With the advance in sensors, data become ubiquitous. To make sense of the data we have to solve higher and higher dimensional problems that would seem intractable to solve. However, many high-dimensional problems have solutions that live in low-dimensional spaces. Sparsity is a way to exploit the low-dimensional structure of solutions to obtain feasible solution methods for high-dimensional problems. Here in this talk, I will introduce regularization methods that enforce sparsity in solutions and their applications to several image reconstruction problems including single-molecule localization microscopy and ground-based astronomy.
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A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs has a long history in discrete mathematics, going back to the 18th century work of Euler on Latin squares. Since then rainbow structures were the focus of extensive research and found numerous applications in design theory and graph decompositions. Many problems in this area can be solved or partially solved by applying probabilistic arguments. In this talk we discuss a few such applications focusing on recent progress on some long-standing open questions.
High-energy particle collisions are one of the main experimental tools for studying particle physics. What fundamental physical questions need to be addressed by this method? What products emerge from collisions and how can they be detected? This lecture will provide a brief introduction to the basic principles of particle physics research and the major international large-scale collision exper...
The moduli space of decorated twisted G-local systems on a marked surface, originally introduced by Fock--Goncharov, is known to have a natural cluster K_2 structure. In particular, we have a canonically defined cluster algebra A and an upper cluster algebra U inside its field of rational functions. In order to investigate the structure of the function ring of that moduli space, we introduce the Wilson lines valued in the simply-connected group G, which are “framed versions” of those studied by myself and Hironori Oya. We see that the function ring of the moduli space is generated by the matrix coefficients of Wilson iines, and some of them are cluster monomials. As an application, we prove that both A and U coincide with the function ring. Time permitting, I will also mention some relations to the skein theory. This talk is based on a joint work with Hironori Oya and Linhui Shen.