Abstract: There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.
Abstract: The design of interpolatory quadrature rules with positive weights is of great interest in approximation theory and scientific computing: Such quadrature rules achieve near-optimal approximation of integrals and are associated with well-behaved Lebesgue constants. Such quadrature rules are used heavily in applications like uncertainty quantification and computational finance.
Computing such quadrature rules in more than one dimension is an arduous task, frequently attempted with non-convex optimization schemes. The success or failure of such approaches typically depends on the type of domain, dimension, or approximation space. We present a new procedure whose implementation is a probabilistic algorithm based on a novel constructive proof of Tchakaloff's theorem. In particular, we can successfully compute size-N quadrature rules in high dimensions for general approximation spaces. The main feature of our algorithm is a complexity that depends algebraically on N, and in some cases is strictly linear in the dimension. We illustrate that our procedure is effective even for quadrature on complex and unbounded multivariate domains.
Abstract: This talk will focus on geometric realizations of non-commutative algebras. I will discuss how representations of conformal vertex algebras encode information about the geometry of algebraic curves. The starting point is the Virasoro uniformization, which provides an incarnation of the Virasoro algebra in the tangent space of a tautological line bundle on the moduli space of coordinatized curves. After briefly reviewing vertex algebras, I will discuss how their representations yield new vector bundles of conformal blocks on moduli spaces of curves and new cohomological field theories. This is joint work with Chiara Damiolini and Angela Gibney.
Abstract: We consider the existence of invariant attractors for the specific case of dissipative systems known as conformally symplectic systems, which are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter. We provide a KAM theorem in an a-posteriori format: assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby. The theorem does not assume that the system is close to integrable.
This method can be also used to get different results: (i) a breakdown criterion for invariant attractors; (ii) an efficient algorithm to generate the solution, which can be implemented successfully in model problems; (iii) the existence of whiskered tori for conformally symplectic systems, (iv) the analyticity domains of the quasi-periodic attractors in the symplectic limit.
The content of this talk refers to works in collaboration with R. Calleja and R. de la Llave.
Abstract: By studying the Borel complexity of countable families of cross-cutting equivalence relations, we are able to prove that (1) if T has uncountably many 1-types, then T has a Borel complete reduct; (2) if T is not small, then Teq has a Borel complete reduct; and (3) if T is not omega-stable, then the elementary diagram of any model of T has a Borel complete reduct. We also show that `not all Borel complete theories are equally complicated' by mentioning a stronger notion that holds of some Borel complete theories but not others. This is joint work with Douglas Ulrich.
Abstract: In this talk I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).