Mathematical and Statistical Sciences

864-656-1284

Martin Hall O229 [Office]

I mainly consider models from physics and study those using a variety of mathematical methods. Methods that range from dynamical systems theory, measure theory, differential geometry, algebra and low dimensional topology. Many of those theories are naturally employed when studying polygonal billiards via (flat) Riemann surface and Teichmueller theoretic methods.

Many systems I have studied are close relatives of the billiard in a polygon such as the dynamics in planar Eaton lens patterns. To get a foot into applied analysis I worked on problems in optimal mass transportation and ergodic optimization, its cousin from dynamics. The rough idea is to show that dynamical systems can be used to show statements in optimal transportation and the other way round. There exist examples in either direction.

For potential students: While some approaches to polygonal billiard theory require a broad and deep mathematical background, there are plenty of research topics, that do not require a big machinery. Moreover some topics can be started with nearly any mathematical background of the ones mentioned above. Depending on particular interest. I supervise topics ranging from optimal transportation, probabilistic dynamics, dynamical systems and differential geometry that can be conquered by self-study based on our standard coursework. An interest in physics, or other applications of mathematics, is a good driving force. We have a quite substantial and still growing local dynamical systems community that meets once a year at the NSF funded Carolina Dynamics Symposium, short CDS.

MATH 4530/6530, Advanced Calculus I, Sp20, S 20, Fa 20

MATH 4540/6540, Advanced Calculus II, Fa20, Sp21

MATH 8370, Calculus of Variations, Sp20, Sp21

MATH 8230, Complex Analysis, Fa19

MATH 9820, Geometric Analysis, Sp19

MATH 8210, Linear Analysis, Sp19

MATH 8250, Dynamical Systems, Fa18

MATH 8220, MeasureÂ Theory, Sp18MATH 2060, Vector Analysis, Sp18, Fa18

(w. David Aulicino) Weighted Siegel-Veech constants for branched cyclic covers. We hope to place the first preprint of a series about summer 2021.

(w. Michael Burr and Christian Wolf) On the computability of rotation sets and their entropies. Ergodic Theory Dyn. Syst. 40, No 2 367-401 (2020).

(w. Krzysztof Fraczek) On ergodicity of Foliations on Z^d covers (...) and applications to periodic systems of Eaton Lenses. Communications in Math Physics Volume 362, Issue 2, pp 609--657;

(w. Chris Johnson) Pseudo-Anosov eigenfoliations on Panov planes. E.R.A.M.S. 21, 89-108, (2014).(w. Krzysztof FrÄ…czek) Directional localization of light rays in a periodic array of retro-reflector lenses.

Nonlinearity 27, No. 7, 1689-1707 (2014).

(w. Omri Sarig) Adic flows, transversal flows, and horocycle flows. De Gruyter Proceedings in Mathematics, 241-259 (2014)