I am an applied mathematician. My research is focused on numerical analysis, graph theory, and data science/machine learning. I am primarily interested in theoretical results and provable guarantees for practical algorithms/problems, which often lead to new and improved algorithms.
I am currently a fifth-year PhD student at MIT math, and have the pleasure of being advised by Michel Goemans. I expect to graduate in Spring 2021.
Here are a few selected publications:
1. Uniform Error Estimates for the Lanczos Method, Preprint (2020) [PDF]
2. Regarding Two Conjectures on Clique and Biclique Partitions, Preprint (2020) [PDF]
3. On the Characterization and Uniqueness of Centroidal Voronoi Tessellations, SINUM (2017) [PDF]
4. Learning Determinantal Point Processes with Moments and Cycles, ICML (2017) [PDF]
5. Spectral Bisection of Graphs and Connectedness, Lin Alg & App (2014) [PDF]
Below, you can find a detailed description of my research, a full list of my publications, and my current and past teaching. Here is a (most likely outdated) CV. My research and teaching statements are available upon request. My email is above, though non-academic emails will receive no response.
Research Interests:
My research largely consists of topics in numerical linear algebra, graph theory, and data science/machine learning, all of which share a foundation in applied linear algebra. Below is a description of each research topic, as well as a brief discussion of my particular interests within each discipline.
Numerical Linear Algebra: The field of numerical linear algebra is fundamentally concerned with the numerical solution of linear systems Ax = b and eigenvalue problems Ax = λx. These computations have applications in engineering, science, and big data/machine learning, and work in this area is focused on producing and analyzing algorithms that output a good approximation to an exact solution relatively quickly. My research is mostly focused on symmetric eigenvalue problems. I am particularly interested in fast algorithms for computing extremal eigenvalues, such as Krylov subspace methods, a dimension reduction technique to approximately solve a linear system or eigenvalue problem by mapping to a well-chosen lower dimensional subspace. One application area of great interest to me is the computation of low energy eigenpairs of graph Laplacians. The low dimensional eigenspaces of the Laplacian of a graph have proven useful in a number of areas in data science, including graph embedding and partitioning.
Sample Publications:
1. Uniform Error Estimates for the Lanczos Method, Preprint (2020) [PDF]
2. A Cascadic Multigrid Alg. for Comp. the Fiedler Vector of Graph Lapl., J. Comp. Math (2015) [PDF]
3. A Space-Time Multigrid Meth. for the Num. Val. of Barrier Options, Comm. in Math. Fin. (2013) [PDF]
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, depending on the interpretation of a pairwise relationship. 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. For instance, extreme eigenvalues of the Laplacian or adjacency matrix are used for partitioning, community detection, dimension reduction for large data sets, data visualization, and a number of other tasks in data science/machine learning theory. I am particularly interested in nodal domains and graph embeddings, among a wide array of research topics in the area.
Sample Publications:
1. Regarding Two Conjectures on Clique and Biclique Partitions, Preprint (2020) [PDF]
2. Discrete Trace Thms. and Energy Min. Spring Embed. of Planar Graphs, Lin Alg & App (2021) [PDF]
3. Spectral Bisection of Graphs and Connectedness, Lin Alg & App (2014) [PDF]
Data Science/Machine Learning: Data science is a broad class of techniques and tools used to extract useful information from (often large) data sets, and the related field of machine learning is concerned with algorithms that improve through experience (i.e., increases in data). This subject is one of the most popular and important research areas of the 21st century. My areas of interest include clustering, data visualization, and probabilistic models for machine learning. Clustering and data visualization have great overlap with graph theory, but I also study a number of approaches that do not have a graphical formulation, such as quantization/k-means. A particularly interesting class of probabilistic models are determinantal point processes (DPPs), a class of repulsive models which originated in quantum physics, and provide a strong alternative to graphical models. A DPP model captures negative correlation between a discrete set of objects, and has proven to be extremely useful in a number of applications, including the subset selection problem. The DPP model has an elegant linear algebraic formulation, as, given a set of n objects, the probability of a subset S appearing as a sample is proportional to the associated principal minor of some matrix. My research on DPPs mostly focuses on learning algorithms with proveable guarantees.
Sample Publications:
1. Maximum Likelihood Estimation of Determinantal Point Processes, Preprint (2020) [PDF]
2. On the Characterization and Uniqueness of Centroidal Voronoi Tessellations, SINUM (2017) [PDF]
3. Learning Determinantal Point Processes with Moments and Cycles, ICML (2017) [PDF]
Preprints:
Publications:
Courses:
Math 18.03: Differential Equations, MIT, Spring 2018
Role: Recitation Instructor
Subject Evaluation Report: 6.9/7 [PDF]
Math 232: Integral Vector Calculus, Penn State, Fall 2013
Role: Lecturer
Student Rating of Teaching Effectiveness: 6.71/7 [PDF]
Math 041: Trigonometry and Analytic Geometry, Penn State, Spring 2013
Role: Lecturer
Student Rating of Teaching Effectiveness: 6.59/7 [PDF]
© 2020 John C. Urschel -- template shamelessly stolen from Tselil Schramm.