Qualifying Exam

Speaker:  Evan Henning, University of Kentucky

Title:  Permutation Hopf algebras

Abstract:

Hopf algebras are a very natural algebraic structure in the study of combinatorics. This is due to the fact that many combinatorial objects admit canonical operations of combination and decomposition. 

One of the most foundational combinatorial objects of mathematics is that of permutations. There are many Hopf algebras which can be put on the linear span of permutations, the most famous of which is the Malvenuto and Reutenauer Hopf algebra of permutations. This is a shuffle-like Hopf algebra that is self-dual and surjects onto the Hopf algebra of quasisymmetric functions. In this talk, we focus on the work of Mingze Zhao and Huilan Li and consider two other Hopf algebras that can be put on permutations. 

These two new Hopf algebras give permutations a tensor and shuffle algebra structure. We will prove the duality between these Hopf algebras and will consider a generalization of these Hopf algebras, which can be extended to juggling.

Qualifying Exam

Speaker:  Evan Henning, University of Kentucky

Title:  Permutation Hopf algebras

Abstract:

Hopf algebras are a very natural algebraic structure in the study of combinatorics. This is due to the fact that many combinatorial objects admit canonical operations of combination and decomposition. 

One of the most foundational combinatorial objects of mathematics is that of permutations. There are many Hopf algebras which can be put on the linear span of permutations, the most famous of which is the Malvenuto and Reutenauer Hopf algebra of permutations. This is a shuffle-like Hopf algebra that is self-dual and surjects onto the Hopf algebra of quasisymmetric functions. In this talk, we focus on the work of Mingze Zhao and Huilan Li and consider two other Hopf algebras that can be put on permutations. 

These two new Hopf algebras give permutations a tensor and shuffle algebra structure. We will prove the duality between these Hopf algebras and will consider a generalization of these Hopf algebras, which can be extended to juggling.

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. 

A multitriangulation of order k, or a k-triangulation, of a convex n-gon is a maximal set of diagonals such that no k+1 of them mutually cross in the interior of the n-gon. First studied in the 1992 paper “A Turán-Type Theorem on the Chords of Convex Polygons” by Capoyleas and Pach, k-triangulations have recently been studied in the context of the multi-associahedron. In this talk, we will prove a result by Pilaud and Santos in the paper “Multi-triangulations as Complexes of Star Polygons”, namely, that k-triangulations are formed by a union of k-stars and “k-irrelevant” edges. Time permitting, we will also discuss our recent work concerning the realization of the multi-associahedron.

Injective modules play an important role in various algebraic questions.  We will introduce the notion of an injective module and show that any module can be embedded in an injective module in a minimal way.  This is a result originally given by B. Eckmann and A. Schopf.