Seminars Sorted by Series

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...

We will discuss diffusion limits for linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion...

We discuss the local properties of the boundaries of sets whose indicator function is a local minimizer of a Sobolev norm of exponent strictly less than 1/2. It turns out that one devise a regularity that parallels very much that of de Giorgi for...

The first aim of Fefferman-Graham ambient metric construction was to write down all scalar invariants of conformal structures. For odd dimensions, the aim was achieved with the aid of the parabolic invariant theory by Bailey, Eastwood and Graham. In...

I shall explain how to obtain Strichartz estimates with no loss for Schrodinger equation in some cases where the geodesic flow has some trapped trajectories, but the flow is hyperbolic. (This is joint work with Burq and Hassell.)

In pioneering work Tian/Viaclovsky initiated the study of the moduli space of Bach-flat metrics. They showed C^0-orbifold regularity and, equivalently, ALE order zero of noncompact finite-energy solutions. By use of Kato inequalities, the full...

In recent years, fully nonlinear versions of the Yamabe problem have received much attention. In particular, for manifolds with boundary, $C^1$ and $C^2$a priori estimates have been proved for a large class of data. To get an existence result, it is...

In 1964 J. Serrin proposed the following conjecture. Let u be a weak solution (in W^{1,1}) of a second order elliptic equation in divergence form, with Holder continuous coefficients, then u is a "classical" solution ( i.e. u belongs to H^1). I will...

In this talk, I will discuss some characterizations of Sobolev spaces, BV spaces, and present some new inequalities in this context. As a consequence, I can improve classical properties of Sobolev spaces such as Sobolev inequality, Poincare...

Conditional on the scattering conjecture of the mass-critical nonlinear Schrodinger equation in spatial dimension one, we show that there exists a blow-up solution to the mass-critical generalized Korteweg de Vries equation (gKdV) with the minimal...

This talk is based on joint work with YanYan Li and Eduardo Teixeira. Some geometric problems in conformal geometry lead naturally to the study of singular solutions to certain PDEs that describe "canonical" conformal metrics. A good example is the...

In this talk, I will discuss the quadruple junction solutions in the entire three dimensional space to a vector-valued Allen-Cahn equation which models multiple phase separation. The solution is the basic profile of the local structure near a...

The "sigma_k Yamabe problem" is a fully nonlinear generalization of the Yamabe problem, in which one attempts to find a conformal multiple of a given metric to make constant the k-th elementary symmetric function of the eigenvalues of the Schouten...

The location of blowup points is often related to critical points of Green function. In fact, Green function plays a important role to understand the solutions structure of mean field equations. We will show how to use the elliptic function and the...

The talk is devoted to the homogenization of a stochastic evolution problem with non Lipschitz forcing. The problem is considered in a sequence of perforated cylindrical domains obtained from the removal of tiny cylinders from a fixed one (the...