VT Algebra Seminar — Fall 2023

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.

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.

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.

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.

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.

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.

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.

(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

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.

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.

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.