VT Algebra Seminar

Fridays 2:30-3:30 eastern time

McBryde 321

Spring 2024

January 26

Quantum supersymmetric pairs and ıSchur duality of type AI-II

Yalong Shen (UVA)

Abstract: Let g be a semisimple Lie algebra and θ be an involution of g. The quantization (Uq(g), Uı) of the symmetric pair (g, gθ) was systematically developed by Letzter where Uq(g) is the Drinfeld-Jimbo quantum group and Uı is a coideal subalgebra of it. We ususally refer Uı as the ıquantum group. Over the last decade, many fundamental constructions in quantum groups have been generalized to ıquantum groups by Wang and his collaborators. In this talk, we will discuss the super analogue of Uı and introduce one specific family which unites ıquantum groups (non-super) of type AI and AII. We will also demonstrate an ıSchur duality between this specific family and the q-Brauer algebra. This duality can be viewed as a quantization of the classical duality between the orthosymplectic Lie superalgebra and the Brauer algebra. This is joint work with Weiqiang Wang.
February 9

Jellyfish, the arithmetic-geometric mean, and elliptic curves

Eleanor Mcspirit (UVA)

In this talk, we discuss a finite-field analogue of the arithmetic-geometric mean sequence. Study of the classical version dates back to work of Lagrange and Gauss, and makes beautiful contact with approximations of pi, elliptic integrals, hypergeometric functions, and elliptic curves. In the study of AGM(F_q), directed graphs called “jellyfish swarms” naturally arise. In studying these graphs, we find connections to both finite-field hypergeometric functions and elliptic curves over finite fields. Such connections give rise to new identities for Gauss’ class numbers of positive definite binary quadratic forms and show that the sizes of “jellyfish” (connected components of these graphs) are in part dictated by the order of a prime above 2 in certain class groups.
February 16

The lookup conjecture and rational smoothness in type $\tilde{A}_2$

William Graham (UGA)

The notion of rational smoothness in Schubert varieties has been of interest since the foundational paper of Kazhdan and Lusztig connecting singularities of Schubert varieties with representation theory. This talk is based on recent joint work with Brian Boe, where we identify the loci of rationally smooth points and of smooth points in Schubert varieties for the Kac-Moody group of type $\tilde{A}_2$. We hope the methods will provide insight applicable to other types, and in particular, to type $\tilde{A}_n$ for $n \geq 2$. Our work continues the study of the lookup conjecture initiated in earlier work of the authors, as well as more recent joint work with Wenjing Li on spiral Schubert varieties in type $\tilde{A}_2$.
March 1

Complexity, Exactness, and Rationality in Continuous and Discrete Polynomial Optimization

Robert Hildebrand (VT)

Abstract: Optimizing non-convex functions over potentially non-convex is a difficult task, and often it can be difficult even to certify a feasible solution. Modern optimization solvers frequently are comfortable providing approximately feasible solutions to such problems. Due to operating in floating-point arithmetic, we are constrained to providing rational solutions. We demonstrate classes of cubic optimization problems where it is NP-Hard to determine if a rational feasible solution exists. We also show how nearly feasible solutions can have super objective values and other related results. Time permitting, we will discuss some recent work on the complexity of finding integer solutions in non-convex sets.

March 27, 4-5 pm, Derring 1076

Rank metric codes, Shellability and Homology

Sudhir R. Ghorpade (IIT Bombay)

The notion of shellable simplicial complexes has proved extremely useful in algebraic combinatorics, commutative algebra, and combinatorial topology. As such, it has been much studied in the past four decades. It is a classical result that matroid complexes. that is, simplicial complexes formed by the class of independent subsets in a matroid, are shellable. This has some bearing on the study of linear block codes, especially in regard to their Betti numbers and generalized weight enumerator polynomials. We now know that q-matroids have close connections with rank metric codes in a manner similar to the connection between matroids and codes. A recent result establishes shellability of q-matroid complexes and also determines the homology of these complexes in many cases. The determination of homology has now been completed for arbitrary q-matroid complexes. We will outline these developments whlie making an attempt to keep the prerequisites at a minimum. The contents of this talk are based on a joint work with Rakhi Pratihar and Tovohery Randrianarisoa (2022) and also with Rakhi Pratihar, Tovohery Randrianarisoa, Hugues Verdure and Glen Wilson (2024).
March 29

Title TBA

Graduate Student talks

TBA
April 5

Gromov-Witten invariants in the Quantum K-theory of maximal Orthogonal Grassmannians

Mihail Tarigradschi (Rutgers)

Consider an even-dimensional complex vector space endowed with a symmetric nondegenerate form. The set of subspaces of maximal dimension forms a projective variety called the maximal orthogonal Grassmannian X=OG(n, 2n). Since X is a smooth projective variety, we will consider its K-theory group K(X). As a group, K(X) is a freely generated by the classes of Schubert subvarieties and the multiplication of such classes can be computed using known computational rules. The quantum K-theory ring, QK(X), is a deformation of K(X) by Z[[q]]. As a group, it is similarly freely generated by classes of Schubert subvarieties but a (potentially) different multiplicative structure. The multiplicative constants can be computed using the Gromov-Witten invariants in the Kontsevich Moduli space. In this talk, I will give a short survey of the constructions in this setup and discuss recent progress of computing Pieri formulas for Gromov-Witten invariants.
April 12

1:30-2:30 and 2:30-3:30

ACTIVIT talks

Alberto Ravagani and Ben Jany

TBA
April 19

Positivity in Weighted Flag Varieties

Scott Larson (UGA)

Abstract: Let H ⊆ B ⊆ G be Cartan and Borel subgroups in a connected reductive complex algebraic group, and let Z be the complement of the zero section of a line bundle on G/B corresponding to a dominant weight λ of H. Let χ be a cocharacter of H such that for every Weyl group element w ∈ W, the pairing wλ ⋅ χ is strictly positive. Let S = χ(C×) and call X = S ∖ Z the weighted flag variety.

The torus T = H/S acts on X, which enables the study of T-equivariant cohomology of X. In the case where X = G/P, Graham proved that the equivariant structure constants with respect to a Schubert basis satisfy positivity with respect to a system of simple roots. In the case where G = C× × GLn and λ restricts to a fundamental weight from GLn, Abe-Matsumura find the existence of a basis of HT*(X) and parameters in HT*(pt) satisfying a similar positivity. We generalize all positivity results to any G and λ, interpret the basis of HT*(X) as Poincaré dual to weighted Schubert varieties, and define the notion of weighted root to interpret geometrically the parameters in HT*(pt).

April 26

Triangular modular curves

Juanita Duque Rosero (Boston University)

Triangular modular curves are a generalization of modular curves and arise from quotients of the complex upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves naturally parameterize abelian varieties, making them interesting arithmetic objects. In this talk we will introduce triangular modular curves via classical modular curves. We will also study the genus of these curves. This is joint work with John Voight.
May 3

Quantum K-invariants of Grassmannians via Quot schemes

Shubham Sinha (ICTP)

In this talk, we will define K-theoretic invariants involving certain virtual Euler characteristics of sheaves over the quot scheme of a curve. We demonstrate that these invariants fit into a topological quantum field theory valued in Z[[q]]. Additionally, we will show that the genus-0 invariants recover the small quantum K-ring of Grassmannians, offering a new approach for finding explicit formulas. In particular, we use torus localisation to obtain a Vafa-Intriligator type formula for the virtual Euler characteristics over the quot schemes. This is based on a joint work with Ming Zhang.

Fall 2023

September 1

A presentation for the quantum K-theory ring of partial flag manifolds

Weihong Xu

Quantum K-theory ring of a smooth projective variety is a deformation of its K-theory ring of algebraic vector bundles. Gu, Mihalcea, Sharpe, and Zou gave a presentation for the (equivariant) quantum K-theory ring of the Grassmannian, where the relations are deformations of the classical K-theoretic Whitney relations. We conjecture a generalization of these quantum K Whitney relations to all partial flag manifolds. If these relations hold, then they give a complete set of relations. We prove this conjecture for the incidence variety Fl(1,n-1;n), and for the full flag manifold, we reduce this conjecture to a conjecture of Buch and Mihalcea on Chevalley-type K-theoretic Gromov--Witten invariants. This is joint with Gu, Mihalcea, Sharpe, Zhang, and Zou.
September 8

Computing endomorphism rings of supersingular elliptic curves

Travis Morrison

Computing the endomorphism ring of an elliptic curve (or more generally, an abelian variety) is a fundamental problem in computational number theory. An efficient algorithm for computing the endomorphism ring of a supersingular elliptic curve would break all practically-instantiable isogeny-based cryptosystems -- therefore the endomorphism ring problem is the central problem in isogeny-based cryptography. In this talk, I will discuss recent work with Fuselier, Iezzi, Kozek, and Namoijam in which we give an algorithm for computing these endomorphism rings using inseparable endomorphisms.
September 15

Some modular forms and Calabi-Yau varieties from irrationality proofs and mirror symmetry

Michael Schultz

After Apéry stunned the mathematics community in 1978 by proving that ζ(3) is irrational, Beukers & Peters showed in 1983 that fundamental ingredients in Apéry's proof could be understood in terms of crucial geometry underlying a certain family of K3 surfaces. Much later, W. Yang applied ideas from mirror symmetry for Calabi-Yau threefolds to this family of K3 surfaces to derive an interesting weight four modular form whose Lambert series expansion has periodic integral instanton numbers. In this talk, we will dive into these explicit results and show that they can be generalized to certain modular elliptic surfaces and closely related families of K3 surfaces. In each case, we find a modular form with periodic integral instanton numbers, whose periodicity appears to be determined by the underlying modular curve. This talk represents ongoing work with A. Malmendier.
October 4

Stable-Limit Non-Symmetric Macdonald Functions

Milo Bechtloff Weising

Non-symmetric Macdonald polynomials play an important role in the representation theory of double affine Hecke algebras. These special polynomials give a basis for the standard DAHA representation consisting of weight vectors for the classical Cherednik operators and exhibit many interesting combinatorial properties related to affine Weyl groups. I will discuss a natural extension of these polynomials to the setting of the stable-limit DAHA of Ion-Wu. In this case we will obtain a basis for the standard stable-limit DAHA representation consisting of weight vectors for the limit Cherednik operators. These generally infinite variable functions exhibit combinatorial properties akin to their finite variable counterparts with some interesting differences. I will also discuss some further directions in this theory including links to the Shuffle Theorem of Carlsson-Mellit.
October 18

Quantum cohomology and mirror symmetry for flag varieties from two perspectives

Joshua Wen (Northwestern)

A key result in the rich theory of rational Cherednik algebras is the deformed Harish-Chandra isomorphism, proposed by Etingof-Ginzburg and proved by Gan-Ginzburg, that identifies the spherical subalgebra of the type A rational Cherednik algebra with a quantized Nakajima quiver variety, the latter of which is defined as a quantum Hamiltonian reduction of a ring of differential operators. I will discuss a multiplicative analogue of this result, wherein the rational Cherednik algebra is replaced with the usual double affine Hecke algebra of GL_n and the quantized quiver variety is replaced with a quantized multiplicative quiver variety, as defined by Jordan. This setting is strange because we are no longer working with rings of differential operators, but rather a less familiar ring of quantum differential operators defined by Varagnolo-Vasserot. Nonetheless, I will explain how, via an idea of Varagnolo-Vasserot, Macdonald polynomials can be used to establish the analogous isomorphism in a manner quite similar to that of the rational case.
October 20

Quantum cohomology and mirror symmetry for flag varieties from two perspectives

Elena Kalashnikov (Waterloo)

Type A flag varieties can be constructed both as homogeneous spaces G/P and as GIT quotients V//H. These two different constructions give different perspectives on the quantum cohomology (different bases and sets of structure constants) and mirror symmetry (the Gu—Sharpe mirror and the Plucker coordinate mirror) of flag varieties. In this talk, I’ll discuss these two different perspectives on these topics, their advantages and disadvantages, and what is known about the relation between them.
November 3

Analogue of Fomin-Stanley algebra on bumpless pipedreams

Tianyi Yu (UC San Diego)

Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart's growth diagram.
November 10

Counting 0-dimensional sheaves on singular curves

Yifeng Huang

(Based on joint work with Ruofan Jiang) The Hilbert scheme of points on a variety X parametrizes 0-dimensional quotients of the structure sheaf. When X is a planar singular curve, its enumerative invariants are closely related to mathematical physics, knot theory and combinatorics. In this talk, we investigate two analogous moduli spaces, one being a direct generalization of the Hilbert scheme. Our results reveal their surprising relations to Hall polynomials, matrix equations, modular forms, etc
November 17

The volume polynomials of zonotopes

Ivan Soprunov (Cleveland State)

Abstract: The volume polynomial is an n-variate homogeneous polynomial of degree d associated with a collection of n convex compact sets in R^d. This notion goes back to the work of Minkowski on Brunn-Minkowski theory of convex bodies. Over the past hundred years volume polynomials have appeared in almost all areas of pure and applied mathematics, including algebraic geometry and combinatorics. I will talk about the problem of describing the space of volume polynomials via coefficient inequalities and its application to tropical intersection numbers. I will also show how one can use the Grassmann-Plücker relations to produce new polynomial inequalities for the coefficients of volume polynomials of zonotopes. This is joint work with Gennadiy Averkov.
December 1

Hyperelliptic Curves mapping to Abelian Surfaces and Applications to Beilinson's Conjecture for zero-cycles

Evangelia Gazaki (UVA)

The Chow group of zero-cycles is a generalization to higher dimensions of the Picard group of a smooth projective curve. When $X$ is a curve over an algebraically closed field $k$ its Picard group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the $k$-points of the Jacobian variety of $X$. In higher dimensions however the situation is much more chaotic, as the Abel-Jacobi map in general has a kernel, which over large fields like $\mathbb{C}$ can be enormous. On the other extreme, a famous conjecture of Beilinson predicts that if $X$ is a smooth projective variety over $\overline{\mathbb{Q}}$, then this kernel is zero. For a variety $X$ with positive geometric genus this conjecture is very hard to establish. In fact, there are hardly any examples in the literature. In this talk I will discuss joint work with Jonathan Love where we make substantial progress on this conjecture for an abelian surface $A$. First, we will describe a very large collection of relations in the kernel arising from hyperelliptic curves mapping to $A$. Second, we will show that at least in the special case when $A$ is isogenous to a product of two elliptic curves, such hyperelliptic curves are plentiful. Namely, we will describe a construction that produces for infinitely many values of $g\geq 2$ countably many hyperelliptic curves of genus $g$ mapping birationally into A.
December 8

Bumpless pipe dreams meet puzzles

Rui Xiong (Ottawa)

In this talk, we will present a combinatorial rule for the product of two double Grothendieck polynomials in different secondary variables with separated descents. This rule generalizes the separated-descent puzzle rules by Knutson and Zinn-Justin, as well as the bumpless pipe dream by Weigandt. We have utilized the formula to partially confirm a positivity conjecture by Kirillov. If time permits, we will also discuss the proof behind the proof. This work is joint with Neil J.Y. Fan and Peter L. Guo.