We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman and Stefan Schreieder.

The SYZ conjecture provides a way to construct the mirror to a log Calabi-Yau manifold by counting specific rational curves inside the initial variety. In recent years, Keel-Yu and Gross-Siebert implemented the SYZ mirror construction using non-archimedean and logarithmic geometry respectively, and encoded the mirrors into a combinatorial object called a scattering diagram. A major open question is to compare these two approaches to the SYZ conjecture. In this talk, after reviewing these constructions and ideas, I will present a comparison result that expresses non-archimedean cylinder counts in terms of log Gromov-Witten invariants. In the surface case, I will explain how to deduce the exponential formula, a remarkable identity which relates the non-archimedean and logarithmic scattering diagrams, and implies the equivalence of the Keel-Yu and Gross-Siebert mirror constructions. The exponential formula is the first explicit formula relating non-archimedean Gromov-Witten invariants to punctured log Gromov-Witten invariants. If time permits, I will discuss a work in progress that extends the comparison to higher dimensions. Joint work with Tony Yue Yu, based on arXiv:2510.18319.

The SYZ approach to mirror symmetry for Fano toric varieties is well understood, with the mirror superpotential governed by Maslov index two holomorphic discs. For non-Fano manifolds, however, additional discs of non-positive Maslov index appear and fundamentally alter the SYZ construction. In this talk, I will use the non-Fano Hirzebruch surfaces F_3 and F_4 to illustrate that the resulting mirror can depend on the choice of perturbation used to regularize disc moduli spaces. For F_4, I will present a perturbation for which the mirror superpotential becomes an explicit infinite Laurent series, and explain how it is related to other superpotentials via a scattering diagram.

The Noncommutative Minimal Model Program, proposed by Halpern-Leistner, is a set of conjectures that relate decompositions of the derived category of a variety to paths in its space of Bridgeland stability conditions. I will review one of these conjectures and explain recent joint work with Tomohiro Karube and Vanja Zuliani verifying it for several classes of Fano varieties.

The quantum connection is defined from genus zero Gromov-Witten invariants, and it carries rich information with a wide range of applications. In this talk, I will talk about a conjecture that predicts that by work over the p-adic numbers, the small quantum connection of a Fano variety should carry a Frobenius structure with distinguished convergence property, and Morita’s p-adic Gamma function plays a central role in the formulation. Time permitting, I will also discuss the analogue in the Calabi-Yau setting, which is related to mirror counterparts of crystalline cohomology. This is based on joint works with Pomerleano and Seidel, and with Lee.

The period-index problem is a classical question about central simple algebras over a field. I will give an introduction to the problem, and survey recent results for Brauer classes over certain algebraic varieties and complex-analytic spaces.

I will discuss how cocycles appear in a graphical network. Furthermore, the Shannon entropy of a finite probability distribution has a natural interpretation in terms of diagrammatics. I will explain the diagrammatics and their connections to infinitesimal dilogarithms and entropy. If there is time, I will discuss how algebraic K-theory is related to cobordism groups of foams in various dimensions. This is joint with Mikhail Khovanov.

Heegaard-Floer theory is about Fukaya categories of symmetric product of surfaces. If the surface can be glued from smaller pieces along strips, we give a gluing formula of the corresponding categories. This is joint work with Vivek Shende. If time permits, I will discuss the relation with categorification of quantum group from Fukaya category.

The quantum connection is a flat connection arising from genus 0 Gromov--Witten theory. They can be defined integrally for sufficiently positive symplectic manifolds, allowing one to consider their characteristic p or p-adic versions which bear similarity to Gauss--Manin connections in arithmetic geometry. We focus on the case of Calabi--Yau 3-folds and explain the arithmetic aspects of the quantum connection, emphasizing the role of quantum power operations. This talk is based on joint work with Shaoyun Bai and Daniel Pomerleano.

The moduli spaces of Higgs bundles and flat connections on a Riemann surface are fundamental examples of holomorphic symplectic manifolds, lying at the intersection of integrable systems, geometric representation theory, and quantum field theory. In this talk, I will describe holomorphic Lagrangian subvarieties in the moduli spaces of Higgs bundles and flat connections, constructed by specifying a sub-line bundle inside a rank-n vector bundle. These Lagrangians are naturally labeled by divisors: on the Higgs side, they arise from generalized Baker–Akhiezer divisors, while on the de Rham side they correspond to apparent singularities of flat connections. I will further explain how these constructions lead to Lagrangian correspondences between the Higgs (respectively, de Rham) moduli spaces and Hilbert schemes of points on the (respectively, twisted) cotangent bundle of the Riemann surface. Connections to the geometric Langlands program and to the Kapustin–Witten framework of mirror symmetry will be briefly discussed.