VT Algebra Seminar

Fridays 2:30-3:30 eastern time

McBryde 321

September 3

Daniel Orr

Difference operators for wreath Macdonald polynomials

Wreath Macdonald polynomials are a generalization of ordinary (type A) Macdonald polynomials where symmetry in one set of variables is replaced by symmetry in several interconnected sets of variables. Their existence was conjectured by Haiman (2002) and proved by Bezrukavnikov and Finkelberg (2014) using deep methods in geometric representation theory. Unfortunately, these methods do not provide the kind of direct, explicit access to wreath Macdonald polynomials which has been so fruitful (and essential) in the development of ordinary Macdonald theory as we know it today. In this talk, I will begin by introducing ordinary and wreath Macdonald polynomials and discussing some aspects of their (vast) significance. Then, I will discuss new explicit results on wreath Macdonald polynomials (in particular, their eigenoperators) from joint work in progress with Mark Shimozono and Joshua Wen.
September 10

Travis Morrison

Random walks in isogeny graphs

In this talk, I'll give an introduction to random walks in graphs. I will introduce Ramanujan graphs, which are (families of) graphs in which random walks mix rapidly, and give an example of such a graph constructed using elliptic curves. Finally I'll discuss how these graphs can be used to compute endomorphisms of elliptic curves.
September 17

Fazle Rabby

Linking Double Conics

In this talk I will briefly introduce the multiplicity structures on space curves. These curves arise naturally as limits of flat families of smooth curves. Using a classic construction of Ferrand, I will describe all double structures on conics. Finally I will show when these curves are self-linked by complete intersection curves. This result extends the result of Juan Migliore on double lines.
September 24

Ben Goodberry and David Oetjen

Title TBA

Abstract TBA
October 1

No seminar

Fall break

October 8

Daniel Valvo and Aidan Murphy

Valvo: Repair schemes for augmented Cartesian codes; Murphy: Norm-trace-lifted codes

Valvo: A distributed storage system stores a file across multiple storage nodes. Methods of encoding the file such that if one or more of the storage nodes fails, the remaining nodes can recover the missing data are highly relevant. In 2017, Guruswarmi and Wooters introduced a linear exact repair scheme for Reed-Solomon codes. Their repair scheme is capable of exactly repairing a single failed node with low bandwidth. From this foundation, many authors, ourselves included, have worked to extend the idea to repair single and multiple erasures in different contexts. In this talk, we will share recent developments in this area, including multiple ways to extend the linear exact repair scheme framework to repair single and multiple erasures in Reed-Muller codes to augmented Reed-Muller codes and certain other families of evaluation codes. We will also discuss ways to optimize the codes and linear exact repair schemes in these contexts as well as compare these schemes to existing schemes in terms of their rate and bandwidth. Murphy: It is useful for codes to be able to correct errors and recover erasures by accessing less information than classical codes allow. Codes with locality are designed for this purpose. Such codes are said to have locality r and availability t if each codeword symbol can be recovered from t disjoint sets of r other symbols. The Hermitian-lifted code construction provides codes from the Hermitian curve over F_{q^2} which have the same locality and availability as one-point Hermitian codes, but the Hermitian-lifted codes have a rate bounded below by a constant independent of the field size. In this talk, we consider this technique applied to codes from norm-trace curves, which are a generalization of the Hermitian curve.
October 15

Speaker TBA

Title TBA

Abstract TBA
October 22

Leonardo Mihalcea

Cotangent Schubert classes

Cotangent Schubert classes are classes in the (co)homology ring of a flag manifold. They deform the usual Schubert classes, and appear in several areas: characteristic classes of singular varieties, the theory of stable envelopes, Kazhdan-Lusztig theory. I will define these classes and discuss some of their properties, including some open problems about them.
October 29

Minyoung Jeon

Mather classes of Schubert varieties via small resolutions

The Chern-Mather class introduced by MacPherson is a characteristic class, generalizing the Chern class of a tangent bundle of a nonsingular variety to a singular variety. It uses the Nash-blowup for a singular variety instead of the tangent bundle. In this talk, we consider Schubert varieties, known as singular varieties in most cases, in the even orthogonal Grassmannians and discuss the work computing the Chern-Mather class of the Schubert varieties by the use of the small resolution of Sankaran and Vanchinathan with Jones’ technique. We also describe the Kazhdan-Lusztig class of Schubert varieties in Lagrangian Grassmannians, as an analogous result.
November 5

Tianyi Yu

Grothendieck-to-Lascoux Expansions

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. This is a joint work with Mark Shimozono.
November 12

Mark Pengitore

Title TBA

Abstract TBA
November 19

Jason Saied

Title TBA

Abstract TBA
November 26

No seminar -- Thanksgiving break

Mr. Turkey

Gobble gobble!
*December 2, 11:00 a.m*

Jana Sotáková

Title TBA

Abstract TBA