VT Algebra Seminar

Fridays 2:30-3:30 eastern time

McBryde 321

Spring 2022

January 28

Ryan Shifler

Curve Neighborhoods of Schubert Varieties of the Odd Symplectic Grassmannian

A degree d curve neighborhood of a subvariety V in a smooth variety X is the closure of the degree d curves that intersect V. Curve neighborhoods where introduced by Buch, Chaput, Mihalcea, and Perrin to study quantum K-theory. The odd symplectic Grassmannian IG is a quasi-homogeneous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in IG was used by Mihalcea and myself to prove an (equivariant) quantum Chevalley rule. I will discuss how to give a full description of the irreducible components of curve neighborhoods of Schubert varieties in IG. One possible application is to use curve neighborhoods to calculate the minimum degree that occurs in a quantum product. This is joint work with Clelia Pech.
March 18

Caleb Springer

Every finite abelian group arises as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$, and $\mathbb{F}_5$

We will show that every finite abelian group arises as the group of rational points of an ordinary abelian variety over a finite field with 2, 3 or 5 elements. Similar results hold over finite fields of larger cardinality. On our way to proving these results, we will view the group of rational points of an abelian variety as a module over its endomorphism ring. By describing this module structure in important cases, we obtain (a fortiori) an understanding of the underlying groups. Combining this description of structure with recent results on the cardinalities of groups of rational points of abelian varieties over finite fields, we will deduce the main theorem. This work is joint with Stefano Marseglia.
March 25

Jordan Disch

Generic Gelfand-Tsetlin Modules of Quantized and Classical Orthogonal Algebras

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide rational matrix coefficients for these infinite-dimensional modules of both $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of $U_q'(\mathfrak{so}_n)$ were given by Gavrilik and Iorgov, and we consider the commutative subalgebra $\Gamma \subset U_q'(\mathfrak{so}_n)$ generated by these elements and the corresponding subalgebra $\Gamma_1 \subset U(\mathfrak{so}_n)$. The images of $\Gamma$ and $\Gamma_1$ under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show $\Gamma$ is a Harish-Chandra subalgebra of $U_q'(\mathfrak{so}_n)$ and $\Gamma_1$ is a Harish-Chandra subalgebra of $U(\mathfrak{so}_n)$.
April 1

Jenny Fuselier

Hypergeometric functions over finite fields

In the 1980s, Greene introduced a finite field version of hypergeometric functions, now called Gaussian hypergeometric functions. Alternate versions of these functions were developed by Katz and McCarthy. In this talk, we will develop Greene’s functions in detail, comparing to classical hypergeometric functions along the way. We will highlight ways they relate to point-counting, traces of Hecke operators, and supercongruences. We then will present a systematic approach for translating some classical hypergeometric identities and evaluations to the finite field setting by an explicit dictionary.
April 8

Rahul Singh

TBA

TBA
April 15

Joshua Wen

The quantum Harish-Chandra homomorphism for GL_n

Abstract: This talk is about comparing two a priori different quantizations of the GL_n-character variety for the torus. One of them is the spherical double affine Hecke algebra at q=t. The other is the combinatorial quantization defined by Alekseev, Grosse, and Schomerus, which has also reappeared in works of Varagnolo-Vasserot and Jordan. Conjecturally, both algebras are isomorphic. However, neither algebra possesses a presentation via generators and relations, and I will present a way to compare the two via an action on characters for GL_n. This approach has a natural t-deformation wherein the characters are replaced with Macdonald polynomials.
April 22

Michael Schultz

TBA

TBA

Fall 2021

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

Coarse embeddings and homological filling functions

In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result.
November 19

Jason Saied

Alcove walk formula for SSV polynomials

SSV polynomials are a recent generalization of Macdonald polynomials due to Sahi, Stokman, and Venkateswaran. These polynomials are constructed using a new "metaplectic" representation of the corresponding double affine Hecke algebra, and certain specializations of them recover Whittaker functions associated to metaplectic covers of reductive p-adic groups. My work uses this Hecke algebra representation and a result of Ram and Yip in order to give a formula for SSV polynomials in terms of combinatorial objects called alcove walks (generalizing Ram and Yip's formula for Macdonald polynomials). In this talk, I will introduce both Macdonald and SSV polynomials. I will then discuss the alcove walk formula, its consequences, and a variety of open questions prompted by this work.
November 26

No seminar -- Thanksgiving break

Mr. Turkey

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

Jana Sotáková

Title TBA

Abstract TBA