Tuesday, September 19, 2023
09/19/2023 - 4:00pm
In this talk, we discuss a method for manifold learning that relies on a symmetric version of the optimal transport problem with a quadratic regularisation. We show that the solution of such a problem yields a sparse and adaptive affinity matrix that can be interpreted as a generalisation of the bistochastic kernel normalisation. We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, discuss geometric interpretations and establish robustness to heteroskedastic noise. The performance on certain simulated and real data examples will be shown. Some open questions will be raised across the talk.
09/19/2023 - 4:15pm
A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.