Research Seminars: Algebra and Combinatorics

Spring 2021

Time & Location: All talks are on Wednesday in Zoom Meeting at 3:00 PM unless otherwise noted.
Organizer: Mahir Can

January 27

Title: Stable Harbourne-Huneke containment and Chudnovsky's Conjecture

Sankhaneel Bisui | Tulane University

Abstract:

February 3

Title: The Least Generating Degree of Symbolic Powers and Ideal Containment Problem

Thai Nguyen | Tulane University

Abstract: What is the smallest degree of a homogeneous polynomial that vanishes to order m on a given finite set of points, or more generally on some algebraic variety in projective space? A classical result of Zariski and Nagata tells us the set of such polynomials is the m-th symbolic power of the defining ideal I of the variety. To bound the generating degree of the symbolic powers of I, we can study containment between symbolic powers and ordinary powers of I. Conversely, knowing bounds for generating degree can help us study containment. My talk will be an introduction to this subject. I will also present some results from our joint work with Sankhaneel Bisui, Eloísa Grifo and Tài Huy Hà.

February 17

Title: A Survey of Classical Representation Theory

Mike Joyce | Tulane University

Abstract: Representation theory is a vast field which has applications in many other areas of mathematics, including algebra and combinatorics. This talk will review some of the classical theory of representations of groups and Lie algebras, with an emphasis thats lead to more modern aspects of representation theory, which will be addressed in a second talk.

Time: 3:30 - 4:30

February 24

Title: Invariants and properties of symbolic powers of edge and cover ideals

Joseph Skelton | Tulane University

Abstract: In this talk I will address several questions about symbolic powers of edge and cover ideals. The containment between ordinary and symbolic powers of edge ideals has been an active area of research for decades. As a result the resurgence number and Waldschmidt constant are of particular interest. The regularity of symbolic powers of edge ideals has been motivated by a conjecture of N.C. Minh which states that $\reg I(G)^{(s)} = \reg I(G)^s$ for any $s\in \NN$.

For cover ideals we are motivated by the results of Villlarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Necessary and sufficient conditions on $S\subset V(G)$ cover ideal of the graph whiskered at $S$, $J(G\cup W(S))$ is Cohen-Macaulay. While symbolic powers of the cover ideal do not necessarily have linear resolution I will show necessary conditions on $S$ such that symbolic powers of $J(G\cup W(S))$ have componentwise linearity.

Time:  3:00 - 4:00

March 3

Title: A Survey of Hopf Algebras and Quantum Groups

Mike Joyce | Tulane University

Abstract: We will introduce Hopf algebras and quantum groups through some of their key properties and some of the simplest examples. We will discuss R-matrices and the Yang-Baxter equation and then survey some of their manifestations in other areas of mathematics.

Time: 3:30 - 4:30

March 10

Title: TBA

Christopher Manon | University of Kentucky

Abstract: TBA

March 17

Title: TBA

Kiumars Kaveh | University of Pittsburgh

Abstract: TBA

March 31

Title: TBA

Robert Walker | University of Wisconsin-Madison

Abstract: TBA