Georgia Institute of Technology College of Sciences

Exploring Parametrized Systems of Real Polynomial Equations with Neural Networks

We investigate the use of machine learning techniques to assist in the solution of parametrized systems of real polynomial equations. In particular, we discuss the problem of jointly predicting the number of real solutions to a problem and approximating the paramter-to-solution map. Throughout, we emphasize interactions between the algebraic geometry of the underlying problem and considerations in deep learning.
This is based on ongoing work with Julianne Barnhart, Jonathan Hauenstein, Ikenna Nometa, Margaret Regan, Trong-Thuc Trang, and Charles Wampler.

In a highly influential paper from 1978, Lovász used topological methods to determine the chromatic number of the Kneser graph of the set of k-element subsets of a set with n elements. In this talk, we will discuss the Kneser graph of the set of triangulations of a convex n-gon and a recent proof that the chromatic number of this graph is n-2. The geometry of the associahedron will play a particularly important role in the argument. Based on a joint work with Anton Molnar, Cosmin Pohoata and Daniel Zhu.

In this talk, I will show how tools from VC-dimension theory can be used to address questions from incidence geometry, enumerating intersection graphs, and graph drawing.