John Urschel
I am a mathematician. My research is focused on graph theory, numerical analysis, and machine learning. I am primarily interested in theoretical results and provable guarantees for practical problems, which often lead to new and improved algorithms.
I am currently a Junior Fellow at Harvard University. Starting in Fall 2023, I will be an assistant professor in the MIT math department. Previously, I was a member of the Institute for Advanced Study. 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, can be found here.
Here are a few selected publications:
 Maximum Spread of Graphs and Bipartite Graphs, Comm. AMS (to appear) [PDF]
 Uniform Error Estimates for the Lanczos Method, SIMAX (2021) [PDF]
 Regarding Two Conjectures on Clique and Biclique Partitions, EJC (2021) [PDF]
 On the Characterization and Uniqueness of Centroidal Voronoi Tessellations, SINUM (2017) [PDF]
 Learning Determinantal Point Processes with Moments and Cycles, ICML (2017) [PDF]
Below, you can find a brief description of my research, a full list of my publications, my current and past teaching, some outreach programs I am involved in, and my contact information. Here is a (most likely outdated) CV.
Research:
My interests largely consist of topics in graph theory, numerical analysis, and machine learning, most of which share a foundation in matrix analysis. Below is a brief description of each topic and my interests within each discipline.
Graph Theory: Graphs are structures that capture pairwise relationships between a discrete set of objects. This abstract formulation makes graphs useful in a wide variety of contexts. The majority of my research in this area focuses on spectral graph theory, the study of matrices associated with a graph. Spectral graph theory has proven useful in a number of applications, such as graph partitioning, community detection, dimension reduction, and data visualization. I am mostly interested in proving theorems about spectral properties of graphs.
Selected Publications:
 Maximum Spread of Graphs and Bipartite Graphs, Comm. AMS (to appear) [PDF]
 Regarding Two Conjectures on Clique and Biclique Partitions, EJC (2021) [PDF]
 Spectral Bisection of Graphs and Connectedness, Lin. Alg. & App (2014) [PDF]
Numerical Analysis: The field of numerical analysis is fundamentally concerned with the efficient computation of approximate solutions to problems in mathematical analysis. My research is mostly focused on numerical linear algebra, i.e., the numerical solution of linear systems Ax = b and eigenvalue problems Ax = λx. These computations have widereaching applications in all areas of science. My research is mostly focused on eigenvalue problems, though I am also quite interested in Gaussian elimination.
Selected Publications:
 Uniform Error Estimates for the Lanczos Method, SIMAX (2021) [PDF]
 A Cascadic Multigrid Alg. for Comp. the Fiedler Vector of Graph Lapl., J. Comp. Math (2015) [PDF]
 A SpaceTime Multigrid Meth. for the Num. Val. of Barrier Options, Comm. in Math. Fin. (2013) [PDF]
Machine Learning: Data science is a broad set of tools used to extract useful information from (large) data sets and machine learning is concerned with algorithms that improve through experience (i.e., increases in data). I'm particularly interested in quantization/kmeans clustering, graphbased data visualization, and determinantal point processes (DPPs). DPP models are an alternative to graphical models, and have an elegant linear algebraic formulation. My research on DPPs focuses on learning algorithms with proveable guarantees.
Selected Publications:
 Multidimensional Scaling: Approximation and Complexity, ICML (2021) [PDF]
 On the Characterization and Uniqueness of Centroidal Voronoi Tessellations, SINUM (2017) [PDF]
 Learning Determinantal Point Processes with Moments and Cycles, ICML (2017) [PDF]
Papers:
✷ = select publication, u = undergraduate research project

Hamilton Powers of Eulerian Digraphs^{u},
with
Enrico Colón.
arXiv:2210.10259, 2022. [PDF]

Maximum Spread of Graphs and Bipartite Graphs^{✷},
with
Jane Breen,
Alex Riasanovsky,
Michael Tait.
Communications of the AMS, to appear. [PDF]

Regarding Two Conjectures on Clique and Biclique Partitions^{u},
with
Dhruv Rohatgi,
Jake Wellens.
Electronic Journal of Combinatorics, 2021. [PDF]

Multidimensional Scaling: Approximation and Complexity,
with
Erik Demaine,
Adam Hesterberg,
Frederic Koehler,
Jayson Lynch.
Proceedings of the 38th International Conference on Machine Learning (ICML 2021). [PDF]

Uniform Error Estimates for the Lanczos Method^{✷}.
SIAM Journal on Matrix Analysis, 2021. [PDF]

Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs,
with
Ludmil Zikatanov.
Linear Algebra and its Applications, 2021. [PDF]

Testing Gap kPlanarity is NPComplete,
with
Jake Wellens.
Information Processing Letters, 2021. [PDF]

Nodal Decompositions of Graphs.
Linear Algebra and its Applications, 2018. [PDF]

Maximum Likelihood Estimation of Determinantal Point Processes,
with
VictorEmmanuel Brunel,
Ankur Moitra,
Philippe Rigollet.
arXiv:1701.06501, 2017. [PDF]

On the Characterization and Uniqueness of Centroidal Voronoi Tessellations^{✷}.
SIAM Journal on Numerical Analysis, 2017. [PDF]

Learning Determinantal Point Processes with Moments and Cycles^{✷},
with
VictorEmmanuel Brunel,
Ankur Moitra,
Philippe Rigollet.
Proceedings of the 34th International Conference on Machine Learning (ICML 2017). [PDF]

Rates of Estimation for Determinantal Point Processes,
with
VictorEmmanuel Brunel,
Ankur Moitra,
Philippe Rigollet.
Proceedings of the 30th Annual Conference on Learning Theory (COLT 2017). [PDF]

On the Approximation of Laplacian Eigenvalues in Graph Disaggregation,
with
Xiaozhe Hu,
Ludmil Zikatanov.
Linear and Multilinear Algebra, 2017. [PDF]

On the Maximal Error of Spectral Approximation of Graph Bisection,
with
Ludmil Zikatanov.
Linear and Multilinear Algebra, 2016. [PDF]

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians,
with
Xiaozhe Hu,
Jinchao Xu,
Ludmil Zikatanov.
Journal of Computational Mathematics, 2015. [PDF]

Spectral Bisection of Graphs and Connectedness,
with
Ludmil Zikatanov.
Linear Algebra and its Applications, 2014. [PDF]

A SpaceTime Multigrid Method for the Numerical Valuation of Barrier Options.
Communications in Mathematical Finance, 2013. [PDF]

Instabilities in the SunJupiterAsteroid Three Body Problem^{u},
with
Joseph Galante.
Celestial Mechanics and Dynamical Astronomy, 2013. [PDF]
Courses:

18.03: Differential Equations,
MIT, Spring 2018

Math 232: Integral Vector Calculus,
Penn State, Fall 2013

Math 041: Trigonometry and Analytic Geometry,
Penn State, Spring 2013
Outreach:
Here are a few programs outside of my core responsibiltiies as an academic mathematician that I take part in. If you are a K12 math teacher or student, some of these programs may be of interest to you.
MathROOTS: MathROOTS is a twoweek summer program hosted by MIT math for highpotential high school students from underrepresented backgrounds or underserved communities. I am in charge of the the academic side of the program. I create the curriculum, give the lectures, and lead the problem solving sessions. More details regarding this program can be found here.
Mathical: The Mathical Book Prize, awarded by MSRI, is an annual award for fiction and nonfiction 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 (MoMath) highlights the role of math in illuminating the patterns and structures all around us. It has dynamic exhibits, galleries, and programs designed to stimulate inquiry, spark curiosity, and reveal the wonders of math to those of all ages. I serve on the board of the museum and am particularly involved in the exhibits and programming of the museum. More details can be found here.
Contact:
Email: urschel AT mit DOT edu (academic* emails only)
Office: 524 Science Center (appointment** required)
© 2020 John C. Urschel  template shamelessly stolen from Tselil Schramm.
^{*}If you are an academic (in math) or student (at MIT) and your email is about math (if an academic, I mean research mathematics), I will try to respond fairly quickly. Otherwise, I will try my best, but I don't always have time to answer every email I get. I hope you understand.
^{**}If you are not affiliated with MIT math or a student at MIT (interested in math), please do not stop by my office without an appointment. I do not open nonacademic mail sent to my office address.
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