John Urschel

urschel AT mit DOT edu

I am an assistant professor in the MIT Math department and a Junior Fellow at the Harvard Society of Fellows. My research is focused on matrix analysis and numerical analysis, with an emphasis on theoretical results and provable guarantees for practical problems.

Previously, I was a member of the Institute for Advanced Study, under the supervision of Peter Sarnak. I completed my PhD in math at MIT in 2021, and had the pleasure of being advised by Michel Goemans. My thesis, Graphs, Principal Minors, and Eigenvalue Problems, was recently awarded the SIAM DiPrima Prize. I also recently received the SIAM Early Career Prize in Linear Algebra.

Here are a few selected publications:

Below, you can find a brief description of my research, a full list of my publications, my current and past teaching, and some outreach programs I am involved in. Here is a (most likely outdated) CV.

Research:

My interests largely consist of topics in matrix analysis and numerical analysis, many of which are motivated by problems from other areas of mathematics, such as combinatorics, machine learning, probability, and theoretical computer science. A brief description of these two topics and my interests follows.

Matrix Analysis: Linear algebra is a fundamental subject, underpinning almost all areas of mathematics. Matrix analysis, broadly defined, is the study of basis-dependent linear algebra. This additional structure is often crucial -- linear maps encountered in practice may have some special property in a given basis, such as sparsity, symmetry, or non-negativity. I am especially interested in spectral graph theory, the study of a discrete graph/network through the analysis of spectral properties of a matrix representation of it.

Selected Publications:
Numerical Analysis: The field of numerical analysis is primarily concerned with finding efficient and accurate approximate solutions to problems in mathematics. Topics include the numerical solution to linear systems, non-linear equations, eigenvalue problems, numerical differentiation or integration, differential equations, and other problems. I am particularly interested in numerical linear algebra and the solution of linear systems Ax = b and eigenvalue problems Ax = λx. My research is mostly focused on matrix factorizations and moment-based algorithms.

Selected Publications:

Papers:

✷ = selected publication, u = undergraduate research project

Courses:


Outreach:

Here are a few programs outside of my core responsibilities as an academic mathematician that I take part in. If you are a K-12 math teacher or student, some of these programs may be of interest to you.

MathROOTS: MathROOTS is a two-week summer program hosted by MIT Math for high-potential high school students from underrepresented backgrounds or underserved communities. I am in charge of the academic side of the program. More details can be found here.

Mathical: The Mathical Book Prize, awarded by MSRI, is an annual award for fiction and non-fiction books that inspire children of all ages to see math in the world around them. I am a chair of the selection committee, and am more broadly involved in the dissemination of these fantastic books. More details can be found here.

MoMath: The National Museum of Mathematics highlights the role of math in illuminating the patterns and structures all around us. I serve on the board of the museum and am involved in the exhibits and programming of the museum. More details can be found here.

© 2020 John Urschel -- template shamelessly stolen from Tselil Schramm.
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