to be accessible to anyone with a basic understanding of Galois theory over the rationals.

Title:  Fixed Points in Partially Ordered Sets

Abstract:  In this talk we will look at what conditions we need to impose on posets to ensure automorphisms of the poset have fixed points. We will prove, and use, a discrete version of the famous Hopf-Lefshetz fixed point theorem from topology. Examples will be emphasized.

Title:  Applying Bernstein's Theorem to Matrices

Abstract:  Bernstein's Theorem is a theorem that bounds the best approximation error of a smooth function in terms of an ellipse where the function is analytic in its interior. In this talk, we will review this theorem and apply it to symmetric banded matrices to show that the entries of the resulting matrix (after a smooth function is applied) are bounded in an exponentially decaying manner away from the main diagonal.

Title:  Strings of composite numbers

Abstract:  Given d and e in {1,...,9} we consider the set of positive integers k that have the following property: k with d appended on the right n times is composite for all positive integers n. We also consider the set of positive integers k that have the following property: k with d and e appended alternately on the left and right (i.e. k,kd,ekd,ekdd...) is always composite regardless of the number of appended digits. We will discuss methods for finding elements in these sets and also consider finding their minimum elements.

Title:  Another Friendly Talk About Cobordism

Abstract:  In this talk I will present some basic ideas about cobordism. In particular, we will discuss manifolds, an equivalence relation, and some rings.

Haven't had Topology I yet, you say? This talk is still for you. You'll be fine.

Title:    An Introduction to the Bieberbach Conjecture

Abstract:  Please see 7^th floor bulletin board for abstract

Title:    An Introduction to the Bieberbach Conjecture

Abstract:  Please see 7^th floor bulletin board for abstract

Title:  Constructing Simplicial Spheres Using Algebra

Abstract:  Simplicial complexes are a central subject of study in combinatorics. They are also of interest geometrically since that abstract simplicial complex can be realized in real space as a geometric object. One interesting question is when is a simplicial complex homeomorphic to a sphere. We call these simplicial spheres. Using the connection of the Stanley-Reisner ring we can study simplicial complexes by looking at a corresponding ring. In particular the Gorenstein property occurs in these rings exactly when they are simplicial spheres. Hence we can construct simplicial spheres by constructing certain Gorenstein rings. This talk is meant to show the usefulness of the Stanley-Reisner ring connection between combinatorics and algebra and should be accessible to a general audience.

(Pizza at 4:00, talk at 4:15)

Title:  Presenting Sperner's Lemma

Abstract:  In this talk I will prove Sperner's Lemma on simplices. This is a well-known result that highlights a connection between the areas of combinatorics and analysis (passing through topology). The talk is meant to be accessible to all graduate students in mathematics regardless of their chosen field.

Title:  Polar Self-Dual Polytopes and Mahler Volume

Abstract:  The dual of a polytope is a well known topic in polyhedral geometry. There are several notions of what it means for a polytope to be self-dual. Perhaps the most common idea of self-dual would be that a polytope is combinatorially equivalent to its dual (face poset isomorphism). In Stephen's recent work it has been helpful to consider the strictest form of self-duality called polar self-dual, where we require that an embedded polytope be exactly equal to its dual as a subset of real space. We will consider some of the results and conjectures relating to polar self-dual polytopes.

Another famous problem from the theory of convex duality is finding the extremizers of the Mahler volume. This is defined as the product of the area of the convex figure and the area of its dual. Using Steiner symmetrization, we will show the maximizer of this volume is the circle.