Abstract: Hodge Theory provides a general way of understanding moduli spaces of algebraic varieties: Given a family of algebraic varieties, one obtains a `period map' by considering the hodge structure on the cohomology. However, these period maps are built from period integrals, and are highly transcendental, which yields challenges when one wants to recover an algebraic structure. 

Famously, this story works really nicely for the moduli space of (principally polarized, g-dimensional) Abelian varieties A_g, where the hodge theory gives an exact moduli space, and the work of Baily-Borel provides a beautiful compactification of this space which can also be understood using hodge theory. 

We explain how this picture generalizes to arbitrary period maps. This has especially nice applications to moduli spaces of Calabi-Yaus, which has proven less accessible to other techniques. Moreover, the same tools yield a resolution of the  b-semiampleness conjecture of Prokhorov and Shokurov. This is joint work with Bakker, Filipazzi, and Mauri.

Abstract : Maulik-Okounkov introduced stable envelopes to realize R-matrices of Yangian on the equivariant cohomology of quiver varieties. Yiqiang Li considered equivariant cohomology of fixed point sets of involutions on quiver varieties, — -quiver varieties, and constructed representations of coideal subalgebras of Yangian, i.e., twisted Yangian. Recently I clarify necessary and unnecessary assumptions that should be imposed or not imposed on Li's construction. In this talk, I will review these stories. 

Abstract: In this talk, I will discuss a version of quantum K-theory introduced by A.Okounkov, which can be defined through quasimap counts. In this framework, the quantum K-theory ring is obtained as a specialization of the equivariant quasimap count at q=1, where q is the equivariant parameter associated with the torus action on the source of the quasimaps. A related, but less explored, structure emerges when $q$ is specialized at the roots of unity. I will outline the key ideas behind this construction and its implications. As an application, I’ll also describe the spectrum of $p$-curvature for the quantum connection, which offers a new proof of a recent result by P.Etingof and A.Varchenko. This talk is based on joint work with P. Koroteev. 

Abstract: Dirichlet related the residue at s=1 of the Dedekind zeta function of a number field F (a slight generalisation of the famous Riemann zeta function) to two important arithmetical notions: the size of the ideal class group and the `volume' of the unit group in the number ring O_F of F. Generalising this surprising connection, the special values of the Dedekind zeta function of a number field F at integer argument n should, according to Zagier's Polylogarithm Conjecture, be expressed via a determinant of F-values of the n-th polylogarithm function. Goncharov laid out a vast program incorporating this conjecture using properties of multiple polylogarithms and the structure of a motivic Lie coalgebra.

In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight,  some in joint work with, or developed by, S.Charlton, D.Radchenko as well as D.Rudenko and his collaborators. 

Abstract: The goal of this talk is to present recent developments in the study of coordinate rings of braid varieties. These are certain algebraic varieties that appear in the study of moduli spaces of Stokes local systems. The main result is that these commutative algebras of regular functions are cluster algebras. The central technique that we use is that of weaves, which allows for a number of additional results to be proven. In particular, we also discuss Donaldson-Thomas transformations in this context, and the categorical counter-parts of these cluster structures, including categorical compactifications and relative Calabi-Yau structures.

Abstract:  Higgs bundles (introduced by N. Hitchin in 1987) are pairs of holomorphic vector bundles and  holomorphic 1-forms taking values in the endomorphisms of the bundle.  The moduli space of Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory. We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration, and recalling how to realize Langlands duality through spectral data. We shall then look at  a natural

construction of families of subspaces which give different types of branes. Finally, by means of spectral data, we shall relate these subspaces to the study of 3-manifolds and surface group representations. We shall conclude with some conjectures related to Langlands duality, and some applications of geometric techniques to problems in other sciences.   

Abstract: After the work of Bădescu (1986), Manetti (1991), and Hacking (2001) on degenerations of rational surfaces, Hacking and Prokhorov (2010) classified all degenerations with only Wahl singularities of the complex projective plane. They correspond to partial Q-Gorenstein smoothings of weighted projective planes P(a^2,b^2,c^2), where (a,b,c) satisfies a^2+b^2+c^2=3abc (the Markov equation). 

In a recent joint work with Juan Pablo Zúñiga (https://arxiv.org/abs/2504.19929), we classify all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree d, for any fixed 1<=d<=9. With that purpose in mind, we introduce del Pezzo Wahl chains with markings, which define marked del Pezzo surfaces. They control all such degenerations and are in one-to-one correspondence with particular fake weighted projective planes, just like in the case of degree 9. 

I plan to introduce marked Wahl chains and their del Pezzo surfaces, and slides, which is the birational tool to understand the correspondence. I will mention some byproducts such as constraints on Wahl singularities for a given del Pezzo degree, and constructions of particular (Hacking's) exceptional collections of vector bundles, which gives a geometric proof of some recent results by Polishchuk and Rains.

 Abstract: I will talk about joint work with Ruddat and Siebert, where we prove enumerative intrinsic mirror symmetry for the log Calabi-Yau surface formed of the pair of projective space and elliptic curve. Time permitting, I will talk about some extensions of our work. 

Abstract: In the past couple of years, many components of K-moduli spaces parameterising K-polystable Fano threefolds with a smoothing in one of the 105 deformation families classified by Mori, Mukai and Iskovskikh have been explicitly described. In many cases, we now know which smooth Fano threefolds in the family are K-polystable and have some geometric information on K-(poly/ semi)stable degenerations. 

The case of prime Fano threefolds of genus 12 remains mysterious – we don’t even know which smooth prime Fano 3-folds are K-polystable (Donaldson’s conjecture). The associated component of the K-moduli space is 6-dimensional. 

In this talk, I will show that general one-nodal prime Fano threefolds of genus 12 are K-polystable. Prokhorov showed that there are 4 families of one-nodal prime Fano threefolds of genus 12 , and I will show that these 4 families correspond to 4 boundary components of the associated K-moduli. 

This is joint work with Elena Denisova.

Abstract: Hodge-Tate is a class of algebraic varieties that from many perspectives look the simplest. In particular, it contains all varieties of obvious combinatorial or representation-theoretic nature, such as spherical varieties, including all toric varieties and partial flags varieties for all simple Lie groups.

The ADE classification is a phenomenon that many kinds of “simple” or “finite” physico-mathematical objects are classified by simply-laced Dynkin diagrams: platonic solids, rotation groups, singularities, elliptic fibrations, Lie algebras, quivers, cluster algebras, 2d CFTs and 4d N=2 quiver gauge theories, to name a few. 

In a joint work in progress with Naichung Conan Leung, Changzheng Li and Rui Xiong we discovered another unexpected instance: Hodge-Tate hyperplane sections of the Grassmannians, the varieties parametrizing k-dimensional subspaces in n-dimensional vector space, are also bijectively classified by ADE Dynkin diagrams.

Time permits, I discuss new simple but powerful criterion for semi-simplicity of quantum cohomology, and some more convenient versions of the quantum Lefschetz principle.

 Abstract: Gamma conjectures were proposed by Galkin, Golyshev and Iritani, and consist of conjecture O, Gamma conjecture I and II. In this talk, we will introduce the original formulae and some counterexamples for conjecture O and Gamma conjecture I. We will also talk about some modifications for Gamma conjecture I, together with an interplay of birational transformations with an extension of Gamma conjecture I over Kahler moduli space. This is based on my joint work with Sergey Galkin, Jianxun Hu, Hiroshi Iritani, Hua-Zhong Ke and Zhitong Su.

 Abstract: the quantum connection is a basic object coming from enumerative geometry. It's fun to study in a hands-on 19th century way, as a linear differential equation, but it is relevant to both algebraic and symplectic geometry, and subject to some of the deepest conjectures in the field. I will try to introduce what questions one can ask about this object, and mention some recent progress. 

Abstract: K-theoretic Gromov--Witten (KGW) theory was introduced by Givental and Y.P. Lee as a generalization of Gromov--Witten theory. Recently, Givental realised that if we want to compute KGW invariants via fixed-point localization methods, we have to consider a more general theory, i.e., the permutation equivariant version of KGW theory.  I would like to give an introduction to this topic and to explain how to compute the invariants in genus-0 for the simplest possible target -- the point. 

Abstract: Hurwitz’s paper ”Ueber die Anzahl der Riemannischen Flächen mit gegebenen Verzweigungspunkten” (1901) started the study of the enumeration of branched coverings of the Riemann sphere. Though more than a century has passed now, there have been many recent developments in the subject that Hurwitz opened. I will explain new results and perspectives on Hurwitz numbers, Hurwitz moduli spaces, and their connections to the moduli space of curves. 

Abstract: I will describe recent work with Vivek Shende and Peng Zhou, which relates the Fukaya category of a multiplicative hypertoric variety to the Fukaya category of the associated toroidal hyperplane arrangement. I will discuss the relation with mirror symmetry for multiplicative Coulomb branches, and how this picture fits into wider conjectures on Fukaya categories of spaces with group actions. 

Abstract: Spin Gromov–Witten invariants were introduced by Kiem and Li to determine the ordinary Gromov–Witten invariants of surfaces with smooth canonical divisors. Conjecturally, these invariants can be expressed as linear combinations of spin Hurwitz numbers, which are themselves computable via representation theory—a relationship known as the spin GW/Hurwitz correspondence. In this talk, I will present a proof of the correspondence in the case of target P^1, and explain how the general case follows from a conjectural degeneration formula for spin GW invariants. Time permitting, I will also discuss new directions related to the Virasoro conjecture for such targets, as well as connections to the GW/PT correspondence.

Abstract: In this talk, we will discuss a new approach to prove irrationality of certain periods. Our method uses rational approximations from the literature and we develop a new framework to make use of these approximations. The key ingredient is an arithmetic holonomy theorem built upon earlier work by André, Bost, Charles (and others) on arithmetic algebraization theorems via Arakelov theory. The history of algebraization theorems goes back to the work of Borel, Dwork, and others. Borel and Dwork gave conditions on when a nice power series with rational number coefficients comes from a rational function in terms of meromorphic convergence radii at all places. Such a criterion was used in Dwork’s proof of the rationality of zeta functions of varieties over finite fields. Later, the work of André, Bost, Charles and many others generalized the rationality criterion of Dwork and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will give a brief discussion of the history and explain our arithmetic holonomy theorem and its applications. This is joint work with Frank Calegari and Vesselin Dimitrov.

Abstract:  Algebraic cycles on Calabi-Yau 3-folds played a starring role in demonstrating the inequivalence of algebraic and homological equivalence, opening mirror symmetry on the quintic, and describing asymptotics of local Gromov-Witten invariants.  In this talk, they play the role of a new testing ground for connections between transcendental and arithmetic geometry.

  On the transcendental side, we have the extensions of mixed Hodge structure that arise from the motivic cohomology / higher Chow / algebraic K-theory groups of a variety over Q.  Their periods define the regulators and height pairings which, for a family of varieties, produce solutions of inhomogeneous Picard-Fuchs equations.  Thinking in terms of families deforming the original variety  turns out to be the key to going beyond “classical” cases, both for constructing such cycles and computing their transcendental invariants.

The arithmetic of a variety over Q is described in part by the L-function attached to the Galois representation on its middle cohomology.  The Bloch-Beilinson conjectures, which posit a relationship between its special values and the generalized periods above, remain one of the deepest open problems in mathematics. In this talk, based on recent work with Vasily Golyshev, I’ll explain how to use hypergeometric families and Hadamard convolutions to produce new numerical evidence for these conjectures.