I just finished my PhD in mathematics from UC Berkeley under the supervision of Bernd Sturmfels. I am interested in algebraic geometry, optimization, machine learning, and the interplay between these fields. In the fall of 2016 I am joining the MIT mathematics department as a Statistics Instructor.
Here is a link to my thesis.
Singular Vectors of Orthogonally Decomposable tensors with Anna Seigal
The Geometry of Positive Semidefinite Rank with Kaie Kubjas and Richard Robinson
Orthogonal and Unitary Tensor Decomposition from an Algebraic Perspective with Ada Boralevi, Jan Draisma and Emil Horobet
Superresolution without Separation with Geoffrey Schiebinger and Benjamin Recht
Decomposing Tensors into Frames with Luke Oeding and Bernd Sturmfels, Advances in Applied Mathematics, 76 (2016), pp. 125-153
Orthogonal Decomposition of Symmetric Tensors, SIAM Journal on Matrix Analysis and Applications, 37 (2016), pp. 86-102
Fixed Points of the EM Algorithm and Nonnegative Rank Boundaries with Kaie Kubjas and Bernd Sturmfels, Annals of Statistics, 43:1 (2015), pp. 422-461.
Robust Toric Ideals with Adam Boocher, Journal of Symbolic Computation, 68 (2015), pp. 254-264
A Tropical Proof of the Brill-Noether Theorem with Philip Cools, Jan Darisma and Sam Payne, Advances in Mathematics, 230 (2012), pp. 759-776
Artificial Intelligence for Bidding Hex with Sam Payne, Games of No Chance, Edited by Richard J. Nowakowski. Mathematical Sciences Research Institute Publications, 63. Cambridge University Press, Cambridge (2015) pp. 207-214.
An Extensive Survey of Graceful Trees, Undergraduate Honors Thesis, Stanford University 2011
Here is a link to my CV.
A nonnegative matrix M has positive semidefinite (psd) rank r if there exist r by r positive semidefinite matrices A_i and B_j such that the (i,j) entry of M is the trace inner product of A_i and B_j. Positive semidefinite rank is a generalization of nonnegative rank. Given a polytope P, the psd rank of its slack matrix is the smallest number r such that P can be written as a linear projection of the r-th psd cone. In joint work with Kaie Kubjas and Richard Robinson, we study the space of matrices of given positive semidefinite rank. This amounts to finding families of nested spectrahedra and spectrahedral shadows.
Orthogonal tensor decomposition has been used in machine learning in the recent years. I have been studying the set of orthogonally decomposable tensors. In particular, I have been able to provide a formula for all eigenvectors of such tensors. I have also conjectured that certain polynomial equations define the set of orthogonally decomposable tensors. Here is a link to the talk I gave at the Simons Institute and a link to my paper.
In the fall of 2013 I worked with Kaie Kubjas and Bernd Sturmfels on studying the fixed points of the EM algorithm on the mixture model of m by n matrices of nonnegative rank r. Some of these fixed points are critical points of the log-likelihood function for the determinantal variety of matrices of rank r. The rest of the fixed points land on the boundary of the mixture model. For this reason, I am interested in studying the components of the boundary of this model.
For supplementary programs to our study, including an algorith for determining whether a matrix has nonnegative rank at most 3, please click here.
In the fall of 2012, I worked with Adam Boocher on classifying robust toric ideals: toric ideals whose universal Grobner basis is also a minimal generating set.
While I was an undergraduate student at Stanford, I worked with Professor Sam Payne on chip-firing on graphs, which is very important for the study tropical Brill-Nether theory. I implemented this chip-firing program, which takes a graph as an input and outputs divisors of given degree d and rank r.
In the summer after my first year at
Stanford, I worked with Professor Sam Payne on the game of Bidding Hex. This lead
to producing and implementing an optimal strategy for Bidding Hex, which beats a
human player 99% of the time.
An online version of the game is here. It might be very slow to run it from a browser, so I recommend downloading the file and running it from one's computer. Have fun!
I love math competitions and I would be happy to get involved with organizing and grading any of those. I have also given two lectures at the Berkeley Math Circle.