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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 CV (pdf)



Teaching:

Spring 2012 I taught two sections of MA266 (Ordinary Differential Equations). Information can be found here. Here is the general course page.
REU:

Summer 2012 I am organizing an REU with three students. They will learn about gravitational lensing, which is the fastest growing and perhaps the most important current area of research in astrophysics. It is also a nice source of problems that can be easily stated purely in terms of complex variables, which is what the students will work on. The course will include some historical background and physics of gravitational lensing and derivation of lensing models, a review of complex variables basics (Cauchy Integral Formula, argument principle, conformal maps, and harmonic functions), as well as more specialized topics (generalized argument principle for harmonic maps, area and arc-length quadrature domains, calculating Cauchy transforms using the Plemelj-Sokhotsky decomposition of the Schwarz function, and Fatou's Theorem from complex dynamics). After the background material, the students will work on problems over a wide range of difficulty. Possible research problems could include deriving their own models for cases that haven't yet been studied using complex variables, studying the maximum number of images possible for a given family of examples, and investigating the magnification relations among images.
Research:

My research is mainly in holomorphic PDEs and potential theory, but what I really love are physically-motivated problems that are simple to state but with solutions requiring "pure" mathematics. I am also delighted by unexpected connections and interactions between different areas.

Publications

Talks

My Ph.D. Thesis (pdf)

Current pursuits: My collaborators and I have made some recent progress on a few investigations:

an overdetermined problem for the Laplace equation (with D. Khavinson and R. Teodorescu),

algebraicity of higher dimensional quadrature domains (with A. Eremenko) preprint,

the self-commutator of Toeplitz operators acting on Bergman space and isoperimetric inequalities (with T. Ferguson),

generalized patterns in permutation theory (with J. Cooper and B. Nagle),

a survey of lemniscates as moving boundaries (with V. Totik) preprint,

and bijectivity of Fischer operators (with M. Cecil, and R. Walker)

I hope to soon post some more preprints.

While trying to branch out during my postdoc, lately I'm learning about minimal surfaces from Allen Weitsman and the interaction of quadrature domains with function spaces from Steve Bell. I look forward to working on interesting problems in those areas. I'm studying random geometry with Antonio Lerario. We hope to run a seminar on random matrix theory and random algebraic geometry.

As a recreational pursuit, this summer I would like to write a short exposition explaining the inverse scattering method by example. For instance, it would be enlightening to see the n-soliton solution of KdV explained with all details. I'm thinking to draw on Novikov's book "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.

I am also 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.


Personal information can be found here.

A custom Purdue homepage can be found here.


Some interesting links:

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.