Title: Introduction to the geometric Langlands conjecture
Abstract: The geometric Langlands program takes its origin from a series of conjectures formulated by Langlands in late 1960's. A geometric version of these conjectures relates two natural spaces associated to a Riemann surface: the space of vector bundles and the space of local systems. In my talks, I will provide an informal introduction to the (global) geometric Langlands conjecture. I will then focus on some recent developments in this area, which combine classical ideas and modern tools. Finally, I will discuss some related `flavors' of the geometric Langlands program: its classical limit and its quantization.
Title: Exponential motives
Abstract: What motives are to algebraic varieties, exponential motives are to algebraic varieties together with a regular function. Such pairs arise in a wealth of contexts: as Landau-Ginzburg models in mirror symmetry of Fano varieties, in the cohomological interpretation of exponential sums over finite fields, or when trying to treat numbers such as exponentials or special values of the gamma function on an equal footing to periods. Following ideas of Kontsevich, Katz, and Nori, one can construct a tannakian category of exponential motives over a subfield of the complex numbers and a realisation functor with values on a suitable subcategory of mixed Hodge modules over the affine line. I will first explain the construction of the category and a useful criterion to decide whether an exponential motive is classical or not. I will then illustrate this criterion with an example where it allows one to study L-functions associated with symmetric power moments of Kloosterman sums. The talks are based on joint work with Peter Jossen (first part) and with Claude Sabbah and Jeng-Daw Yu (second part).
Title: Generalized Riemann-Hilbert correspondence
Abstract: Let P be a compact holomorphic Poisson manifold with an open dense symplectic leaf M. Then, under certain convergence assumptions, one can define two triangulated categories over complex numbers. The first category (A-model) is the compact Fukaya category of M considered as a real symlectic manifold endowed with a with B-field. The second category is the category of perfect modules over the non-perturbative deformation quantization of P, with the vanishing restriction to P-M. Generalized Riemann-Hilbert correspondence is a conjectured (by Y. Soibelman and me) equivalence between these two categories. I'll explain in details this conjecture and related companion conjectures. Also I'll illustrate it in the case of usual holonomic D-modules (when M is a cotangent bundle), and of q-difference and elliptic difference equations.
Title: Variations of BPS structure and enumerative geometry
Talk 1: A “variation of BPS structure” is a nice name for the kind of infinite dimensional bundle with connection one can construct, at least formally, starting from the Donaldson-Thomas type invariants of a Calabi-Yau threefold. In the first part of the talk I will offer an introduction to this circle of ideas, pointing to a lot of references. Then I will focus on the concrete example of what happens in this construction when we start with the DT invariants counting 1-dimensional torsion sheaves, or more generally sheaf-theoretic Gopakumar-Vafa invariants. The answer is closely related to the Gromov-Witten partition function. This second part is based on work of Bridgeland and on some work in progress.
Talk 2: In the first part of the talk I will describe what happens when we construct, at least formally, the “variation of BPS structure” starting from DT invariants which are no longer torsion, but framed, i.e. Pandharipande-Thomas stable pairs. The Gromov-Witten partition function reappears in a different limit. In the second part of the talk I will go back to torsion invariants and explain how the corresponding “variation of BPS structure” can be described in terms of much more familiar differential equations of hypergeometric type. Based on arXiv:1705.08820 and arXiv:1712.01221.
Title: The foliated topology and higher differential Galois theory
Abstract: The foliated topology is a direct analog of the étale topology in the category of schematic foliations. In the same way that étale topology is related to Galois theory, the foliated topology is related to differential Galois theory. However, in the latter context, a new phenomenon appears: differential fields tend to have higher differential Galois groups. I will report on some computations of higher differential Galois groups and, if time permits, I will describe a interesting open question.
Title: Coherent Satake functor
Abstract: To a reductive group G one can associate the so-called Langlands dual group G^. The geometric Satake equivalence is the statement that the category of representations of G^ can be recovered as the category of G(O)-equivariant D-modules on the affine Grassmannain of G. (In fact, this can be taken as the definition of G^ via the Tannakian formalism). I will discuss an ongoing project with D. Arinkin where we aim at constructing the quasi-classical limit of the Satake equivalence, laying foundations for the local Hitchin-Langlands duality.
Title: Quantum D-modules and toric flips
Abstract: In this talk, I will describe how the quantum D-modules of toric orbifolds change under toric birational transformations. The analysis is based on mirror symmetry for toric orbifolds studied in joint work with Coates, Corti and Tseng. I will also discuss how the gamma integral structures are related in some special cases. This suggests a certain functorial relationship of quantum D-modules under birational transformations.
Title: Toward a construction of 2-parameter family of Painlevé tau-function via the topological recursion
Abstract: Painlevé equations are 2nd order non-linear ODEs with many interesting properties (Painlevé property, isomonodromy deformation, space of initial conditions…). In our previous work with O. Marchal and A. Saenz, it was shown that the tau-function corresponding to a particular solution of Painlevé equations (called 0-parameter solution) can be constructed as a partition function of the topological recursion applied to a family of singular elliptic curves parametrized by isomonodormic time (based on the idea of earlier work by G. Borot and B. Eynard). In this talk, I will present a conjectural expression of the tau-function corresponding to the general solution (called 2-parameter solution) of the first Painlevé equation through the topological recursion applied to a family of smooth elliptic curves.
Title: Duality of surface graphs and CohFT
Abstract: We will present an alternative formulation of cohomological field theories based on the categories of surface graphs. The aim of this formalism is to visualize the classification theorem due to Givental and Teleman. Surface graph duality leads to a Frobenius-Hopf correspondence, which illuminates the structure theorem of semi-simple CohFT. Talk is based on my joint work with O. Dumitrescu.
Title: Wild Hitchin moduli spaces
Abstract: We consider a wild Hitchin moduli space on P1 with a wild singularity at 0 and a possible tame singularity at infinity. This moduli space has rich structures, some of which are not true for the usual cases. Moreover it is expected that this geometry is connected with representation theory of affine Lie algebras at admissble level. Based on the joint work arXiv:1809.043638 with Dedushenko, Gukov, Pei and Ye.
Title: Higgs bundles for the Geometric Langlands correspondence
Abstract: We present current joint work with Donagi and Pantev, on the construction of some local systems entering into the geometric Langlands correspondence, by constructing the corresponding parabolic logarithmic Higgs bundles. We look at the case of a compact genus 2 curve. The key feature, as predicted by the program of Donagi and Pantev, is that the spectral variety of the Higgs bundle on Bun is identified with a blow-up of the Hitchin fiber.
Title: Sandwich resolution of a dual free associative algebra
Abstract: Let D be the graded Q vector space generated by motivic multiple zeta values modulo “π2”. The depth filtration is defined as the subspaces of D generated by MZV's whose lengths are less than or equal to given numbers. Broadhurst and Kreimer gave a conjecture on the two variable generating function of the dimensions of weight n and depth d part. This conjecture suggests the existence of an influence of mixed elliptic motives on mixed Tate motives. The Hopf algebra classifying the mixed elliptic motives is given by the relative bar complex defined by Hain. In this talk, we introduce a certain resolution, called a sandwich resolution of a dual free associative algebra motivated by the Broadhurst-Kreimer's generating function and the relative bar complex.
Title: Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
Abstract: In this talk, I will discuss birational geometry for Joyce's d-critical loci, by introducing notions such as 'd-critical flips', 'd-critical flops', etc. I will show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry, e.g. certain wall-crossing diagrams of Pandharipande-Thomas stable pair moduli spaces form a d-critical minimal model program. I will also show the existence of semi-orthogonal decompositions of the derived categories under simple d-critical flips satisfying some conditions. This is motivated by a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also gives a categorification of wall-crossing formula of Donaldson-Thomas invariants.