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

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.

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.

Convex polytopes are the traditional combinatorial engines of toric geometry. Irrational polygons are already naturally represented by quantum toric stacks, so irrationality alone does not require a new geometric object. The genuinely new phenomenon appears when the moment region is no longer polyhedral—for example, when it is an ellipse or, more generally, a smooth convex body.

I will explain how the tropical wavefront of a convex body canonically determines a quantum horizon. For a polygon, this construction equips the corresponding quantum toric stack with an additional canonical horizon structure. For a nonpolyhedral convex body, successive polygonal wavefronts produce a compatible limiting system of quantum toric stacks, retaining geometric information for which no finite fan is available.

I will then describe how divisor classes and intersection products can be organized on this horizon. The construction extends quantum toric intersection theory in the polygonal case and produces genuinely new limiting intersection invariants for smooth convex bodies. Ellipses provide the basic example: their affine symmetry makes the horizon explicit and reveals a close relationship between the resulting intersection theory, tropical wavefronts, and equiaffine geometry.

This is joint work with Laurent Meersseman and Mikhail Shkolnikov.

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.

The talk is devoted to several real enumerative problems. We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed enumeration of real algebraic curves of genus 1 and 2. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the genus zero case. We also suggest an extension of the invariants under consideration to a non-toric setting using a reformulation of the quantum index in terms of Arnold-Rokhlin surfaces. This is a joint work with Eugenii Shustin.

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.

Bridgeland and the author construct from the cluster combinatorics of an ADE quiver a twistor space whose finite fibres are known explicitly, leaving the fibre over $ϵ=\infty$ open. We determine this fibre for the $A_2$ quiver. It is selected by the degeneration of the twistor lines that we study in terms of the deformed cubic oscillator. It consists of two Donaldson–Thomas fixed points together with an escaping chart that is an explicit affine quotient singularity arising from the automorphism group of the equianharmonic curve appearing in the outer scaling – and the fiber is non-separated! The proof is a matched-asymptotics theorem in exact WKB resting on a rigid local model.

The properties of the Dubrovin connection are fundamental for the recent remarkable applications of quantum cohomology in birational geometry. I will discuss a conjectural Riemann-Hilbert type correspondence between the connection behavior at singular loci and the associated categorical semi-orthogonal decompositions.

We review the theory of super Riemann surfaces and their supermoduli spaces, and describe some new results regarding their non-projectedness and the super Schottky problem.

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.

Quantum cohomology is an interesting object to both mathematicians and physicists due to its relation to string theory. Its product structure, called the quantum product, deforms the product structure of the ordinary cohomology ring via 3-pointed Gromov-Witten invariants. The associativity equations is called WDVV equations. In a similar way, one can use 2|1-quasimap invariants, which are similar to Gromov-Witten invariants, to define such a quantum deformation. It defines no longer a product structure, but a quantum module structure. This is an analogue of the WDVV equations. The semipositive conjecture asks what is the relations in the quantum module for all smooth projective toric semipositive varieties. It turns out that all relations are the same as the Batyrev relations. In this talk, a proof will be given.

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.

Generalized complex structures interpolate between symplectic and complex geometries. We study these on almost abelian Lie groups, whose geometry is entirely governed by a single matrix $A$. We show how the Jordan blocks of $A$ obstruct specific geometric types, providing a complete classification of generalized complex and generalized Calabi-Yau structures when $A$ is diagonalizable, as well as all non-extremal cases in dimension six. Finally, we study the associated generalized cohomologies to uncover further obstructions to these geometries.