The moduli space of holomorphic curves with boundary on a Maslov zero Lagrangian in a 3-dimensional symplectic Calabi-Yau has codimension one walls. The wall crossings can be identified with the HOMFLYPT skein relations in the Lagrangian. This leads to invariant counts of open curves with values in the skein module of the Lagrangian boundary conditions. In this first lecture we will discuss this construction and show how it leads to a proof of the Ooguri-Vafa conjecture. The talk reports on joint work with Shende.

In this talk we outline how to construct exotic spheres systematically by considering different group actions on classical spheres, cutting invariant subsets, and pasting them back. This construction lifts well for cotangent bundles and might be helpful on topics related to symplectic geometry.

In this talk, I will review the basic properties of definable analytic varieties of Fujiki class C and explain how to endow their cohomology with functorial mixed Hodge structures. This talk is based on joint work with Ludmil Katzarkov, Ernesto Lupercio, and Laurent Meersseman.

We observe that on a compact $\mathrm{Spin}(7)$ manifold with $\nabla$-closed torsion 3-form the torsion connection $\nabla$ is a $\mathrm{Spin}(7)$ instanton if and only if the torsion 3-form is parallel with respect to the torsion connection. We show that on a compact $\mathrm{Spin}(7)$ manifold with closed torsion 3-form the torsion connection is a $\mathrm{Spin}(7)$-instanton exactly when the torsion 3-form is parallel with respect to the torsion connection.

Calabi-Yau 3-folds, i.e. complex 3-dimensional Kahler manifolds with holonomy group contained in $SU(3)$, play a central role in complex, symplectic and algebraic geometry, mathematical physics, and more. It is natural to seek extensions beyond the setting of Kähler geometry. In this talk, I will discuss compact non-Kahler Hermitian 3-folds with pluriclosed fundamental 2-form, whose Bismut-Ricci form is identically zero. These geometries appeared in the 1980’s in mathematical physics as string backgrounds extending the Kahler Calabi-Yau condition. It turns out that compact non-Kahler Hermitian manifolds with pluriclosed fundamental form and zero Bismut-Ricci form have a canonical symmetry reduction to complex co-dimension one. In complex dimension 3, the transversal geometry is actually Kähler and governed by a single 6th-order scalar PDE for the Kahler potential of the underlying Kahler metric. In the quasi-regular case, i.e. when the 3-fold fibers over a compact complex orbifold surface, the PDE has an infinite dimensional momentum map interpretation, similar to the much studied Kahler metrics of constant scalar curvature. We use this to obtain obstructions for the existence of solutions in terms of the automorphism group, paralleling results by Futaki and Calabi-Lichnerowicz–Matsushima in the cscK case. As an application, we characterize the standard Samelson’s examples, i.e. the Hermitian 3-folds which are covered by either $SU(2)\times {\mathbb{R}}^3$ or $SU(2)\times SU(2)$ equipped with a bi-invariant metric and a left invariant complex structure, as the only regular examples having Bott-Chern number $h_{BC}^{1,1}=2$. We also obtain explicit solutions of the PDE on orthotoric Kähler orbifold surfaces, which yield infinitely many non-Kahler 3-folds with pluriclosed fundamental form and zero Bismut-Ricci form on $S^3\times S^3$ and $S^1\times S^2\times S^3$, which are not locally isometric to a Samelson geometry. These appear to be the first such examples. This talk is based on joint works with Barbaro, Lee and Streets, and Lahdili and Lee, available on arXiv:2408.09648 and arXiv:2601.04937.

In this talk we present the notion of Gromov-Hausdorff distance, a field of study that aims to understand the topology of the moduli space of Riemannian metrics on a given manifold. We will then present new results on the subject concerning the collapse of principal bundles onto reductive homogeneous spaces.

In this talk we use the procedure outlined in the first talk to study Spherical T-duliaty between spheres and exotic spheres. Joint work with L. Grama and L. Katzarkov.

We associate to a Lagrangian in a cotangent bundle of a 3-manifold that branch covers the zero section a 7-manifold with a $G_2$ structure that fibers over the cotangent bundle with fibers that degenerate over the Lagrangian. We find a positive 3-form with the property that invariant associatives in the 7-manifold are unions of fibers over holomorphic curves in the cotangent bundle with boundary in the Lagrangian and the Lagrangian itself. There is in particular an invariant count of such objects with values in the skein of the Lagrangian. We show that in the limit when the Lagrangian collapses to the zero section the holomorphic curves converge to flow graphs. Viewing the 7-manifold as a fibration with $A_n$ surface fibers this gives a direct analogue of Donaldson-Scaduto flow trees for adiabatic K3-fibrations. The talk reports on joint work with Esfahani, Shende, and Wang.

I will discuss a conjectural limit of the local geometric local Langlands correspondence which identifies the category of coherent sheaves on the moduli of Higgs bundles on a formal disk with the category of coherent sheaves on the moduli of framed Higgs bundles on the formal disk. As in the global case, the correspondence should intertwine Willson and Hecke symmetries. I will explain how the appropriate symmetries can be constructed and matched by utilizing Fedorov’s mixed Hodge module refinement of the geometric Satake correspondence. I will also discuss how a version of Contou-Carrére duality leads to a construction of the Dolbeault limit of the local Langlands correspondence over the open part of the moduli parametrizing Higgs bundles or framed Higgs bundles with smooth cameral covers. This is a joint work with Gurbir Dhillon and Roman Fedorov.

I will outline work in progress on an algebraic approach to deriving a multiplicative blowup formula for the quantum Chen-Ruan cohomology of global quotient projective orbifolds.

In topology, one studies the boundary at infinity of a nice topological space in terms of the limit of complements of compact subsets. In algebraic geometry, there are at least two different approaches to defining the boundary at infinity: via some form of rigid analytic geometry (for example, solid algebraic geometry of Clausen-Scholze), and another via noncommutative geometry of differential graded categories, due to Efimov. We give a unified construction of the boundary at infinity compatible with both the analytic and noncommutative points of view. We motivate our construction by focusing on an explicit example that can be viewed from both topological, symplectic, and algebraic points of view, namely that of local systems on the circle. This is joint work in progress with Yuan Gao and Yingdi Qin.

I will report on the classification of surfaces of general type with the lowest possible invariant $\chi =1$, which means $q=p_g$. If $q=p_g$ then we have the upper bound 4, which holds exactly for a product of 2 curves of genus 2, and the case $q=p_g=3$ was settled some 25 years ago due to work of several authors, Ciliberto, myself, Mendes Lopes, Pirola, Pardini and Hacon. Still open is the case $q=p_g=2$, which was considered by several authors. Here $4\le K^2\le 9$, and the case $K^2=4,5,6$ was recently settled by Du, Jiang, Zhang; there are many examples for $K^2=8$, and the case $K^2=9$ is suspected not to happen (but the published proofs of this assertion are badly wrong).In the case $K^2=7$ there are two known examples, where the Albanese map has degree 2, respectively 3. I will report on joint work in progress with Matteo Penegini, in the case of degree 3. Theorem. If $K^2=7$, $d=3$, we have only one irreducible family, consisting the CFPP surfaces (some surfaces of the family were discovered by Cancian-Frapporti, and a larger family was constructed by Pignatelli and Polizzi).

Two holomorphic invariants will be defined for pairs of linking algebraic cycles on a smooth projective variety. One of these is an infinitesimal bi-invariant and the other identifies the bi-extension line bundle with the classical notion of incidence equivalence of cycles. Three types of linking invariants: arithmetic (height), topological (monodromy), and analytic (Hodge theoretic) will be defined their relation briefly discussed.

We review some old results about Torelli theorems for projective hypersurfaces, weighted hypersurfaces, and elliptic surfaces, and discuss some new results for elliptic fibrations.

We review the motivations for the study of the moduli spaces of Riemann surfaces together with a meromorphic 1-form, and then discuss some recent developments in understanding the geometry of these spaces and their compactifications.

We prove the existence of Lagrangian torus fibrations on compact Calabi-Yau hypersurfaces in toric Fano manifolds associated with reflexive polytopes in all dimensions. The base is diffeomorphic to a sphere, the discriminant in the base has real codimension two, all singular fibers are half-dimensional Lagrangian skeleta, the critical set has real codimension four in the hypersurface, and the fibration is a smooth torus bundle away from the discriminant. The Arnold-Liouville affine structure is the one expected from the SYZ conjecture. As a byproduct we obtain two Lagrangian sections whose difference generates the symplectic monodromy of certain one-parameter families determined by a divisor on the mirror dual whose holomorphic Euler number matches the intersection number of the Lagrangians. This is joint work with Cheuk Yu Mak, Diego Matessi, and Ilia Zharkov.

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.

The integration of AI and Machine Learning techniques has had a transformative effect in scientific discovery, with applications ranging from numerical computations to complex abstract reasoning. In the talk we discuss some recent advances with a specific focus on applications within research mathematics. We survey several recent use cases, examining the deployment of AI agents and specialized ML algorithms across various problems. Building on these examples, we reflect on how such tools could empower researchers and enable new modes of collaborative efforts.

I will propose a notion of Hermitian structures on complexes of holomorphic vector bundles, and compatible (Chern) connections.

In this talk, we provide necessary and sufficient conditions for the existence of non-Kähler SYZ mirror pairs, in the sense of Lau, Tseng, and Yau, for solvmanifolds, following a construction proposed by Bedulli and Vannini. We further investigate the relationship between the cohomology of these pairs and, time permitting, propose new cohomological frameworks tailored to this setting. This is joint work in progress with L. Cavenaghi, L. Grama, and L. Katzarkov.

Einstein metrics on complex projective spaces provide a natural setting for exploring the interplay between curvature and symmetry in Riemannian geometry. In this talk, we study Einstein metrics that are invariant under cohomogeneity one actions of compact connected Lie groups, under the assumption that the singular orbits are totally geodesic. Such actions were classified by Takagi into five distinct models. We discuss the resulting system of ordinary differential equations arising from the Einstein condition and analyze the smoothness requirements imposed by the geometry of the singular orbits. This is a joint work with Anderson Araujo and Brian Grajales.

I will review first an older scaling limit theorem realizing the class of simplistically extremal marked tropical analytic curves as the zero-mesh limit of deviation loci of several-point perturbations of the maximal stable state on planar convex domains established jointly with Nikita Kalinin. Then, we will pass to the novel construction which starts with the result of the first scaling limit by sending the number of perturbation points, chosen independently at random according to a fixed probability measure with support isolated from the boundary of the underlying convex domain, to infinity. After the rescaling by the square root of the number of perturbation points of the tropical series defining the deviation tropical curve, one gets a deterministic limit solving the Monge-Ampère equation with the source given by the perturbation density and the Dirichlet boundary condition, which was established very recently, both theoretically and experimentaly, in the joint work with Nikita Kalinin, Higinio Serrano, and Ernesto Lupercio. The talk doesn’t assume any prior knowledge of Abelian or Tropical sandpiles, the essential basics of which will be covered.

Classical toric geometry transforms rational combinatorial data into algebraic and symplectic geometry. LVM and LVMB manifolds reveal a wider toric world. These compact complex manifolds, often non-Kähler, carry canonical holomorphic foliations whose transverse geometry is encoded by calibrated fans. Their leaf stacks provide natural models for quantum toric spaces, encompassing classical toric varieties and stacks while also accommodating irrational and noncommutative directions.

In this survey, I will present this theory from our point of view, beginning with Gale duality and the LVM construction, passing through canonical foliations and quantum toric stacks, and arriving at the motivic geometry of nonalgebraic compact complex manifolds. A recurring theme is the interplay between the visible polytope, which controls much of the topology, and the complex weights invisible to that polytope, which govern Dolbeault and genuinely non-Kähler phenomena. Hodge polynomials and related motivic invariants make this interaction computable and reveal structures that ordinary topological invariants cannot detect.

The resulting picture brings together toric combinatorics, complex foliations, Hodge theory, and motives as parts of a single extension of toric geometry beyond the Kähler setting.

This is joint work with Ludmil Katzarkov, Kyoung-Seog Lee, and Laurent Meersseman.