From Fall 2011 until Spring 2014, I was a Golomb Visiting Assistant Professor of Mathematics.
During that time, I worked with the Analysis group, Real Algebraic Geometry group, and Analytic Combinatorics group at Purdue.
In Fall 2014, I begin a tenure-track position at Florida Atlantic University in Boca Raton.
This joint work with my friend and collaborator Antonio Lerario
began while we were both postdocs at Purdue.
I was immediately captivated when Antonio gave a seminar talk describing open problems in the topology of random real algebraic sets.
Planar harmonic mappings, minimal surfaces, and harmonic polynomials.
Quadrature domains and free boundary problems.
Coauthors of papers published and submitted:
Steven R. Bell, Daoud Bshouty, Joshua N. Cooper, Alexandre Eremenko, Brett Ernst, Sean Fancher, Timothy Ferguson,
Rodrigo Ferraz de Andrade,
Yan V. Fyodorov, Jonathan Hauenstein, Charles R. Keeton, Dmitry Khavinson,
Abi Komanduru, Ludwig Kuznia,
Seung-Yeop Lee, Antonio Lerario, Dhagash Mehta, Brendan Nagle, Hermann Render,
Razvan Teodorescu, Vilmos Totik, Allen Weitsman.
(my Erdos number is 2)
REU in gravitational lensing:
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 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).