IST Austria

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

We present a kinetic version of the optimal transport problem for probability measures on phase space. The central object is a second-order discrepancy between probability measures, analogous to the 2-Wasserstein distance, but based on the minimisation of the squared acceleration. We discuss the equivalence of static and dynamical formulations and characterise absolutely continuous curves of measures in terms of reparametrised solutions to the Vlasov continuity equation. This is based on joint work with Giovanni Brigati (ISTA) and Filippo Quattrocchi (ISTA).

First we recall the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we discuss a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.

In 1985, McDowell introduced a family of parabolically induced Whittaker modules over a complex semisimple Lie algebra, which includes both Verma modules and the nondegenerate Whittaker modules studied by Kostant. Many classical results for Verma modules and the Bernstein--Gelfand--Gelfand category O have been generalized to the category of Whittaker modules introduced by Milicic--Soergel, including the classification of irreducible objects and the Kazhdan--Lusztig conjectures. Contravariant forms on Verma modules are unique up to scaling and play a key role in the definition of the Jantzen filtration. In this talk I will discuss a classification of contravariant forms on parabolically induced Whittaker modules. In a recent result, joint with Anna Romanov, we show that the dimension of the space of contravariant forms on a parabolically induced Whittaker module is given by the cardinality of a Weyl group. This result illustrates a divergence from classical results for Verma modules, and gives insight to two significant open problems in the theory of Whittaker modules: the Jantzen conjecture and the absence of an algebraic definition of duality.