That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will argue that such direct solvers have several compelling qualities, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

After a general introduction to the field, particular attention will be paid to a set of recently developed randomized algorithms that construct data sparse representations of large dense matrices that arise in scientific computations. These algorithms are entirely black box, and interact with the linear operator to be compressed only via the matrix-vector multiplication.

In this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d using cohomological Donaldson-Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. We will see that the BPS cohomology does not depend on d. This is a generalization of the Hausel-Thaddeus conjecture to non-coprime case. I will also explain that Toda's χ-independence conjecture (and hence the strong rationality conjecture) for local curves can be proved in the same manner. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda.

Hilbert-Schmidt independence criterion (HSIC) is among the most widely-used approaches in machine learning and statistics to measure the independence of random variables. Despite its popularity and success in numerous applications, quite little is known about when HSIC characterizes independence. I am going to provide a complete answer to this question, with conditions which are often easy to verify in practice.

This talk is based on joint work with Bharath Sriperumbudur.