VT Algebra Seminar — Fall 2025

(Joint work in progress with Ki Fung Chan, Kwokwai Chan and Chin Hang Eddie Lam) I will discuss a module action of the trigonometric degeneration of DAHA on the equivariant quantum cohomology of $T^*G/P$. It is constructed using Gromov-Witten invariants and can be described explicitly in terms of the stable envelopes. I will also discuss its relation to the “quantum equals affine” phenomenon.

Certain isogeny-based cryptographic protocols rely on a class group action. We study a large family of generalized class groups of imaginary quadratic orders . We prove that they act freely and (essentially) transitively on the set of primitively O-oriented elliptic curves over a field k equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin, and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder O’ of O on the set of O’-oriented elliptic curves and discuss several other examples. This is joint work with W. Castryck, J. Komada Eriksen, G. Lorenzon, and F. Vercauteren, with ongoing follow-up work joint with J. Macula and E. Orvis.

In [1] it was shown that the minimal genus of irreducible curves lying on an abelian surface influences the bound on the minimum distance of algebraic geometry codes constructed from it: the highest the minimal genus, the better the bound. In this talk, we will study abelian surfaces defined over finite fields containing no irreducible curves of genus less than or equal to 3. Our starting point will be the characterisation of isogeny classes of abelian surfaces without curves of genus less than or equal to 2, initiated in [1] and completed in [2]. We will then show that, for simple abelian surfaces, containing a curve of genus 3 is equivalent to admitting a polarisation of degree 4. Hence, we will characterise the isogeny classes of abelian surfaces without curves of genus ≤ 2 containing no abelian surface with a polarisation of degree 4, using the tools developed in [3,4]. Finally, if time permits, we will describe absolutely irreducible curves of genus 3 lying on abelian surfaces containing no curves of genus less than or equal to 2, and show that they have few rational points.

Bibliography:

[1] Y. Aubry, E. Berardini, F. Herbaut, and M. Perret, Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus, Finite Fields and Their Applications, 70 (2021), p. 101791.

[2] E. Berardini, A. J. Giangreco-Maidana, S. Marseglia, Abelian surfaces over finite fields containing no curves of genus 3 or less, arXiv preprint 2408.02493v2

[3] E. W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., 347 (1995), pp. 2361--2401.

[4] E. W. Howe, Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom., 5 (1996), pp.583--608.

Many important families of polynomials in algebraic combinatorics including Schubert polynomials, Demazure characters, and chromatic symmetric polynomials satisfy the Saturated Newton Polytope property (SNP). This convexity property introduced by Monical-Tokcan-Yong is directly related to other important notions like M-convexity, generalized permutahedra, and Lorentzian polynomials. In this talk, I will discuss the recent resolution to a 2019 conjecture of Monical-Tokcan-Yong that the non-symmetric Macdonald polynomials have SNP. I will discuss the combinatorial and convex-geometric tools used in the proof of this result as well as some consequences regarding affine Demazure characters, affine Bruhat intervals, and affine Grassmannian Schubert varieties. This is joint work with Alex Black.

We discuss techniques to bound the cost of the group action used in the post-quantum protocol CSIDH. By formulating the problem of finding an optimal algorithm for CSIDH group action evaluation as a mathematical optimization problem and computing the dual, we provide a method to produce the first known lower bounds on the optimal cost of evaluating the CSIDH group action using a “point-pushing strategy”-based algorithm—such algorithms are used in all implementations of CSIDH. We apply our techniques to the CSIDH-512 parameter set to bound the extent to which the best known strategies are suboptimal. Finally, we consider applications of our techniques to new schemes that build on the ideas of CSIDH.

In geometry, the quantum K theory of Grassmannians is a ring with a product deforming the usual K theory product. In (mathematical) physics, it is the coordinate ring of an affine variety given by the Bethe Ansatz equations. I will discuss a dictionary between these two perspectives, with emphasis on geometric interpretations. In particular, the graphical calculus from a 5-vertex lattice model yields Pieri-type rules, to quantum K multiply Schubert classes by Hirzebruch lambda_y classes of tautological bundles. One may also construct eigenvectors of the previous quantum multiplication operators, called Bethe vectors, which quantize the usual classes of torus fixed points. I will discuss how the existence of these Bethe vectors leads to a theory of quantum equivariant localization for Grassmannians. This is joint work with V. Gorbounov and C. Korff, following earlier work with W. Gu, E. Sharpe, and H. Zou.

The structure of isogenies between supersingular elliptic curves is fundamental to the study of isogeny-based post-quantum cryptography. In this talk, we will discuss "oriented" elliptic curves and the structure of isogenies between them. The structure of "horizontal" isogenies between such curves is well understood. The main result is that we can relate N-isogenies between supersingular elliptic curves oriented by an order of discriminant D, to solutions of equations involving positive definite binary quadratic forms of discriminant D. In the case that -DN < 2p, we characterize when "non-horizontal" N-isogenies arise between oriented curves. This talk is intended to be accessible to grad students; in particular, if you do not know what an elliptic curve is, most of it can be understood through graphs and explicit equations.