EXPOSÉS
Amina Abdurrahman (Stony Brook / IHES)
TBC
Résumé
TBC
Amadou Bah (Columbia)
Variation of the Swan conductor of an l-adic sheaf on a rigid disc
Résumé
The radius of convergence of a $p$-adic differential equation on a p-adic disc has interesting variational properties. It turns out that its $\ell$-adic analog, the Swan conductor of an $\ell$-adic local system $\mathcal{F}$ on a closed rigid unit disc $D$, has similar properties: using the ramification theory of Abbes and Saito, we show that the Swan conductor function ${\rm sw}_{\mathcal{F}}:\mathbb{Q}\to\mathbb{Q}$ (of variable the radius of a sub-disc of $D$) is continuous, convex and piecewise linear; its slopes are integers that can be computed in terms of local characteristic cycles of $\mathcal{F}$ (analogs of the characteristic cycle of a holonomic $\mathcal{D}$-module).
Roman Bezrukavnikov (MIT)
Derived divided power operators on a stack
Résumé
Crystalline differential operators on a smooth variety X in positive characteristic form an Azumaya algebra over the Frobenius twist of its cotangent bundle. The natural extension to the case of a smooth algebraic stack X has derived nature: it involves a gerbe over the cotangent of X which is a derived stack. I will discuss an attempt to define a similar derived enhancement for divided power differential operators on a stack, focusing on the example where X is the classifying stack of a maximal unipotent subgroup in a reductive algebraic group G; the construction is motivated by the study of G-modules via derived localization. Time permitting I will explain related structures in the topology of the affine Grassmannian, connected to the above by the geometric Satake equivalence. The results are partly joint with Pablo Boixeda Alvarez.
Ekaterina Bogdanova (Harvard)
Non-vanishing of quantum geometric WhiPaker coefficients
Résumé
We will discuss the functor of geometric WhiPaker coefficients in the context of quantum geometric Langlands. Concretely, we will prove (modulo the spectral decomposi1on conjecture) that for good levels the functor of quantum geometric WhiPaker coefficients is conserva1ve. The proof will combine generaliza1ons of representa1on-theore1c and microlocal methods from the preceding works of Faergeman-Raskin and Nadler-Taylor respec1vely.
Alexander Braverman (Toronto)
TBC
Résumé
TBC
Yagna Dutta (Leiden)
A relative intermediate Jacobian
Résumé
Hélène Esnault (Harvard/Copenhague)
A non-abelian version of Deligne’s Fixed Part Theorem
Résumé
We formulate and prove a non-abelian analog of Deligne’s Fixed Part theorem on Hodge classes, revisiting previous work of Jost–Zuo, Katzarkov–Pantev and Landesman–Litt. To this aim we study algebraically isomonodromic extensions of local systems and we relate them to variations of Hodge structures, for example we show that the Mumford-Tate group at a generic point stays constant in an algebraically isomonodromic extension of a variation of Hodge structures. Joint with Moritz Kerz.
Joakim Færgeman (Yale University)
Motivic realization of rigid G-local systems on curves
Résumé
A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. While a classification of motivic local systems is evidently out of reach, Simpson conjectured that for a reductive group G, rigid G-local systems with suitable finiteness conditions at infinity are motivic. This was proven for curves when G=GL_n by Katz who classified such rigid local systems. In this talk, we discuss our generalization of Katz’ theorem to a general reductive group. Our proof goes through the (tamely ramified) categorical geometric Langlands program in characteristic zero.
Quantum Hikita conjecture
Résumé
The quantum Hikita conjecture relates two D-modules coming from dual symplectic resolution. The first D-module is the quantum D-module coming from quantum cohomology of one variety. The second D-module controls graded traces for the resolution of the dual variety. I will present this conjecture and present some examples.
Vasily Krylov (MIT)
Representations for quantized ADHM spaces via the D-modules on the nilpotent cone
Résumé
Over the past twenty years, mathematicians and physicists have shown increasing interest in studying certain Poisson varieties, known as “symplectic singularities,” and the quantizations of their rings of functions. These objects naturally arise in the study of 3D Topological Quantum Field Theories (TQFTs) and also within the framework of 3D mirror symmetry. In this talk, we will focus on an important family of examples of symplectic singularities — namely, “ADHM spaces” or « moduli spaces of instantons on R^4 ». We will explain how to compute characters of « minimally supported » irreducible representations over the quantizations of the ADHM spaces and, in particular, will provide formulas for the dimensions of all finite-dimensional representations over these quantizations. Our main tools include SL_n-equivariant D-modules on the nilpotent cone of SL_n and the representation theory of rational Cherednik algebras. The talk is based on joint work with Pavel Etingof, Ivan Losev, and José Simental.
Daniel Litt (University of Toronto)
Integrality and algebraicity of solutions to differential equations
Résumé
Eisenstein proved, in 1852, that if a func8on f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the converse of this theorem holds. Namely, we conjecture that if f(z) sa8sfies a (possibly nonlinear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Zalgebra, then f is algebraic. For linear ODE, we prove this conjecture when (A) f(z) sa8sfies a Picard-Fuchs equa8on, with ini8al condi8ons the class of an algebraic cycle, or (B) f(z) sa8sfies an ODE of geometric origin and order at most 2. For non-linear ODE, we prove it when f(z) sa8sfies an « isomonodromy » ODE with « Picard-Fuchs » ini8al condi8ons.
Mircea Mustata (Ann Arbor)
Invariants of singularities and Hodge modules
Résumé
I will give an overview of some applications of Saito’s theory of Hodge modules to invariants of hypersurface (and, more generally, complete intersection) singularities.
Sebastian Olano (Toronto)
Singularities of Secant Varieties from a Hodge theoretic perspective
Résumé
Given a smooth projective variety $X$ and a sufficiently positive embedding into $\mathbb{P}^N$, we can define the associated secant variety. These varieties have been extensively studied, yet many aspects of their singularities remain to be understood. The singular locus admits a simple description since it is the original variety $X$. However, its singularities are more intricate, as the variety is generally not a local complete intersection. In this talk, I will explore the singularities of secant varieties from a Hodge-theoretic perspective, focusing on generalizations of notions well-understood in the context of local complete intersections. Specifically, I will describe the du Bois complex and its relationship with the sheaves of differential forms, linking it to the generalization of the classical notions of du Bois and rational singularities. Additionally, I will present results concerning the local cohomology module of the secant variety in projective space.
Rachel Ollivier (UBC)
Rigid dualizing complexes for affine Hecke algebras
