Georgia Institute of Technology College of Sciences

Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. In this talk I will focus on discrete linear problems: giving a general overview of the well-established class of solvers called Krylov subspace methods and its regularizing properties; as well as flexible variants that make them suitable to solve more challenging optimization tasks. I will show results and examples in different imaging applications.

This talk will delve into a method specifically designed for
constructing high-order normal forms in Poincaré maps with high-order
precision and without any major assumption or structure of the
dynamical system itself. We will use the result to generate explicit
twist maps, calculating invariant tori, and determining the flying
time expansions around an elliptic fixed point of a Poincaré map. In
particular, this approach is able to check some non-degenerate
conditions in perturbation theory.

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

The Acyclic Edge Coloring Conjecture, posed independently by Fiam\v{c}ik in 1978 and Alon, Sudakov and Zaks in 2001, asserts that every graph can be properly edge colored with $\Delta+2$ colors such that there is no bicolored cycle. Over the years, this conjecture has attracted much attention. We prove that the conjecture holds asymptotically, that is $(1+o(1))\Delta$ colors suffice. This is joint work with Michelle Delcourt and Luke Postle.