**Number Theory**

Faculty with research interests in this area:

- Dr. Ayla Gafni
- Dr. Rizwanur Khan
- Dr. Thái Hoàng Lê
- Dr. Micah B. Milinovich

Number Theory is the field of mathematics that studies arithmetic properties and patterns in the integers. It is one of the oldest branches of mathematics and comprises a vast array of problems and methods. The Number Theory Group at the University of Mississippi includes four tenure-track faculty, one visiting faculty, and several PhD students. We have an active research seminar which meets weekly during the semester. Below is a brief description of each faculty’s interest areas.

Ayla Gafni studies the ways in which harmonic analysis methods can be applied to number theoretic problems. Her particular area of expertise is the Hardy-Littlewood circle method, which is a powerful technique with applications to a multitude of problems. She has used the circle method to develop a systematic method for enumerating certain restricted integer partitions. Gafni is also interested in the distribution of sequences, using harmonic analysis to investigate the relationships between the arithmetic properties of the integers and the pseudorandom characteristics of associated sequences on the unit interval.

Rizwanur Khan is interested in the analytic theory of L-functions and automorphic forms, which are generalizations of the Riemann Zeta function and modular forms, respectively. This includes studying moments, non-vanishing, and subconvexity bounds of L-functions, and the distribution of automorphic forms, with applications to Arithmetic Quantum Chaos.

Thái Hoàng Lê is interested in number-theoretic problems with an additive or combinatorial flavor. He likes to find structures in dense subsets, sumsets, and difference sets, in the integers and the primes. He also studies these phenomena in other settings such as finite fields, function fields, and general abelian groups. To tackle these questions, he also uses tools from other areas such as Fourier analysis and ergodic theory.

Micah B. Milinovich is interested in the distribution of the primes, the theory of the Riemann zeta-function and other L-functions, the connections between L-functions and arithmetic, and the interplay between Fourier analysis and number theory.