Qualifying Exam

Speaker:  Williem Rizer, University of Kentucky

Title:  Combinatorics of the Positroidal Stratification of the Totally Nonnegative Grassmannian

Abstract:

The Grassmannian has been an object of much interest in algebra, geometry, and combinatorics. We can decompose the Grassmannian into matroid strata, in which each element of the stratum has the same set of nonzero Plücker coordinates corresponding to the bases of a matroid. If we restrict to the elements of the Grassmannian with nonngeative coordinates, the corresponding matroids are called positroids. Postnikov revealed a family of combinatorial objects that can be used to parameterize positroidal cells. Recently, various authors have studied objects (X-diagrams and LACD colored permutations) that are in correspondence with this family, though the bijections exhibited do not commute with one another. In this talk we discuss all of these objects, the bijections between them, the information they reveal about positroids, and the possible connections and generalizations we can make to fold these newer objects neatly into the family.

Title: Ferrers Diagram Rank-Metric Codes

Abstract: Our codes of interest are subspaces of $F_q^{m\times n}$ in which every nonzero matrix has rank at least $\delta$, and conforms to the shape of a given Ferrers diagram. In 2009, Etzion and Silberstein proved an upper bound for the dimension of such codes, and conjectured that it was achievable for any given parameters. In particular, the case for unrestricted matrices was solved in 1985 by Gabidulin, predating the complications brought on by nontrivial Ferrers diagram shapes. In this talk, we will prove the bound and discuss several known cases of the conjecture, including two new cases.

Title: Some Algebraic Properties of Hierarchical Models

Abstract: We are going to introduce  certain toric ideals that correspond to structures of Hierarchical Models in Algebraic Statistics and explore the properties of these ideals.  One  challenge will be to describe ideals that have infinitely many generators in a finite way, which we do using the symmetry group action on the set of indices. We will also show a formula for calculating the dimension of the variety defined by these ideals, and make progress in calculating the corresponding Hilbert function.

Title: Matrix Tree Theorem

Abstract: The Laplacian matrix of a graph is a well studied object in mathematics. One of its more prominent roles comes in the famous Matrix Tree Theorem. In this talk, we first look at a classical proof of the Matrix Tree Theorem. Then we carefully explore a more recent proof through the work of Dall and Pfeifle using polytopes. Finally I will outline my research plan and present some preliminary results. This talk serves as my Qualifying Exam.

Title:  Stability of Ground States for cubic NLS in 1-D

Abstract:  In this talk, we discuss orbital stability of ground state (equilibrium) solutions for a 1D cubic nonlinear focussing Schrödinger equation under perturbations of H1(R) initial data for cNLS. Stability of equilibrium solutions is an important condition for being able to use theoretical models in physical applications. The method of proof is a generalization of Lyapunov stability theory for nite dimensional systems. We rely on global time existence of a unique H1(R) solution to the cNLS (Ginibre-Velo, 1977).

Title:  Homogenization of Elliptic Operators with Random Coefficients

Abstract:  In this talk, we mainly consider the homogenization problems of elliptic equations with rapidly-oscillating random coefficients. The homogenization theorem is proved for this case. We  also prove the general theorem of individual homogenization when the bounded coefficients are ergodic.

Title:  On Bounded Point Derivations and Analytic Capacity

Abstract:  Let X be a compact subset of the complex plane and let R(X) denote the uniform closure of the space of rational functions whose poles lie off X. We say that there is a bounded point derivation on R(X) at x if and only if there exists a constant k such that |f t(x)| ≤ k||f ||X for all f ∈ H(X), where H(X) is the space of all functions that are holomorphic in some neighborhood of X. In this talk we will give necessary and sufficient conditions for the existence of a bounded point derivation on R(X) at x.
 

Title:  Non-Vanishing Homology of the Matching Complex

Abstract: A matching on a graph G is any subgraph where the maximum vertex degree is 1.  Since edge-deletion preserves the property of being a matching, the set of all matchings on G forms a simplicial complex M(G).  We will survey results on the lowest non-vanishing homology group for M(K_n) and discuss the extension of these results to more general graphs, specifically the r-stable ones.  Prior familiarity with simplicial complexes and homology is assumed.