Georgia Institute of Technology College of Sciences

For a given generic form, the problem of finding the nearest rank-one form with respect to the Bombieri norm is well-studied and completely solved for binary forms. Nonetheless, higher-rank approximation is quite mysterious except in the quadratic case. In this talk we will discuss such problems in the binary case.

In this talk I will present my recent result about the ergodic properties of nonequilibrium steady-state (NESS) for a stochastic energy exchange model. The energy exchange model is numerically reduced from a billiards-like deterministic particle system that models the microscopic heat conduction in a 1D chain. By using a technique called the induced chain method, I proved the existence, uniqueness, polynomial speed of convergence to the NESS, and polynomial speed of mixing for the stochastic energy exchange model. All of these are consistent with the numerical simulation results of the original deterministic billiards-like system.

A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present a self-contained version of Moser's proof and also to present some efficient numerical algorithms.

Spielman and Teng (2004) showed that linear systems in Laplacian matrices can be solved in nearly linear time. Since then, a major research goal has been to develop fast solvers for linear systems in other classes of matrices. Recently, this has led to fast solvers for directed Laplacians (CKPPRSV'17) and connection Laplacians (KLPSS'16).Connection Laplacians are a special case of PSD-Graph-Structured Block Matrices (PGSBMs), block matrices whose non-zero structure correspond to a graph, and which additionally can be expressed as a sum of positive semi-definite matrices each corresponding to an edge in the graph. A major open question is whether fast solvers can be obtained for all PGSBMs (Spielman, 2016). Fast solvers for Connection Laplacians provided some hope for this. Other important families of matrices in the PGSBM class include truss stiffness matrices, which have many applications in engineering, and multi-commodity Laplacians, which arise when solving multi-commodity flow problems. In this talk, we show that multi-commodity and truss linear systems are unlikely to be solvable in nearly linear time, unless general linear systems (with integral coefficients) can be solved in nearly linear time. Joint work with Rasmus Kyng.