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
(Alex Eremenko and his graduate student Koushik Ramachandran, Allen Weitsman, and Steve Bell), Real Algebraic Geometry group (Saugata Basu and Antonio Lerario),
and Analytic Combinatorics (Mark Daniel Ward and his graduate student Rodrigo Andrade).
My background is in holomorphic PDEs and potential theory, 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.
Areas 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).
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: The mapping problem for polygons posed by T. Sheil-Small (this is related to Jenkins-Serrin minimal surfaces), growth of minimal surfaces that are graphs over unbounded domains, and Wilmshurst's problem on the valence of harmonic polynomials.
Interaction of Algebraic Geometry and PDE: Algebraicity of quadrature domains, free boundary problems for Laplace's equation, and algebraic boundary value problems for Laplace's equation and the heat equation.
Functional analysis: Measuring the "abnormality" of a Toeplitz operator. Upper and lower bounds have led to "isoperimetric sandwiches", estimates that imply isoperimetric inequalities as a by-product. So far, the area-perimeter and Saint-Venant's isoperimetric inequalities have been recast in this way, while giving sharp bounds for the norm of an operator's self-commutator. The methods combine geometric and functional analysis and have some interaction with PDE.
Combinatorics: Avoidance and frequency of generalized patterns and related topics in permutation statistics: the excedence set and stretching pairs. I am especially interested in asymptotics (include multi-variate asymptotics) taking the point of view of both probabilistic and analytic combinatorics.
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.