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Algebra and Combinatorics 2017 Fall

Fall 2017
Time & Location: All talks are on Thursdays in Norman Mayer 200-A at 12:30 PM unless otherwise noted.
Organizer: Tai Ha

September 21
Determinants in Wonderland
Tewodros Amdeberhan - Tulane University
Determinants are found everywhere in mathematics and other scientific endeavors. Their particular role in Combinatorics does not need any cynical introduction or special advertisement. In this talk, we will illustrate certain techniques which proved to be useful in the evaluation of several class of determinantal evaluations. We conclude this seminar with open problem(s). The content of our discussion is accessible to anyone with "an intellectual appetite".

September 28
Involution Schubert Polynomials and Some Ordinary Schubert Polynomial Identities
Michael Joyce - Tulane University
Abstract: Ordinary Schubert polynomials are algebraic manifestations of a certain orbit structure on the variety of complete flags. By considering two other orbit structures, we obtain involution and fpf-involution Schubert polynomials, respectively. We will discuss some of their properties and give an application for an identity involving ordinary Schubert polynomials.

October 5
Locally compact p-groups and their challenges
Karl Hofmann - TU Darmstadt and Tulane University

October 19
Lech's inequality and its improvements
Dr. Ilya Smirnov - University of Michigan
In 1960 Lech found a simple inequality that relates the colength and the multiplicity of a primary ideal in a local ring. Unfortunately, Lech's proof also shows that his inequality is almost never sharp. After explaining the necessary background, I will present a stronger form of Lech's inequality and an even stronger conjecture that will make the inequality sharp.

October 26
Beyond Perfect Graphs: Hypercycles and Perfect Hypergraphs
Jonathan O'Rourke - Tulane University
In attempting to extend the notion of perfect graphs to the class of hypergraphs, my research partner and I studied a class of hypergraphs which bear some resemblance to cyclic graphs.  We studied the associated primes of the cover ideals associated to this class of hypergraphs,
as well as their index of stability. This study resulted in an easy-to-describe class of hypergraphs which answer a question of Francisco, Van Tuyl, and H\`a regarding the relationship between the index of stability and the chromatic number of a family of hypergraphs, and in fact proving a stronger result.  I will explain the preliminaries necessary to understand the problem and some of the techniques used to solve it.

November 2
The Genesis of Involution

Ozlem Ugurlu - Tulane University
Let G be a complex semisimple algebraic group and B be a Borel subgroup of G.  In many situations, it is necessary to study the Borel orbits in G=G, where is an involutory automorphism. This is equivalent to analyze K = G orbits in the agvariety G=B. In fact, their geometry is of importance in the study of Harish-Chandra modules. The focus of the talk will be enumeration problem of Borel orbits in the polarizations (SL(n;C); S(GL(p;C) GL(q;C))). Its combinatorial relation to the lattice paths will be analyzed. In particular, it will be shown that the generating function for the dimensions of Borel orbits is expressible as a sum over lattice paths (in a p + 1 by q + 1 grid) moving by horizontal, vertical and diagonal steps weighted by an appropriate statistic.

November 9
Depth and Stanley Depth of monomial ideals
Prof. Yan Gur - Institution

November 16
Symmetric functions in superspace
Miles Jones - University of California San Diego
The theory of symmetric functions is a well-studied field that has many applications in mathematics and beyond.  Recently, mathematicians found a promising extension of this field in the setting of superspace whose definition was inspired by a physical phenomenon involving bosons and fermions and how they interact. It seems as though this generalization may lead to better understanding of classical symmetric function theory.  In this talk, I will introduce basic topics of symmetric function theory and their analogues in superspace. I will share some results and projects that I am working on now.

December 7
Cayley Grassmannian
Ustun Yildirim - Michigan State University
A 3-fold cross product operation exists only in dimensions 4 and 8. A 4-dimensional subspace of an 8 dimensional vector space is called a Cayley plane if it is closed under the 3-fold cross product operation. Cayley grassmannian is the space of all Cayley planes. It is naturally a homogeneous space with an algebraic torus action, and it resides in Gr(4,8). Over complex numbers, this space is not compact. In this talk, after I talk about the necessary background, I will explain some of the results I obtained on the minimal compactification of Cayley grassmannian.