Gromov–Witten invariants can be computed in GKM spaces via torus localization methods. These invariants can then be used to determine the eigenvalues of the small quantum multiplication by the first Chern class. In this talk, I will review the basic notions of GKM spaces and introduce the concept of Hodge atoms, introduced by Katzarkov-Kontsevich-Pantev-Yu. I will then explain how localization techniques can be implemented to compute the atoms of complete intersections on flag varieties. Joint work with Leonardo Cavenaghi, Giovane Galindo, and Ludmil Katzarkov.

We quickly introduce the theory of G-equivariant atoms, discuss enhancements and questions. Joint work with L. Katzarkov and M. Kontsevich.

In this talk, we will show how the theory developed in the previous lecture can be specialized to flag varieties. We will also present a software that applies these results to compute the small quantum multiplication of complete intersections in flag varieties. Several examples will be discussed, illustrating how atom theory can be used to establish the non-rationality of various algebraic varieties. This is joint ongoing work with Leonardo Cavenaghi, Ludmil Katzarkov, and Pedro Muniz.

In this talk I will give a panoramic overview of tropical zeta functions of convex domains introduced in the recent joint work with Nikita Kalinin and Ernesto Lupercio, focusing on known explicit examples, motivations and parallels with other zeta functions, structural results on analytic continuation and geometric meaning of residues at some persistent locations of the complex plane, conjectures and a few currently established applications. In particular, I will explain how the knowledge of pole profile of a tropical zeta allows to derive asymptotic expansions of various geometric and number-theoretic quantities associated with convex domains. I will also share with the audience the most recent outcomes of computations of concrete families of tropical zetas by Kevin Calderon allowing to speculate on the conjectural shape of the first concrete numerical bridge between tropical optics and quantum toric varieties.

I will describe a framework in which derived analytic geometry is constructed relative to exact categories and completeness data. It includes the nearly equivalent bornological and Ind-Banach descriptions, as well as condensed and completed settings of Scholze and Clausen, and provides comparison functors between them. The resulting machinery supplies analytic affines, cotangent complexes, blowups, and deformation to the normal bundle over general Banach rings. Its compact projective objects are explicit and sometimes have measure-theoretic descriptions. I will then present two spectral calculations illustrating what these foundations make visible. Arithmetic localizations of Wiener and overconvergent algebras are organized by a prime filter spectrum over the Berkovich spectrum of the integers and exhibit fracture phenomena at finitely many primes. For a compact Riemann surface with boundary, the finite rational topology has a prime-filter spectrum with three kinds of points: ordinary points, analytic half-branches, and residual chamber choices. The residual points can be represented by arcs and associated continuous germs. The talk is part of the program developed with Jack Kelly and Kobi Kremnizer in A Perspective on the Foundations of Derived Analytic Geometry, available as arXiv:2405.07936, together with several companion projects.

I will first survey older and very recent results concerning the existence of an irrational pencil on a Kaehler manifold, from the classical results of Castelnuovo-de Franchis, Siu-Catanese-Beauville, Green-Lazarsfeld, to recent extensions using the profinite completion of the fundamental group. I shall then illustrate the application to VIP’s = Varieties isogenous to a product, and explain then how this class of varieties allows to see a faithful action of GAL on the components of moduli spaces, indeed by the action on the fundamental groups of VIP’s.

We discuss the mathematical setup of superRiemann surfaces and their moduli. Superstring scattering amplitudes are then certain integrals over moduli spaces of superRiemann surfaces together with extra data, and we discuss a rigorous determination of these from first principles, in genus zero.

We observe that on a compact ACYT 8-manifold which Nijenhuis tensor is parallel with respect to the torsion ACYT connection $\nabla$ if its the three Ricci tensors vanish then the torsion 3-form is closed. Consequently, on a compact CYT 8-manifold if the three Ricci tensors of the Strominger-Bismut connection vanish then its torsion is closed (pluriclosed CYT). We deduce that a compact ACYT 8-manifold with $\nabla$-parallel Nijenhuis tensor has closed torsion if and only if the trace of the exterior derivative of the torsion vanishes. We present exact conditions an almost hyperhermitian 8-manifold to be an AHKT and show that an AHKT 8-manifold is HKT exactly if and only if the three Lee forms coincide.

Using the notion of covariantly constant Hermitian structures on complexes of holomorphic vector bundles on projective manifolds, I will describe a Hermitian-Einstein condition, which implies that holomorphic sections are covariantly constant.

In this talk, I will discuss several analogues of the classical Hodge polynomials of complex algebraic varieties in the setting of definable complex analytic varieties and their motivic aspects. This talk is based on joint work with Ludmil Katzarkov, Ernesto Lupercio, and Laurent Meersseman.

I will report on recent joint work with Shamil Shakirov/Chiarello-Hausel on a surprising link of the $P=W$ equivalence for Higgs bundles and a special solution of a certain system of Bethe equations.