Program

Pieter Belmans

Title: Fano 4-folds from subspace quiver moduli

Abstract: The classification of Fano 4-folds is still largely open. Whilst moduli of stable quiver representations are varieties with very special properties, it turns out that subspace quivers give rise to four interesting Fano 4-folds (2 of which appear to be new) which are rigid, have no vector fields, have high Picard rank, and have an interesting structure from an MMP perspective. The tools for studying moduli of quiver representations work particularly well for these examples, and I will describe their intricate geometry. This is joint work with Markus Reineke.

Alessandro Chiodo

Title: Forgetting (Ramond, Ramond)
Abstract: Spin structures are square roots of the canonical bundle, or theta characteristics. On curves with stack-theoretic structure, marked points and nodes may carry trivial or nontrivial local characters; these give the two local sectors, Ramond and Neveu-Schwarz. The NS sector has led to Norbury’s Theta classes, recursive formulae, and the Brézin-Gross-Witten tau function. Ramond insertions are in some sense simpler, but also more elusive: insertions with psi-classes often vanish, while a single Ramond marking cannot be forgotten within the spin theory.

Recent work shows that Ramond insertions nevertheless produce the generalized BGW deformation and super Weil-Petersson volumes with Ramond markings. I will explain the genus-zero geometry behind the basic operation that remains available: forgetting a pair of Ramond markings. We establish a K-theoretic forgetful formula with vertical boundary corrections. After symmetrization, these corrections collapse to a simple expression involving cotangent-line classes and Ramond-node divisors. This provides a local geometric mechanism for the forgetful functor, as well as a cohomological formula for (Ramond, Ramond) pairs.

Joint work with Paul Norbury.

Giordano Cotti

Title: Enumerative geometry of flag varieties and prime numbers

Abstract: Enumerative geometry, as formulated in Gromov--Witten theory, encodes curve-counting information on smooth projective varieties. Such data can be organized in different ways, giving rise to rich geometric structures and invariants, including quantum cohomology and quantum spectra. In the work G.Cotti, ``Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers,'' IMRN, 2022, an unexpected connection was observed between the quantum cohomology of Grassmannians and the distribution of prime numbers. In this talk, I will present recent progress extending this perspective to the enumerative geometry of more general partial flag varieties, highlighting how the relation with prime numbers persists in a broader setting.

Luca Giovenzana

Title: Boomerangs, elliptic curves and del Pezzo surfaces

Abstract: We introduce boomerangs in the derived category of an elliptic curve C, defined as filtrations of the zero object whose factors are polystable objects with increasing phase. The numerical invariants of a boomerang are the Chern characters of the direct summands of its factors, which determine a lattice polygon. When this polygon is reflexive, we show that the moduli space of boomerangs with prescribed polystable factors is the complement of the anti-canonical embedding of C inside a del Pezzo surface Z. The proof relies on exceptional collections on Z. This is joint work in progress with Pierrick Bousseau and Tom Bridgeland.

Jérémy Guéré

Title: Rational Cubic Fourfolds and Their Relation to K3 Surfaces

Abstract: I will first review the construction of atoms introduced by Katzarkov--Kontsevich--Pantev--Yu, specifically addressing the behavior of Hodge structures under Iritani's blow-up formula. Then, I will introduce a new atomic invariant to prove the following theorem: if a smooth complex cubic fourfold is rational, then its primitive cohomology is isomorphic - as a rational Hodge structure - to the shifted middle cohomology of a projective K3 surface.

Giulia Gugiatti

Title: Kuznetsov components in mirror symmetry

Abstract: In this talk I will explore how the structural features of Kuznetsov components of Fano varieties are reflected in the geometry of their mirror Landau-Ginzburg models. I will focus on Fano varieties arising as branched covers along Calabi-Yau hypersurfaces and propose a mirror description of their Kuznetsov component. I will discuss the case of the quartic double solid in detail. The talk is based on work in progress with Ilaria Di Dedda, Danil Kozevnikov, and Nick Sheridan.

Tyler Kelly

Title: Open Mirror Symmetry for Landau-Ginzburg Models

Abstract: I will describe an open enumerative theory for certain Landau-Ginzburg models. Its intersection numbers are constructed by taking certain multivariate integrals of a multisection of an open analogue of Witten’s bundle over a moduli space of orbidisks. This so-called open FJRW theory satisfies mirror symmetry. If time permits, I will explain the B-model. This is joint work with Mark Gross and Ran Tessler.

Chunyi Li

Title: Bridgeland stability conditions on projective varieties

Abstract: Bridgeland stability condition on a projective variety can be viewed as a generalized slope stability on curves to higher-dimensional varieties in a more unified and robust way, combining advantages of both slope and Gieseker stability. A central question in the theory has been to determine which varieties admit Bridgeland stability conditions. In this talk, I will discuss recent progress on the existence of stability conditions on all projective varieties.

Antonios-Alexandros Robotis

Title: Augmented stability conditions and the NMMP

Abstract:  I will survey recent work explaining how certain paths in the space of Bridgeland stability conditions Stab(X) associated to a variety X give rise to semiorthogonal decompositions of its derived category D(X). The basic mechanism is the notion of a quasi-convergent path in Stab(X), which can be interpreted as a convergent path in a certain partial compactification of Stab(X)/C, called the space of augmented stability conditions AStab(X). I will explain some expectations of how one can obtain canonical decompositions of D(X) by studying topological properties of AStab(X).

This is based on joint works with Daniel Halpern-Leistner, Jeffrey Jiang, Tomohiro Karube, and Vanja Zuliani.

Mark Shoemaker

Title: A topological Chern character for matrix factorizations

Abstract: The Grothendieck—Riemann—Roch theorem famously relates, for a map f: X —> X’ between varieties, the induced pushforwards in K-theory and cohomology.  The goal of this talk is to generalize this story to the setting of Landau—Ginburg models, which consist, roughly, of a space together with a function.

Let Y be a smooth quasi-projective complex variety and w: Y —> \CC a regular function.  Associated to the pair (Y, w) is the category MF(Y, w) of matrix factorizations of w, whose objects are “twisted" complexes of vector bundles, where the square of the differential is equal to multiplication by w.  Let Y_- = Re(w)^{-1}(-\infy, 0) denote the set of points in Y such that the real part of w(y) is negative.  I will construct a Chern character from the Grothendieck group of MF(Y, w) to the relative cohomology H*(Y, Y_-), and explain how a Grothendieck—Riemann—Roch type theorem can be deduced.  If time permits I will describe an application to enumerative geometry.

Jacopo Stoppa

Title: A toric case of the Thomas-Yau conjecture

Abstract: We consider a class of Lagrangian sections L contained in certain

Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce.

Tony Yue Yu

Title: F-bundles, mirror symmetry and birational invariants

Abstract: There are two main themes in algebraic geometry research: classification and enumeration. In this talk, we will bring them together by applying enumeration to birational classification, more precisely, we will construct new birational invariants from curve counting invariants. I will introduce F-bundles, the spectral decomposition theorem and the equivariant unfolding theorem. I will discuss applications to quantum cohomology, mirror symmetry and birational geometry, in particular, to the proof of irrationality of very general cubic fourfolds. The talk is based on arXiv preprints 2411.02266, 2505.09950, 2508.05105 and 2508.15770.