Title: Constructing the p-adic Integers
Abstract: For a prime p, the p-adic integers Z_p is the valuation ring of the field of p-adic numbers Q_p. In this talk, we will explicitly construct Zp as a ring of coherent sequences and explore its algebraic and topological properties. We will then explore the multiplicative structure of Q_p using Hensel's Lemma.
Title: Reflexive Polytopes and the Reflexive Dimension
Abstract: Reflexive polytopes were first introduced in the context of theoretical physics and have since played a role in Mirror Symmetry, construction of Calabi-Yau varieties and Gorenstein polytopes. Applications aside, reflexive polytopes are interesting combinatorial objects. In this talk we define what it means for a polytope P to be reflexive and characterize P according to its Ehrhart series. Then we look at the work done by Haase and Melnikov in defining the reflexive dimension of a polytope and producing lower and upper bounds.
Title: Zonotopes and Zonal Diagrams
Abstract: A zonotope is the Minkowski sum of a finite number of closed line segments in some Euclidean space. Peter McMullen's 1971 paper "On Zonotopes" describes the technique of zonal diagrams as a tool for studying combinatorial properties of zonotopes. Zonal diagrams, which are analogous to Gale diagrams for general polytopes and to central diagrams for centrally symmetric polytopes, have proved more effective than these techniques for the particular study of zonotopes.
In particular, zonal diagrams yield relationships between zonotopes with n zones of dimensions d and n-d, and lead to the enumeration of all combinatorial types of d-zonotopes with at most d+2 zones. In this master's talk, we will discuss zonotopes and zonal diagrams, describe the combinatorial results mentioned above, and briefly consider the relationship between zonotopes and hyperplane arrangements.
December 4, Thursday
Title: Graph Products, Fourier Analysis, and Spectral Techniques
Abstract: Selected results will be discussed from the paper "Graph Products, Fourier Analysis, and Spectral Techniques" by Noga Alon, Irit Dinur, Ehud Friedgut, and Benny Sudakov.
Title: An approach to enumerating lecture hall partitions.
Abstract: We will introduce and define lecture hall partitions. We will then investigate two different proofs of the Lecture Hall Theorem.
Title: Poset Fiber Theorems
Abstract: Suppose that f: P --> Q is a poset map whose fibers are sufficiently well connected. We present a result of Bjorner, Wachs, and Welker which gives the homotopy type of P in terms of Q and the fibers, generalizing Quillen's famous fiber lemma. Applications to homology and subspace arrangements will be covered. Necessary topological definitions and constructs will be presented.
Title: Homogenization of Parabolic Operators
Abstract: Operators that oscillate rapidly on a domain present challenges to estimating a solution. In homogenization, we hope to replace such an operator with one that is uniform in the domain. In this talk, we will explore the problem of homogenizing a periodic parabolic operator with Dirichlet boundary conditions. We find a form for the operator using a formal expansion and then use Tartar's method of oscillating test functions to prove the result.
Title: Bounds on the Entries of Matrix Functions
Abstract: Consider a banded matrix A and a smooth function f defined on the spectrum of A. We use Bernstein's Theorem to show that the entries of f(A) are bounded in an exponentially decaying manner away from the main diagonal. Then by representing the entries of f(A) in terms of Riemann-Stieltjes integrals and using Gaussian quadrature rules, we can attain approximations for the entries of f(A). We will also look at some possible applications to preconditioning.
Title: On Certain Convex Sets of Measures and on Phases of Reacting Mixtures.
Abstract: In the 1970's, Walter Noll developed a modern version of the elements of Gibbsian thermostatics for describing the equilibrium of mixtures. In this talk, we will see how we can use mathematical tools to explain why Gibbs said "it does not seem probable that r can ever exceed n+2" where n stands for number of components in a mixture and r stands for the number of coexistent phases. We will also exploit some other properties of a mixture when a it is in equilibrium.
