Federico Galetto — April 18

Equivariant resolutions of some squarefree monomial ideals

Federico GalettoCleveland State University

Free resolutions are algebraic constructs that encode interesting invariants, called Betti numbers, of rings, ideals, and modules. Under mild assumptions, a group action on the object being resolved lifts to its resolutions, which enables the use of representation theory to describe resolutions. I will illustrate these concepts using the ideal generated by all squarefree monomials of a given degree as an example, and show how representation theory leads to a nice combinatorial description of Betti numbers.

Link: http://: https://clemson.zoom.us/j/98217715119?pwd=a1hNY2NTN0lNRzNUOWJDMlBnU2M4dz09

Brigitte Servatius — April 4

Binary delta-matroids and canonical forms

Brigitte ServatiusWorcester Polytechnic Institute

The handle slide operation, originally defined for ribbon graphs, was extended to delta-matroids by  I. Moffatt and E. Mphako-Bandab, whoshow that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form analogous to the canonical form for one-vertex maps on a surface.

We provide a canonical form for binary delta-matroids without restriction on the feasibility of the empty set. This is joint work with Remi Cocu Avohou.

Zoom recording: https://clemson.zoom.us/rec/share/rvFLiEIhTUg0C1kl-dIQA_dnKsqG6qAN2M2eNTLkGQPT7_JsoPWhl72wCRIaSFAK.14TVBRUBia7IfN0r

Gabriel Sosa — March 16

The Koszul property and multiRees algebras of strongly stable Ideals. 

Gabriel SosaColgate University

This talk is divided into three parts. The first part introduces the Koszul property, why it is a desirable property for an algebra, along with methods and difficulties in determining when it is satisfied.

The second part introduces Strongly Stable Ideals through examples. We then describe some of their combinatorial properties and what benefits one obtains from studying them. 

In the third part, we discuss a historical overview of results regarding the characterization of Rees and multiRees algebras of Strongly Stable Ideals that are Koszul, starting with a result from Emmanuela DeNegri (1994) up to the most recent result from Selvi Kara, Kuei-Nuan Lina and myself (2020).

Recording: https://clemson.zoom.us/rec/share/mXEn9HPw0G9GcFASrX-tpT5noR7pBeFkOgDn6EfvwfnZNGUm_qnlc6lOJSgHwA.ZvSBWYQygi4UKNbe

Songling Shan — March 11

Some recent progress towards the overfull conjecture

Songling ShanIllinois State University

Let G be a simple graph with maximum degree Delta(G). A subgraph H of G is overfull if |E(H)|>Delta(G) is greater than the floor function of |V(H)|/2. Chetwynd and Hilton in 1985 conjectured that a graph G  with \Delta(G)>|V(G)|/3 has chromatic index \Delta(G) if and only if G contains no overfull subgraph. In this talk, we will survey some recent progress towards the conjecture. In particular, we will mention the confirmation of the conjecture on graphs with a small core degree, dense quasi-random graphs of odd order, and large graphs of order 2n and minimum degree at least (1+\varepsilon)n for any 0<\varepsilon<1.

Recording: https://clemson.zoom.us/rec/share/zBkey2FhupBPuzbbyK0kv4ZdJL32eFX2LNY3hsPRVHfvZgjcBi8EIvZzP0JdCCRp.Ij3_hurxcuCPU4Vh

Rachelle Bouchat — March 7

Betti numbers of domino ideals

Rachelle BouchatBerea College

Domino ideals are a class of monomial ideals whose generating sets correspond to the sets of domino tilings of a2 x n tableau. It is well known that the number of domino tilings of a 2 x n tableau is given by a Fibonacci number. This talk will consider the graded Betti numbers from the minimal free resolution of a domino ideal, and the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal. Additionally, the application of the Eliahou–Kervaire splitting of a monomial ideal will be discussed within the context of these domino ideals.

Recording: https://clemson.zoom.us/rec/share/pmp-RS1UBB5VlLNmSVkOsrYdxmRygdb34VgoK0Ovl_eNKQGnP7aGb0iKBTmdFgAx.btugum6eOzmMwiEQ?startTime=1646669858000

Carolina Naim: March 4

Private Multi-Group Aggregation

Carolina NaimRutgers University

We consider the private multi-group aggregation (PMGA) problem. This setting involves n users, and each user belongs to one of k distinct groups and holds an integer value. A central server wants to find the aggregate (sum) of the values in each group (with high accuracy) under communication and local differential privacy constraints. The privacy constraint guarantees that the user’s group remains private. This is motivated by applications where a user’s group can reveal sensitive information, such as religious and political beliefs, health conditions, or race.  
In this talk, we will introduce the Query and Aggregate (Q&A) scheme for PMGA. The novelty of Q&A is that it is an interactive aggregation scheme. In Q&A,  each user is assigned a random query matrix, to which he sends the server an answer based on his group and value. We will compare Q&A to the Randomized Group (RG) scheme, which is non-interactive and adapts existing randomized response schemes to the PMGA setting.

Recording: https://clemson.zoom.us/rec/share/H6gmK-8OhMBXXud37s4HZS3uycW5VgV68Ll8gCJYyPezkrgdBcQOUWGazKXbiNzu.4sba-zUe0LGamFQO