ICTS

Logarithmic Gromov--Witten (log GW) theory is about counting curves in a smooth projective variety (or closed symplectic manifold) which have prescribed tangency orders with a given simple normal crossings divisor. The foundations of algebraic log GW theory were established by Abramovich--Chen--Gross--Siebert (2013-14). Compactified moduli spaces suitable for symplectic log GW theory were found by Farajzadeh-Tehrani (2022). I will discuss work-in-progress (joint with M. Farajzadeh-Tehrani) which completes the foundations of symplectic log GW theory by constructing virtual fundamental classes on these compactified moduli spaces.

I will discuss ideas towards a categorical analogue of Kähler geometry, in which Bridgeland stability conditions play the role of Kähler classes and admit refinements involving metric structures and flows. Examples and applications will be presented. This is based on joint work in progress with F. Haiden, L. Katzarkov, and M. Kontsevich.    

In this talk I will report on joint work with Hennion extending the contractibility of the stack parametrizing  local Darboux models for (-1) symplectic derived Artin stacks, extending previous results with Hennion and Holstein for schemes and DM stacks Applications will be discussed.

I will explain how constructions of 2d, 3d, and 4d ADHM quiver representations can be recovered via Lagrangian Floer theory. The main ingredients are formal deformations of Lagrangian Floer cohomology and homological mirror symmetry for local SYZ fibrations. This talk is based on joint work in progress with Lau and Tan.

The correspondence between spaces on the one side, and algebraic structures on the other side,  is a very general mathematical phenomenon. The purpose of this series of lecture is to revisit this correspondence from the modern point of view of derived algebraic geometry and to present some of its recent developments.

In this first lecture, I will mainly focus on some of the historical aspects of this correspondence, starting from the seminal works of Tannaka and Gelfand,  concerning the reconstruction of a compact topological group in terms of its linear representations, and the reconstruction of a compact space in terms of its ring of continuous functions.  I will explain how Grothendieck reinterpreted and pursued these original ideas using the language of categories, and how this has led him to a new vision: a striking correspondance between spaces and categories. This notably drove him to his famous theory of motives, and to envision his program on infinity-categories and infinity-stacks as a general context to express general duality statements. In the final part of the talk, I will briefly mention some modern incarnations of the duality spaces categories: the first one concerning moduli of sheaves and enumerative geometry, and a second one concerning algebraic models of homotopy types. These two examples, as well as others,  will be discussed and explained in more detail in the following lectures of the series.

Moduli spaces of stable sheaves on K3 surfaces are holomorphic symplectic varieties, which include the Hilbert schemes of points on such surfaces. Addington showed that the latter carry certain exotic derived symmetries called P-twists, and conjectured that all such moduli spaces of sheaves admit these P-twists. In this talk, we describe a large family of examples of moduli spaces of sheaves on K3s and P-twists on them. We also discuss another related conjecture. This is joint work with Eyal Markman.

The correspondence between spaces on the one side, and algebraic structures on the other side,  is a very general mathematical phenomenon. The purpose of this series of lecture is to revisit this correspondence from the modern point of view of derived algebraic geometry and to present some of its recent developments.

In this first lecture, I will mainly focus on some of the historical aspects of this correspondence, starting from the seminal works of Tannaka and Gelfand,  concerning the reconstruction of a compact topological group in terms of its linear representations, and the reconstruction of a compact space in terms of its ring of continuous functions.  I will explain how Grothendieck reinterpreted and pursued these original ideas using the language of categories, and how this has led him to a new vision: a striking correspondance between spaces and categories. This notably drove him to his famous theory of motives, and to envision his program on infinity-categories and infinity-stacks as a general context to express general duality statements. In the final part of the talk, I will briefly mention some modern incarnations of the duality spaces categories: the first one concerning moduli of sheaves and enumerative geometry, and a second one concerning algebraic models of homotopy types. These two examples, as well as others,  will be discussed and explained in more detail in the following lectures of the series.

The problem of deriving a formula for the double ramification (DR) class, originally proposed in symplectic topology in 2001, was resolved by Janda, Pandharipande, Pixton, and Zvonkine in 2017. In this joint work with K. Armini, Y.-C. Chou, L. Herr, D. Holmes, and I. Huq-Kuruvilla, we introduce and study a $K$-theoretic version of logarithmic DR classes. We show that while these classes satisfy many expected properties, they also exhibit a few (to us) unexpected features.

The correspondence between spaces on the one side, and algebraic structures on the other side,  is a very general mathematical phenomenon. The purpose of this series of lecture is to revisit this correspondence from the modern point of view of derived algebraic geometry and to present some of its recent developments.

In this first lecture, I will mainly focus on some of the historical aspects of this correspondence, starting from the seminal works of Tannaka and Gelfand,  concerning the reconstruction of a compact topological group in terms of its linear representations, and the reconstruction of a compact space in terms of its ring of continuous functions.  I will explain how Grothendieck reinterpreted and pursued these original ideas using the language of categories, and how this has led him to a new vision: a striking correspondance between spaces and categories. This notably drove him to his famous theory of motives, and to envision his program on infinity-categories and infinity-stacks as a general context to express general duality statements. In the final part of the talk, I will briefly mention some modern incarnations of the duality spaces categories: the first one concerning moduli of sheaves and enumerative geometry, and a second one concerning algebraic models of homotopy types. These two examples, as well as others,  will be discussed and explained in more detail in the following lectures of the series.

On projective space, every coherent sheaf admits a resolution by line bundles. This is implied by a theorem of Beilinson which shows that the derived category of projective space has a full strong exceptional collection of line bundles. King’s Conjecture stated that Beilinson’s theorem should generalize to smooth projective toric varieties. This was false and remained false after various iterations. I will describe a new modification of King’s conjecture which holds and implies that every coherent sheaf on a smooth projective toric variety admits a resolution by line bundles. This is based on joint works with Ballard, Brown, Berkesch, Cranton-Heller, Erman, Ganatra, Hanlon, Huang, and Sapronov.

The correspondence between spaces on the one side, and algebraic structures on the other side,  is a very general mathematical phenomenon. The purpose of this series of lecture is to revisit this correspondence from the modern point of view of derived algebraic geometry and to present some of its recent developments.

In this first lecture, I will mainly focus on some of the historical aspects of this correspondence, starting from the seminal works of Tannaka and Gelfand,  concerning the reconstruction of a compact topological group in terms of its linear representations, and the reconstruction of a compact space in terms of its ring of continuous functions.  I will explain how Grothendieck reinterpreted and pursued these original ideas using the language of categories, and how this has led him to a new vision: a striking correspondance between spaces and categories. This notably drove him to his famous theory of motives, and to envision his program on infinity-categories and infinity-stacks as a general context to express general duality statements. In the final part of the talk, I will briefly mention some modern incarnations of the duality spaces categories: the first one concerning moduli of sheaves and enumerative geometry, and a second one concerning algebraic models of homotopy types. These two examples, as well as others,  will be discussed and explained in more detail in the following lectures of the series.

In this talk, I will explain how the categorification of Donaldson-Thomas theory leads to a new formulation of the Dolbeault geometric Langlands conjecture of Donagi-Pantev, namely an equivalence of categories associated with moduli stacks of Higgs bundles on curves.
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. 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 will explain that these (Ind-)limit categories are compactly generated, admit Hecke operators, and carry semiorthogonal decompositions into quasi-BPS categories, which categorify BPS invariants in Donaldson-Thomas theory. This is joint work with Tudor Padurariu, arXiv:2508.19624.
Finally, I will discuss a proof of this equivalence for GL_r and SL_r/PGL_r over the locus of the Hitchin base where the spectral curves have at worst type A singularities. This gives nontrivial cases in which the relevant moduli stacks are not quasi-compact, and where the use of limit categories is essential both for the formulation and for the proof.