Moduli space constructions of birational morphisms
This series of lectures will focus on application of Category Theory to Birational Geometry. We will discuss how to construct a class of birational morphisms as moduli spaces of objects in the derived categories of coherent sheaves on algebraic varieties.
One has to consider ‘perverse’ t-structures and suitably defined moduli of ‘point’ objects in them. We will discuss the relevant t-structures and their properties, in particular, how they can be defined by (relative) tilting objects. We will also discuss the conditions on point objects that allow to construct the required moduli spaces.
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.
Rationalities and Galois symmetries in open-closed Gromov-Witten Floer theory
TBA
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.
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.
Deligne's conjecture after Lurie
We will present an introduction to the strategy used by Jacob Lurie in his proof of Deligne's conjecture and to some of the tools from higher category theory which he relied on. If time permits, we will also give a glimpse of Brav-Rozenblyum's recent results on the cyclic Deligne conjecture.
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.