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

Spring 2014: I am currently teaching MA266 (Ordinary Differential Equations). Here is the course page.

Fall 2013: I taught MA303 (Second semester in differential equations including an introduction to PDE).

Spring 2013: I taught MA265 (Linear Algebra).

Fall 2012 I taught MA266 (Ordinary Differential Equations).

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:**
Recent work includes: (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:**
Recent work includes: (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)

In the Summer of 2012 I organized an REU with three students, Brett Ernst, Sean Fancher, and Abi Komanduru. Steve Bell and I served as mentors. They learned about gravitational lensing, and complex variable methods (the generalized argument principle for harmonic maps, area and arc-length quadrature domains, Cauchy transforms, the Plemelj-Sokhotsky decomposition of the Schwarz function, and Fatou's Theorem from complex dynamics).

They developed a model for the lensing effect imposed by a spiral galaxy which assumes the mass density (projected to the lensing plane) is constant on each ellipse in a family of ellipses that rotate as they are scaled. This set up is motivated by Lin and Shu's density wave theory which assumes that spiral arms are a rather stationary effect of offset orbits (the stars move in and out of the spiral arms like cars moving through a stationary traffic jam). Check out this wikipedia page on this topic which includes a hypothetical animation of stars moving while the spiral arms persist: density wave theory.

Our (preprint) following the students' work derives a lensing equation which extends the ''arcsine lens'' for elliptical galaxies. An important feature of our model is that, even though the spiral structure of mass is sophisticated, we integrated the deflection term in closed form using a Gauss hypergeometric function. This lensing equation can be used to model the position, magnification, and orientation of images lensed by a spiral galaxy.

** 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.)

**Carl Bender's lecture videos in mathematical physics and perturbation theory**