# Prize Spotlight: Thomas Trogdon

The award recognizes Trogdon for his versatility in combining orthogonal polynomials and special functions in new and creative ways to deduce results in a variety of fields, such as rational approximation, random matrices, and Riemann-Hilbert problems. As part of the award, he delivered a lecture, “The High Oscillation of Special Functions,” at the meeting.

The SIAM Activity Group on Orthogonal Polynomials and Special Functions (SIAG/OPSF) awards the Gábor Szegő Prize every two years to a researcher in their early career for outstanding research contributions in the area of orthogonal polynomials and special functions. The prize is awarded to a researcher who has at most 10 years involvement in mathematics since receiving their PhD, or who, in the opinion of the selection committee, is at an equivalent stage in their career. The contributions must be contained in a paper or papers published in English in peer-reviewed journals.

Thomas Trogdon received his PhD in applied mathematics from the University of Washington under the supervision of Bernard Deconinck in 2013. He was awarded the 2014 SIAM Richard C. DiPrima Prize for his dissertation titled *Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions*. SIAM later published his book by the same title, co-authored with Sheehan Olver.

Dr. Trogdon is an assistant professor of mathematics at the University of California, Irvine. Before that, he was an NSF Postdoctoral Fellow under the supervision of Percy Deift at the Courant Institute of Mathematical Sciences, New York University, where his research focused on applications of random matrix theory to numerical analysis (and vice versa).

**Q:** *Why are you excited about winning this prize?*

**A:** I feel very honored to have been selected to receive the 2017 Gábor Szegő Prize. The study of orthogonal polynomials and special functions (OPSF) is in many ways at the heart of modern applied and computational mathematics, and certainly at the core of my research. And to be recognized by the SIAM Activity Group on OPSF is a special honor.

**Q:** *Could you tell us a bit about the research that won you the prize?*

**A:** I have long been interested in the numerical solution of oscillatory and singular integral equations posed on infinite domains. Such integral equations, usually being equivalent to a so-called Riemann-Hilbert problem, arise in the solution of integrable partial differential equations, in special function theory, and in Weiner-Hopf factorization problems. These integral equations present a series of issues, some more obvious (singularities and oscillation) and some less obvious (coefficient functions that eliminate the possibility of the truncation of the infinite domain). The work recognized by the 2017 Gábor Szegő Prize aims to expand a set of computational tools for such integral equations and to analyze the error of these tools. Specifically, the tools and their analysis involve a combination of ideas from infinite-dimensional linear algebra, interpolation, orthogonal polynomials and special functions (Laguerre polynomials and hypergeometric functions), and Fourier analysis on Sobolev spaces.

**Q:** *What does your research mean to the public?*

**A:** The research presents the construction and analysis of new methods in approximation theory. These methods allow for the descriptive numerical solution of problems in inverse scattering and multi-sensor Wiener filtering. We hope that the study of these applications through this descriptive method will give an increased understanding of both physical and biological phenomena, including the ability of the auditory cortex to filter noise.

**Q:** *What does being a SIAM member mean to you?*

**A:** I take pride in being a SIAM member. SIAM provides a high quality platform to share and discuss research via academic journals and conferences. SIAM is an important component of my academic identification.