Seminars Sorted by Series

This talk will be an exposition of a recent paper of Bezrukavnikov-Gaitsgory-Mirkovic-Riche-Rider giving an Iwahori-Whittaker model for the Satake category. The main point is that their argument works for modular coefficients. I will give some...

Before the "geometric Satake equivalence" there was a decategorified version of it which however contained most of its essential features. In my talk I will talk about some of the ideas which have led to this theory. In particular I will explain the...

Motivated by a formal similarity between the Hard Lefschetz theorem and the geometric Satake equivalence we study vector spaces that are graded by a weight lattice and are endowed with linear operators in simple root directions. We allow field...

The talk is about convolution in the setting of geometric representation theory. What are its formal properties? As a starting point, let $G$ be a group and let $D(G)$ be the derived category of constructible sheaves on it. Convolution turns $D(G)$...

Smith theory is a type of equivariant localization with respect to a cyclic group of prime order $p$, with coefficients in a field of the same characteristic $p$. It has been the source of various recent advances in modular representation theory and...

The linkage principle says that the category of representations of a reductive group $G$ in positive characteristic decomposes into "blocks" controlled by the affine Weyl group. We will discuss the beautiful geometric proof of this result that Simon...

Wall-crossing functors on the principal block of category $O$ give an action of the (finite) Hecke category. If one knows enough about the Hecke category, one can deduce the Kazhdan-Lusztig conjectures from the existence of this action. This is a...

In the upcoming series of talks we discuss some equivalences by Bezrukavnikov between categories of coherent sheaves on Steinberg varieties and of perverse sheaves on the affine flag variety. In this talk I discuss the decategorified isomorphism...

This is the second in a series of talks on "two realizations." We discuss equivariantization and de-equivariantization for a group $G$, which relate categories with $G$-actions and categories with RepG-actions, and how this relates to the notion of...

A theorem of Bernstein identifies the center of the affine Hecke algebra of a reductive group $G$ with the Grothendieck ring of the tensor category of representations of the dual group $G^\vee$. Gaitsgory constructed a functor which categrorifies...

We will discuss the following theorems concerning colimits taken in the infinity-category of monoidal DG-categories. (No familiarity with infinity-categories will be required or assumed.) The affine Hecke category is the monoidal colimit of its...

The anti-spherical modules over the affine Hecke algebras admit two different realizations: one realization is in terms of the space of Whittaker functions on the affine flag manifolds and the other realization, due to Kazhdan-Lusztig, is in terms...

In this talk we will begin the discussion of the results in Bezrukavnikov's "On Two Geometric Realizations of the Affine Hecke algebra". We will put all the previous tools described in this series of talks together to construct the equivalence of...

I will start with a few comments on the proof of the equivalence presented in the previous talks. Then I will focus on the description of the abelian category of perverse sheaves on the affine flag variety on the coherent side, where the answer is...

The goal of this talk is two-fold. We state a parabolic version of the two realizations theorem and sketch a proof. This version relates Iwahori-constructible sheaves on parabolic affine flag variety to coherent sheaves on a parabolic version of the...

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore...

For homogeneous affine Springer fibers (those with $G_m$ symmetry), we realize them as Lagrangian cycles inside ambient symplectic varieties, and make sense of microlocal sheaves supported on these affine Springer fibers. We also propose a...

I will report on a project joint with Roman Bezrukavnikov (and partly with Laura Rider) aiming at constructing a variant for positive-characteristic coefficients of the equivalence constructed by Bezrukavnikov in "On two geometric realizations of an...

I will report on a joint work in progress with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic $p > 0$ admits a fiber functor to the Verlinde category (semisimplification of $Rep(Z/p)$) if and only if it has...

I will discuss some aspects of the proof of (most of) the unramified geometric Langlands duality conjecture for GL(n) by Gaitsgory (based on his work with Frenkel, Vilonen etc), and also the work in the direction of local tamely ramified duality.

In this talk I will discuss the construction of blowing up solutions to the Yamabe Problem in high dimensions. I will show how one can use the methods of S. Brendle to produce smooth counterexamples to the Weyl Vanishing Conjecture also.

We will discuss the introduction of the scattering operators associated with a conformal manifold by Graham and Zworski. Then we will discuss some uses of such family of operators in describing conformally invariant global quantities for conformal...

We discuss Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds from the viewpoint of existence results and stability issues. We prove that the Einstein-scalar field Lichnerowicz equation is stable when $n \le 5$, and...

We will discuss topological information carried by weakly differentiable maps and its applications in an existence theory for absolute minimizers of the Faddeev knot energies in higher dimensions.

The special Lagrangian equations define calibrated minimal Lagrangian surfaces in complex space. These fully nonlinear Hessian equations can also be written in terms of symmetric polynomials of the Hessian, giving a minimal surface interpretation to...

In this talk, we shall review the convexity of solutions of elliptic partial differential equations; we concentrate on the constant rank theorem for the hessian of the convex solution. As for the interesting from geometry problems, recently we have...

In optimal transport theory, one wants to understand the phenomena arising when mass is transported in a cheapest way. This variational problem is governed by the structure of the transportation cost function defined on the product of the source and...

We address the problem of building a body of specified shape and of specified mass, out of materials of varying density so as to minimize the first Dirichlet eigenvalue. It leads to a free boundary problem and many uniqueness questions, The...

In 1979, in collaboration with D. Bennequin, we started a direction of research around the study of periodic trajectories of the Reeb vector field $\xi$ on a contact manifold $(M^3, \alpha)$. We will describe in this talk where this direction of...

The de Rham complex is a prototype for a large class of sequences of differential operators often called (generalized) Bernstein-Gelfand-Gelfand BGG sequences. Conformal manifolds admit such sequences and on locally conformally flat manifolds the...

In this talk, we prove existence and uniqueness of vortexless solutions for Chern-Simons-Higgs equation in nonselfdual case.

In this talk we describe some recent result as well as the work in progress on the neck pinching of surfaces under under mean curvature flow.

Given two differential equations it is often useful to know invariants which guarantee that there exists a transformation of variables (independent, dependent or both) that transforms one of the equations into the other. Recently it has been...

In this talk, I will describe the blow-up profile for Q-curvature equations and related work.

If a solution (M,g(t)) of Ricci flow develops a local singularity at a finite time T , then there is a proper subset S of M on which the curvature becomes infinite as time approaches T . Existing approaches to Ricci-flow-with-surgery, due to...