Georgia Institute of Technology College of Sciences

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.

Determinantal point processes (DPPs) have attracted a lot of attention in probability theory, because they arise naturally in many integrable systems. In statistical physics, machine learning, statistics and other fields, they have become increasingly popular as an elegant mathematical tool used to describe or to model repulsive interactions. In this talk, we study the geometry of the likelihood associated with such processes on finite spaces. Interestingly, the local behavior of the likelihood function around its global maxima can be very different according to the structure of a specific graph that we define for each DPP. Finally, we discuss some statistical consequences of this fact, namely, the asymptotic accuracy of a maximum likelihood estimator.

There is no general h-principle for Legendrian embeddings in contact manifolds. In dimension 3, however, Legendrian knots in the complement of an overtwisted disc, which are called loose, satisfy an h-principle. We will discuss the high dimensional analog of loose knots.