Golomb Assistant Professor
Department of Mathematics, Purdue University
150 N. University Street
West Lafayette, IN 47907-2067
Office: MATH 410
E-mail: elundber AT math.purdue.edu
My CV (pdf)
My Ph.D. Thesis (pdf)
At Purdue, I am working with the Analysis group, Real Algebraic Geometry group, and Analytic Combinatorics group.
I developed my background initially working in areas of Analysis and holomorphic PDEs, potential theory, asymptotics, complex variables,
and dynamical systems, but what I really love are nice problems (physically-motivated and/or simple to state).
I am also delighted by unexpected connections and interactions between separate areas of mathematics.
Problems I have worked on include harmonic function theory (quadrature domains, free boundary problems, harmonic mappings), mathematical physics (Laplacian growth, gravitational lensing, minimal surfaces, integrable evolution equations), expected topology of random real algebraic sets, combinatorics, and functional analysis (Hilbert spaces of analytic functions).
Upcoming Research in Pairs: Antonio Lerario and I will spend June 2014 in France for an intense month of work continuing the theme of our recent collaborations investigating the topology of random real algebraic varieties. The problems we are working on are enticing and have required a nontrivial combination of techniques including Antonio's background in algebraic topology, my background in harmonic analysis, and also topics we learned together, such as random matrix theory and integral geometry.
Here are some links to short descriptions of recent work:
Random Geometry: Probabilistic study of the geometry and topology of real algebraic sets. Antonio Lerario and I recently showed that the expected number of components of a hypersurface given by a random homogeneous polynomial in the Fubini-Study ensemble has maximal order (preprint). This was inspired by a hand-written letter of P. Sarnak and used the "barrier method" developed by Nazarov and Sodin for spherical harmonics. Our more recent work (preprint) uses random matrix theory to study the expected Betti numbers of an intersection of quadrics.
Harmonic Mappings: Including: (1) A complete solution to the mapping problem for polygons posed by T. Sheil-Small (1989) related to Jenkins-Serrin minimal surfaces. (2) Growth of minimal surfaces that are graphs over unbounded domains. (3) Counterexamples to Wilmshurst's conjecture (1995) on the valence of harmonic polynomials.
Interaction of Algebraic Geometry and PDE: Including: (1) An answer to H. S. Shapiro's question (1992) on algebraicity of quadrature domains. (2) Free boundary problems for Laplace's equation (and an answer to a recent question of Hauswirth, Helein, and Pacard). (3) Algebraic boundary value problems for Laplace's equation and the heat equation.
Functional analysis: Estimates for the "abnormality" of a Toeplitz operator lead to a novel operator-theoretic proof of Saint-Venant's isoperimetric inequality from elasticity theory. The methods combine geometric and functional analysis.
Combinatorics: Avoidance and frequency of generalized patterns and the excedence set statistic, taking the point of view of both probabilistic and analytic combinatorics. Enumerative results and asymptotics with an answer to a question of S. Elizalde on avoidance of 12-34 and an answer to a question of R. Ehrenborg and E. Clark on asymptotics of the extremal excedance set statistic.
Coauthors of papers published and submitted: Steven R. Bell, Daoud Bshouty, Joshua N. Cooper, Alexandre Eremenko, Brett Ernst, Sean Fancher, Timothy Ferguson, Dmitry Khavinson, Abi Komanduru, Ludwig Kuznia, Seung-Yeop Lee, Antonio Lerario, Brendan Nagle, Hermann Render, Razvan Teodorescu, Vilmos Totik, Allen Weitsman. (my Erdos number is 2)
Learning Seminars at Purdue: I have helped organize several small learning seminars including postdocs, graduate students, and undergraduates: one in Random Matrix Theory (using Tao's book), one in Analytic Combinatorics (using Flajolet's book), one in Complex Dynamics (using Milnor's book), and a seminar in random algebraic geometry (using papers of Edelman, Kostlan, Fyodorov, Shub and Smale, Burgisser, Nicolaescu, Nazarov and Sodin, Gayet and Welschinger).
Personal information can be found here.
A page of important Purdue links can be found here (I use this as my home page.)
Terence Tao's blog
Cool Math and Physics Applets
MIT open courses
Carl Bender's lecture videos in mathematical physics and perturbation theory
Although I disagree with him on many points, I find that Doron Zeilberger is refreshingly opinionated.