Kyoto University, Research Institute for Mathematical Sciences, fourth floor, room 420
| Monday 13 | Tuesday 14 | Wednesday 15 | Thursday 16 | Friday 17 | ||
|---|---|---|---|---|---|---|
| 09:30 | Nakajima | Fantechi | Fukaya | Keller | Fukaya | 09:30 |
| 10:30 | tea break | tea break | tea break | tea break | tea break | 10:30 |
| 11:00 | Yong-Geun Oh | Yong-Geun Oh | Nakajima | Fukaya | Keller | 11:00 |
| 12:00 | lunch break | lunch break | lunch break | 12:00 | ||
| 14:00 | Fantechi | Fantechi | Keller | 14:00 | ||
| 15:00 | tea break | 15:00 | ||||
| 15:30 | Nakajima | 15:30 |
| Monday 20 | Tuesday 21 | Wednesday 22 | Thursday 23 | Friday 24 | ||
|---|---|---|---|---|---|---|
| 09:30 | Kapranov | Göttsche | Mochizuki | Toda | Jeongseok Oh | 09:30 |
| 10:30 | tea break | tea break | tea break | tea break | tea break | 10:30 |
| 11:00 | Vasserot | Ohta | Polishchuk | Khan | Bryan | 11:00 |
| 12:00 | lunch break | lunch break | lunch break | 12:00 | ||
| 14:00 | Bandiera | poster session | Fiorenza | 14:00 | ||
| 15:00 | tea break | in room 110 | tea break | 15:00 | ||
| 15:30 | Iacono | Lee | 15:30 |
L-infinity liftings of semiregularity maps
After reviewing the approach to deformation theory via DG Lie algebras, we will consider deformations of a coherent sheaf F on a complex manifold X: these are controlled by the DG Lie algebra of derived endomorphisms of F, and in particular the obstructions live in the space Ext^2(F,F). In this situation, Buchweitz and Flenner introduced a family of linear maps, which they call semi-regularity maps, from the obstruction space Ext^2(F,F) with values in the cohomology of X, and showed that they vanish on certain special obstructions called simple. In a joint work with Emma Lepri and Marco Manetti, we showed that semi-regularity maps can be lifted to L-infinity morphism from the controlling DG Lie algebra of derived endomorphisms to an abelian target, and therefore vanish on all obstructions.
The Pardon algebra of 1-cycles on Calabi-Yau orbifolds and the Gromov-Witten Crepant Resolution conjecture
In 2023, John Pardon defined an algebra of “curve enumeration problems” on smooth threefolds and its dual algebra of “curve enumeration theories”. He showed that his algebra is freely generated by “equivariant local curves” which he then used to prove the famous Gromov-Witten/Donaldson-Thomas correspondence. We define a version of Pardon’s algebra for orbifold Calabi-Yau threefolds (of type A), and we show this algebra is generated by equivariant local orbifold curves. As a corollary, we prove the Gromov-Witten crepant resolution conjecture for local orbifold surfaces. This is work in progress with Felix Thimm.
Deformations of singular varieties
Lecture 1: After briefly reviewing functors over Artinian algebras (over a fixed algebraically closed field) we focus on deformations of singular varieties, and in particular those which have lci singularities.
Lecture 2: We review Tziolas’s result on formal smoothability, describe applications to smoothening of stable surfaces (joint work with Franciosi and Pardini) and a similar, independent smoothability result (joint with Miró Roig).
Lecture 3: We relate what done so far with algebraic stacks and complex analytic geometry; if time allows we discuss work in progress with Siao Chi Mok on the stack of Fulton-MacPherson expansions.
Homotopy fibers in classical algebraic geometry
Homotopy fibers, usually regarded as genuinely derived objects, are in fact already implicit in classical algebraic geometry: familiar spaces such as projective spaces, Grassmannians, and flag varieties all arise as homotopy fibers of natural morphisms between moduli stacks of principal bundles. This perspective suggests that many classical parameter spaces arise from universal constructions that are naturally homotopical. In characteristic zero, the equivalence between formal moduli problems and differential graded Lie algebras provides a bridge to describe the infinitesimal geometry of moduli spaces presented as homotopy fibers through the cohomology of mapping cones of dgla morphisms. Several examples from the works of Bandiera, Iacono, Lepri, Manetti, Martinengo, Meazzini, and myself will be presented.
Rationalities and Galois symmetries in open-closed Gromov-Witten Floer theory
In this talk, I will discuss some foundational issues of Gromov-Witten-Floer theory over the field of rational numbers $\mathbb{Q}$.
In joint work with Abouzaid, Oh, Ohta, and Ono (to appear soon on arXiv), we have constructed a foundation for this theory based on de Rham theory. A key development in this work is the establishment of the cyclic symmetry of the operations. I will explain why cyclic symmetry becomes a more significant challenge when working with chain models other than de Rham theory.
I will also address a less technical but fundamental question: what it means for the theory to be defined over $\mathbb{Q}$. There are important cases where a specific object—such as a Lagrangian submanifold equipped with a bounding cochain—is not itself defined over $\mathbb{Q}$. Nevertheless, we can employ ‘Galois symmetry’ to show that the global structure remains defined over $\mathbb{Q}$. This implies, for instance, that the closed-open and open-closed maps are defined over $\mathbb{Q}$. I expect these results to be related to some Hodge-theoretic aspects of mirror symmetry, where the $\mathbb{Q}$-structure plays a central role in the study of Hodge theory.
Computations and symmetries of Vafa-Witten invariants
Vafa-Witten invariants are invariants of algebraic surfaces defined using moduli of Higgs bundles. Based on the physics paper of Vafa and Witten, they were given a mathematical definition by Tanaka and Thomas. We will review these invariants and then explain two generalizations: (1) invariants with $\mu$-classes, which interpolate between the Donaldson invariants and the Vafa-Witten invariants, and (2) Vafa-Witten invariants twisted by a line bundle.
The connection of these invariants conjecturally allows to compute them in many cases in terms of modular functions and uncovers remarkable symmetries.
Deformations of morphisms of coherent sheaves
In this talk, we devote our attention to infinitesimal deformations of morphisms of coherent sheaves over a smooth projective variety. In particular, we describe a differential graded Lie algebra controlling these deformations over any algebraically closed field of characteristic zero. This is based on a collaboration in progress with Emma Lepri and Elena Martinengo.
Virtual classes of shifted Lagrangians
The virtual fundamental class is the key tool for extracting enumerative invariants from quasi-smooth moduli spaces. Some enumerative invariants, such as those arising in DT4 theory and GLSM, require virtual classes beyond the quasi-smooth setting. I will explain how these can be understood as instances of a conjecture of Joyce, which predicts virtual invariants associated to (-1)-shifted Lagrangians in shifted symplectic geometry. This also allows one to define other algebraic structures, such as a Fukaya-type category of algebraic Lagrangians, and 3-dimensional cohomological Hall algebras. I will sketch a proof of Joyce’s conjecture, based on joint work in progress with Tasuki Kinjo, Hyeonjun Park, and Pavel Safronov.
Supersymmetry, differential operators of infinite order and theta-functions
Differential operators of infinite order (DOI) are infinite series in derivatives with holomorphic coefficients decaying so fast that the action on holomorphic functions converges and preserves the domain of definition. Thus $\exp(d/dx)$ (shift operator) is not a DOI but $\cos(\sqrt{d/dx})$ is.
Starting from 1973, Sato, Kashiwara, Kawai, Takei, Yoshida and others developed a characterization of theta-zerovalues by manifestly modular invariant systems of DOI in the modular variables alone, thus deducing modularity from local conditions.
I will present a “supersymmetric” interpretation of this theory based on a natural super-thickening of the Lagrangian Grassmanian and two observations:
(1) Any odd supersymmetry generator $D$ has order $1/2$ in a natural sense and so $e^D$ is a DOI.
(2) In some cases such odd generators, acting “on-shell” (in the space of solutions of equations of motion) satisfy even-style commutation relations.
Cluster algebras, Higgs categories and braid group actions
Cluster algebras are certain commutative algebras with a very rich combinatorial structure. They were invented by Sergey Fomin and Andrei Zelevinsky in 2002, who were motivated by problems from Lie theory, in particular the study of Lusztig’s canonical bases and his notion of total positivity. Since then, it has turned out that cluster algebras also appear in a wide range of other subjects, from representation theory to mathematical physics, mirror symmetry and symplectic topology. Among the many methods developed to study cluster algebras is “additive categorification”, where the combinatorics of cluster algebras are lifted to the level certain (stably) 2-Calabi-Yau categories. In the case of cluster algebras with coefficients, these are the Higgs categories recently introduced by Yilin Wu. In this minicourse, after a historical survey and a basic introduction to cluster algebras, we will present the framework of additive categorification via Higgs categories and then study classes of examples appearing in Lie theory and in higher Teichmuller theory as approached by Fock-Goncharov, Jiarui Fei, Ian Le, … and Goncharov-Shen. In these examples, braid group actions play an important role and we will show how the presence of coefficients yields a natural way of lifting them to the categorical level. The results we will present were obtained in several joint projects notably involving, in chronological order, Chris Fraser, Yilin Wu, Alessandro Contu, Miantao Liu, Haoyu Wang and Xiaofa Chen.
Quantum Elliptic Cohomology
This is a progress report outlining an ongoing study of quantum elliptic cohomology, conducted in collaboration with Emile Bouaziz and Irit Huq-Kuruvilla.
Boundedness of meromorphic flat bundles with bounded irregularity
Boundedness is an important notion related to moduli problems in algebraic geometry. Roughly speaking, we call a family of objects bounded if it is a part of a bigger family algebraically parameterized by a variety. It is much weaker than the representability of the moduli space. However, for an algebraic operation, we may divide a bounded family into finite subfamilies, and members in each family behave similarly with respect to the operation. In this sense, bounded families are well controlled. For example, in the classical moduli problem of semistable bundles, the boundedness is important as a preliminary to the GIT construction.
In this talk, we shall discuss the boundedness of meromorphic flat bundles with bounded irregularity. By non-abelian Hodge theory, we may attach a meromorphic Lagrangian cover to a meromorphic flat bundle, which plays an important role in our discussion.
Coulomb branches and Relative Langlands
In the first talk, I will explain a mathematically rigorous definition of Coulomb branches of 3d SUSY gauge theories given jointly with Braverman and Finkelberg, and then explain a definition of S-dual Hamiltonian spaces, as its variant. In the second and third talks, I will explain the relevance of S-dual in relative Langlands by Ben-Zvi, Sakellaridis, and Venkatesh.
Virtual cycles via Fulton classes
The virtual cycle of a quasi-smooth scheme coincides with that of its (−2)-shifted cotangent bundle. For a derived scheme whose tangent complex has three terms – which may be regarded as the mildest non-quasi-smooth case – we define the equivariant virtual cycle using the Fulton class. We then prove the analogous result in the equivariant setting.
A non-equivariant cycle can also be defined, but it need not coincide with that of the (−2)-shifted cotangent bundle. We explain how the two differ.
Presymplectic stratifications of generic closed two-forms and stratificed L-infinity spaces
Motivated by the question on ‘defining a symplectic form’ on the Gromov-Witten moduli spaces of symplectic manifolds, we prove that there exists a residual subset of closed 2-forms such that any element therefrom admits a Whitney stratification each of whose strata is a presymplectic manifold. We then associate an L-infinity space to each stratum (and to its tubular neighborhood) and glue the collection of L-infinity spaces to a global stratified L-infinity space by the coordinate atlas consisting of L-infinity morphisms, which is a collection of L-infinity morphisms, not necessarily of quasi-isomorphisms. This is a joint work with Taesu Kim.
Stokes curves, adiabatic limit and Lagrangian Floer theory
The WKB analysis is asymptotic analysis for solutions of certain ODE on a Riemann surface with a parameter. The Stokes curves on the Riemann surface play an important role in the WKB analysis. In this talk, I will discuss some aspect of the Stokes curves from the point of view of Lagrangian Floer theory. This is based on joint work with Tatsuki Kuwagaki.
Schwartz spaces and L2-stacks
This talk is based on joint works with Alexander Braverman and David Kazhdan. The motivation comes from the Analytic Langlands program, which is an analog of the usual Langlands correspondence involving a curve over a local field. Given an algebraic stack over a local nonarchimedean field, one can define natural (twisted) Schwartz spaces.
In the case of the Schwartz space of half-densities, there is a natural proposal for a hermitian form on this space, which converges for the class of stacks we call L2-stacks.
I will discuss some examples including the stack of rank 2 bundles on the projective line with parabolic points.
The Dolbeault geometric Langlands conjecture via limit categories
I will propose a precise formulation of the Dolbeault geometric Langlands conjecture, introduced by Donagi–Pantev as the classical limit of the (de Rham) geometric Langlands correspondence. It asserts an equivalence between certain derived categories of coherent sheaves on moduli stacks of Higgs bundles on a smooth projective curve.
On the automorphic side, I introduce limit categories, which may be viewed as classical limits of categories of D-modules on moduli stacks of bundles over curves. Their definition is based on noncommutative resolutions due to Špenko–Van den Bergh and on magic windows in the sense of Halpern-Leistner–Sam. Our formulation states an equivalence between the derived categories of moduli stacks of semistable Higgs bundles and the limit categories associated with moduli stacks of all Higgs bundles. I show that the (ind-)limit categories are compactly generated, admit Hecke operators, and carry semiorthogonal decompositions into quasi-BPS categories, categorifying BPS invariants in Donaldson–Thomas theory (joint work with Tudor Pădurariu, arXiv:2508.19624).
Finally, I will explain a proof of this equivalence for GL_2 over the locus of the Hitchin base where spectral curves are reduced. This provides the first nontrivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential both for the formulation and for the proof.
BPS Lie algebras and $\chi$-independence for symplectic surfaces
We prove the $\chi$-independence conjecture of Toda for the BPS cohomology of quasiprojective symplectic surfaces, relative to the Chow variety. We do this by constructing an action by Hecke operators of the cohomological Hall algebra of zero-dimensional sheaves.