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Research Seminars: Algebra and Combinatorics

Fall 2022

Time & Location: All talks are on Wednesday in Gibson Hall room 126A at 3:00 PM unless otherwise noted.
Organizer: Daniel Bernstein




August 31

Title: Quaternary Lattices, Modular Forms and Elliptic Curves

Daniel Fretwell

Abstract: A common theme in modern Number Theory is to find non-trivial links between objects coming from very different places, by relating their arithmetic. In this talk we will (hopefully) see a surprising example of this, connecting the first and third objects in the titleā€¦using the second to bridge the gap. Time permitting, we will sketch the proof, motivated by a hidden 1.5th object (Clifford algebras). (Based on joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)

Location: DW 103

Time: 3:00


September 14

Title: Generalized Wilf Conjecture

Naufil Sakran | Tulane University

Abstract: The Wilf Conjecture is a longstanding conjecture regarding the complement finite submonoids of N, the monoid of natural numbers. There have been several attempts to generalize the conjecture for higher dimensions. In this talk, we will extend the conjecture for complement finite submonoids of unipotent group with entries from N. This extension gives a better bound compared to the previous generalizations. Also, we will prove our conjecture for certain subfamilies (thick and thin) of the unipotent groups.

Location: Gibson Hall 126A

Time: 3:00


September 21

Title: Weakly Complete Universal Enveloping Algebras of Profinite-Dimensional Lie Algebras

Karl Hofmann | Tulane University


Location: Gibson Hall 126A

Time: 3:00


October 12

Title: Explicit non-Gorenstein R=T via rank bounds

Cathy Hsu | ???

Abstract: In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

Location: Gibson Hall 126A

Time: 3:00


November 2

Clifford Lectures


December 9

Last Day Of Class