Thanks for visiting my website. I am a postdoctoral fellow at Math+ the Berlin Mathematics Research Center as part of an Incubator Project with Myfanwy Evans, Frank Lutz, and Bernd Sturmfels, and at Max Planck Institute for Mathematics in the Sciences in the Nonlinear Algebra research group directed by Bernd Sturmfels. Starting January 1, 2021, I will be a postdoctoral fellow at The Fields Institute for Research in Mathematical Sciences in Toronto as part of a thematic program on geometric constraint systems, framework rigidity, and distance geometry. My research supervisors will be Anthony Nixon and Pablo Parrilo. Here you will find information about my research, teaching, and other topics relevant to mathematics. Please feel free to contact me via email at alexheaton2 at gmail dot com. For some recent work, see below:
September 2020, Catastrophe in Elastic Tensegrity Frameworks
June 2020, Logarithmic Voronoi cells
April 2020, Symmetry adapted Gram spectrahedra
February 2020, Epsilon local rigidity and numerical algebraic geometry
September 2019, Computing a logarithmic Voronoi cell
August 2019, Nonlinear algebra with tensegrity frameworks
I am working on getting computations and code publicly available for all my projects. This means that major computations in each paper can be checked independently by anyone. In particular, you can check exactly what calculations were made. So far you can find code for three projects on the MPI MiS mathrepo, available at the links below.
In this Jupyter notebook we verify the sum of squares counterexample of https://arxiv.org/abs/1909.00081 by explicit computation. Dominance or majorization order fails to capture non-negativity relationships amongst the homogeneous symmetric functions.
A statistical model is a set of probability distributions on the sample space. For discrete random variables this can be thought of as a subset of the probability simplex in a suitable Euclidean space. One result of https://arxiv.org/abs/2006.09431 states that, for linear models, the logarithmic Voronoi cells are always polytopes. One might hope that finite unions of linear models would admit logarithmic Voronoi cells which are polytopes. However, this is not the case. The log-normal spaces from two disjoint linear models can meet in such a way that the boundary created on a logarithmic Voronoi cell is nonlinear. For an explicit example with two linear models, we present the computations in this notebook.
As a (very) special case of the main theorem of https://arxiv.org/abs/2007.08909 , the set of two by two matrices of rank 1 is a minimal submanifold of real four-dimensional space with the Frobenius norm. Minimal submanifolds are a mathematical model of soap films, since they locally minimize volume given a fixed boundary. A minimal submanifold can be defined by the requirement that its mean curvature vector field be everywhere vanishing. In fact, the set of all tensors of any fixed multilinear rank forms a minimal submanifold of its ambient Euclidean space, which yields matrices of fixed rank as the special case we examine computationally in this notebook. The notebook also computes the mean curvature vector field of the independence model on two binary discrete random variables, of interest in statistics. This is displayed below.