Erik Lundberg
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 research 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. Specific areas I have worked on include problems in quadrature domains and free boundary problems (and relation to fluid dynamics and inverse potential problem, two-dimensional case vs. higher dimensions), harmonic mapping problems (both from the degree-theory point of view applied to multivalent maps arising in gravitational lensing as well as univalent harmonic maps arising in minimal surfaces), mathematical physics (Laplacian growth, gravitational lensing, random waves), nonlinear PDE (minimal surface equation and integrable evolution equations), combinatorics (probabilistic and analytic combinatorics, problems in pattern avoidance in permutations), real algebraic geometry (geometry and topology of random real algebraic sets), functional analysis in Hilbert spaces of analytic functions, polynomial solutions to PDE, and analytic continuation and singularity analysis of solutions to PDE.
Some elaboration on my most recent projects is given below (some topics have links to a short page).
Random Algebraic Geometry:
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 used a method developed by Nazarov and Sodin for spherical harmonics,
and was further inspired by a hand-written letter of P. Sarnak.
We also give some discussion of a random version of Hilbert's sixteenth problem.
Here is Antonio Lerario's webpage.
Together with Saugata Basu we also had a small seminar last semester in order to learn a variety of topics
relevant to the topology of random algebraic sets.
This semester, we are focusing on random matrix theory.
Harmonic Mappings and Minimal Surfaces:
I've been learning about minimal surfaces from
Allen Weitsman, and working on problems involving minimal surfaces and harmonic maps.
Related to this, D. Bshouty, A. Weitsman, and I recently worked on the mapping problem for polygons posed by T. Sheil-Small
(this is related to Jenkins-Serrin minimal surfaces).
We found a complete solution showing that any polygon bounding a Jordan domain can be mapped univalently by the Poisson integral of a step function
(preprint).
Algebraicity of Quadrature Domains:
Alex Eremenko and I recently gave a negative answer (preprint) to a question of H. S. Shapiro,
by showing that higher-dimensional quadrature domains are not always algebraic.
Norm Estimates of Commutators:
Timothy Ferguson, Steve Bell, and I have recently obtained (preprint) lower estimates
for the self-commutator of a Toeplitz operator acting on Bergman space.
The methods combine geometric and functional analysis, and
encountered the classical isoperimetric inequalities from mathematical physics (the Faber-Krahn inequality for
the base tone of a drum, and the Saint-Vennant inequality for torsional rigidity).
We made a conjecture about Putnam's inequality not being sharp within the setting of an analytic Toeplitz operator acting on Bergmann space.
Generalized Patterns in Permutations:
Joshua Cooper, Brendan Nagle, and I just completed a study (preprint)
on generalized patterns in permutations.
We used probabilistic methods to study avoidance and frequency of generalized patterns,
and we also answered a question of S. Elizalde on avoidance of 12-34.
Free Boundary Problem for Laplace Equation:
Dmitry Khavinson, Razvan Teodorescu, and I recently worked on a problem of
(preprint) conjecture of L. Hauswirth, F. H\'elein, and F. Pacard,
to characterize all "exceptional domains" (domains that admit a harmonic function with zero Dirichlet data and constant Neumann data).
Under a priori assumptions, an exceptional domain
in the plane must be one of three types, exterior of the disk, half-plane, or a certain nontrivial example.
The Heat Equation and Bijectivity of Fischer Operators:
For those that have not seen the shockingly simple proof
that Dirichlet's problem with polynomial data on an ellipsoid always has a polynomial solution,
stop reading this page and go here.
D. Khavinson and H. S. Shapiro also showed that entire data gives entire solution.
Matt Cecil, Ron Walker, and I are investigating parallel questions about the solutions
of boundary value problems for the heat equation.
An Electrostatic Skeleton Problem:
V. Totik and I have a recent preprint giving a survey of lemniscates as moving boundaries.
We encountered an interesting problem posed by E. Saff regarding electrostatic skeletons.
The problem can be found restated in Section 4 of our paper.
Integrability of the KdV Equation:
As a recreational pursuit, I'm writing a short exposition explaining the inverse scattering method by example,
because I think it would be enlightening to see the n-soliton solution of KdV explained WITH ALL THE DETAILS.
I'm drawing on the book by Novikov et al "Theory of Solitons", Tao's survey "Why are solitons stable?",
and the last Chapter (on Riemann-Hilbert problems) from the Complex Analysis textbook by Ablowitz and Fokas.
Book on Holomorphic PDE: I am working with D. Khavinson on writing a book: "holomorphic partial differential equations".
We gave a mini-course on the subject matter at the conference
HCAA 2012.
Here are the slides: Four Lectures.
Learning Seminars: This semester I am part of a few learning seminars, one in Random Matrix Theory (using Tao's book), one in Analytic Combinatorics
(using Flajolet's book),
and one in Complex Dynamics (using Milnor's book).
Personal information can be found here.
A custom Purdue homepage can be found here.
Carl Bender's lecture videos in mathematical physics and perturbation theory