Qualifying Exam

Speaker:  Maxwell Hosler, University of Kentucky

Title:  TBA

Abstract:

Qualifying Exam

Speaker:  Maxwell Hosler, University of Kentucky

Title:  TBA

Abstract:

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

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.

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.