John Urschel

Class of 1956 Career Development
Assistant Professor
MIT Department of Mathematics

I am an Assistant Professor at MIT Math. Before that, I was a Junior Fellow at the Harvard Society of Fellows and a Member of the Institute for Advanced Study. I did my PhD at MIT Math.

My research is focused on applied linear algebra, broadly defined. This includes topics like matrix analysis, numerical linear algebra, random matrix theory, spectral graph theory, etc. I am particularly interested in direct methods for linear systems, discrete nodal statistics, extremal eigenvalue problems, Krylov subspace/moment problems, and matrix recovery/recognition problems.

Here are a some of my favorite papers:

• On the Exponential Rate of the Condition Number of Fourier Submatrices..., Preprint
• Nodal Count for Orthogonally Invariant Ensembles, Preprint
• Average Nodal Count and the Nodal Count Condition for Graphs, Preprint
• A New Upper Bound for the Growth Factor in Gaussian Elimination..., Bull. LMS (2025)
• Maximum Spread of Graphs and Bipartite Graphs, Comm. AMS (2022)

I am grateful for support from the National Science Foundation, through DMS-2513687
(and DMS-2218846 and DMS-2522789, for the MIT PRIMES program).

Here are some honors I've received: SIAM DiPrima Prize, SIAM Linear Algebra Early Career Prize, ILAS Brualdi Prize, Dartmouth College Doctor of Letters, Penn State Distinguished Alumni Award

Below you'll find what I am teaching/what I have taught, a list of my students, all my papers, and some outreach I do these days. Here’s a (most likely outdated) CV.

Teaching:

• 18.S995: Matrix Analysis & Computations, MIT, Sp. '26
• 18.06: Linear Algebra, MIT, Fall '25
• 18.335: Introduction to Numerical Methods, MIT, Sp. '24
• 18.330: Introduction to Numerical Analysis, MIT, Fall '23
• Math 232: Integral Vector Calculus, Penn State, Fall '13
• Math 041: Trigonometry and Analytic Geometry, Penn State, Sp. '13

Students:

Papers / Preprints / Theses:

• On the Exponential Rate of the Condition Number of Fourier Submatrices and Vandermonde Matrices
  with R. Shah. arXiv, 2026.
• Spectral Density Estimation for Normal Matrices
  with C. Musco, C. Musco, R. Shah, N. West. arXiv, 2026.
• What is Jackson's Constant?
  with R. Shah, N. West. arXiv, 2026.
• A Class of Optimal Directed Graphs for Network Syncronization
  with J. Liu, S. Lu. arXiv, 2026.
• The Largest 5th Pivot may be the Root of a 61st Degree Polynomial
  with J. Chen, A Edelman. IMA Journal of Numerical Analysis, 2026.
• Numerical Stability in Gaussian Elimination
  Notices of the AMS, 2025.
• Nodal Count for Orthogonally Invariant Ensembles
  with L. Alon, D. Mikulincer. arXiv, 2025.
• A New Upper Bound for the Growth Factor in Gaussian Elimination with Complete Pivoting
  with A. Bisain, A. Edelman. Bulletin of the LMS, 2025.
• Estimating the Matrix p -> q Norm
  with L. Guth, D. Maldague. SIAM Journal on Matrix Analysis, 2025.
• On the Maximum Spread of Non-Negative Matrices
  with S. Lu. Linear Algebra and its Applications, 2025.
• On the Frobenius Norm of the Inverse of a Non-Negative Matrix
  with E. Frankel. Linear Algebra and its Applications, 2025.
• Estimating the Numerical Range with a Krylov Subspace
  with C. Chen. arXiv, 2024.
• Average Nodal Count and the Nodal Count Condition for Graphs
  with L. Alon. arXiv, 2024.
• Nodal Decompositions of a Symmetric Matrix
  with T. McKenzie. International Math Research Notices, 2024.
• Some New Results on the Maximum Growth Factor in Gaussian Elimination
  with A. Edelman. SIAM Journal on Matrix Analysis, 2024.
• Recovering a Magnitude-Symmetric Matrix from its Principal Minors
  with VE. Brunel. Linear Algebra and its Applications, 2024.
• Hamilton Powers of Eulerian Digraphs
  with E. Colón. Electronic Journal of Combinatorics, 2024.
• On a Perturbation Analysis of Higham Squared Maximum Gaussian Elimination Growth Matrices
  with A. Edelman, B. Zhu. arXiv Technical Report, 2024. (see Bowen's associated thesis here)
• Representing the Special Linear Group with Block Unitriangular Matrices
  PNAS Nexus, 2023.
• Maximum Spread of Graphs and Bipartite Graphs
  with J. Breen, A. Riasanovsky, M. Tait. Communications of the AMS, 2022.
• Graphs, Principal Minors, and Eigenvalue Problems
  PhD Thesis, MIT, 2021.
• Multidimensional Scaling: Approximation and Complexity
  with E. Demaine, A. Hesterberg, F. Koehler, J. Lynch. ICML 2021.
• Uniform Error Estimates for the Lanczos Method
  SIAM Journal on Matrix Analysis, 2021.
• Regarding Two Conjectures on Clique and Biclique Partitions
  with D. Rohatgi, J. Wellens. Electronic Journal of Combinatorics, 2021.
• Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs
  with L. Zikatanov. Linear Algebra and its Applications, 2021.
• Testing Gap k-Planarity is NP-Complete
  with J. Wellens. Information Processing Letters, 2021.
• Nodal Decompositions of Graphs
  Linear Algebra and its Applications, 2018.
• Maximum Likelihood Estimation of Determinantal Point Processes
  with VE. Brunel, A. Moitra, P. Rigollet. arXiv Technical Report, 2017.
• On the Characterization and Uniqueness of Centroidal Voronoi Tessellations
  SIAM Journal on Numerical Analysis, 2017.
• Learning Determinantal Point Processes with Moments and Cycles
  with VE. Brunel, A. Moitra, P. Rigollet. ICML 2017.
• Rates of Estimation for Determinantal Point Processes
  with VE. Brunel, A. Moitra, P. Rigollet. COLT 2017.
• On the Approximation of Laplacian Eigenvalues in Graph Disaggregation
  with X. Hu, L. Zikatanov. Linear and Multilinear Algebra, 2017.
• On the Maximal Error of Spectral Approximation of Graph Bisection
  with L. Zikatanov. Linear and Multilinear Algebra, 2016.
• A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians
  with X. Hu, J. Xu, L. Zikatanov. Journal of Computational Mathematics, 2015.
• Spectral Bisection of Graphs and Connectedness
  with L. Zikatanov. Linear Algebra and its Applications, 2014.
• Graph-Based Topics in Applied Mathematics
  Master's Thesis, Penn State, 2013.
• A Space-Time Multigrid Method for the Numerical Valuation of Barrier Options
  Communications in Mathematical Finance, 2013.
• Instabilities in the Sun-Jupiter-Asteroid Three Body Problem
  with J. Galante. Celestial Mechanics and Dynamical Astronomy, 2013.

Outreach:

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