AMS Seminar: Bala Krishnamorthy (Washington State University) @ Whitehead 304

Title: Optimization and Topology: Two Stories
Abstract: Algebraic topology and optimization are typically not considered as closely related fields of mathematics. We will present two stories of fruitful interaction between these two fields, with the implications going the opposite way in the two cases.
In the first result, we consider the question of the existence of certain nice decompositions of generalized surfaces called currents in geometric measure theory. In the finite setting, we could use tools from algebraic topology to pose this question as that of the existence of integer solutions to a certain linear programming (LP) problem. Following classical results on LP that rely on total unimodularity (TU) of matrices, the answer is known in codimension 1. We develop tools to push this result to the infinite case, showing that under certain assumptions the TU result from LP implies the existence result for codimension 1 currents in general.
In the second story, we consider new approaches to characterize the robustness of solutions to a system of nonlinear equations. This problem arises in many applications such as the power grid and other infrastructure networks. We use techniques from algebraic topology (topological degree theory) to characterize the robustness margin of such systems of equations. We then cast the problem of checking for the specified conditions as a nonlinear optimization problem. Based on this formulation, we develop efficient computational techniques to estimate lower and upper bounds for the robustness margin.