Georgia Institute of Technology College of Sciences

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

1. What are the closed subsets of M invariant under both f and g?
2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.  

The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are interesting and important in various areas of applied mathematics and engineering in the sense that they are the shortest sizes of the decompositions of vectors into combinations of simple vectors.



In this talk, we focus on the ranks of points with respect to the rational normal curves, i.e. Waring ranks of binary forms. We introduce an algorithm that produces random points of given rank r. (Note that if we choose points randomly, we expect the rank of the points is just the generic rank.) Moreover, we check some known facts by Macaulay 2 computations. Lastly, we discuss the maximal and minimal rank of points in linear spaces.

Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1649360107625?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical method. Two of the fundamental tools in this field are the polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets put forth by Sommese and Wampler as numerical representations for non-isolated solution components. In this talk, we will describe a stratified polyhedral homotopy method that will bridge the gap between these two largely independent area. Such a homotopy method will discover numerical representations of non-isolated solution components as by-products from the process of computing isolated solutions. We will also outline the pipeline of numerical algorithms necessary to implement this homotopy method on modern massively parallel computing architecture.

Given a collection of functions in a Banach space, typically called a dictionary in machine learning, we study the approximation properties of its convex hull. Specifically, we develop techniques for bounding the metric entropy and n-widths, which are fundamental quantities in approximation theory that control the limits of linear and non-linear approximation. Our results generalize existing methods by taking the smoothness of the dictionary into account, and in particular give sharp estimates for shallow neural networks. Consequences of these results include: the optimal approximation rates which can be attained for shallow neural networks, that shallow neural networks dramatically outperform linear methods of approximation, and indeed that shallow neural networks outperform all continuous methods of approximation on the associated convex hull. Next, we discuss greedy algorithms for constructing approximations by non-linear dictionary expansions. Specifically, we give sharp rates for the orthogonal greedy algorithm for dictionaries with small metric entropy, and for the pure greedy algorithm. Finally, we give numerical examples showing that greedy algorithms can be used to solve PDEs with shallow neural networks.