Wednesday, February 22, 2023
02/22/2023 - 1:00pm
Abstract: In this talk, we present a multilevel kernel-split dual-space (MKDS) framework for fast transforms. The MKDS framework applies "kernel splitting" to avoid the well-separateness condition in the fast multipole method, needed to obtain low-rank approximation of far field interactions. Instead, it uses a hierarchy of Fourier transforms and separation of variables to accelerate the far-field calculation. The framework is dimension-independent and applicable to a broad class of translation invariant kernels. Its preliminary performance is illustrated by comparison with the state-of-art FMM3D and PVFMM libraries for the Laplace kernel in three dimensions. This is ongoing joint work with Leslie Greengard.
02/22/2023 - 4:15pm
A matrix is said to be totally positive, if all its minors are positive. The notion of total positivity plays an important role in different areas of mathematics. Lusztig has generalized this notion to all split real Lie groups. He also introduced a notion of positivity in the corresponding flag varieties, that played a key role in the work of Fock and Goncharov on higher Teichm\”uller spaces.
In this talk I will introduce a new notion of positivity in flag varieties that includes but generalizes Lusztig’s definition. I will discuss some applications of this new notion to higher Teichm\”uller spaces, as well as the geometric and dynamical properties of the corresponding representations of surface groups.