# Mathematics doctoral student wins two awards

Congratulations to WSU mathematics doctoral student, Yatin Patel, for two recent awards: Yatin received the Outstanding Graduate Student Talk Award at this year's Automorphic Forms Workshop hosted by Brigham Young University and Yatin also won the first-place award for students giving poster presentations on March 9 in WSU's 2022 Graduate Research Symposium!

Yatin's presentation was titled "Minimal integral models for principal series Weil characters," which is the subject of his doctoral thesis.

## Presentation summary

The Automorphic Forms Workshop is the premier annual international conference on modular forms, Yatin's area of expertise. Modular forms are complex-valued functions that satisfy a rich collection of symmetries and transformation laws. They are ubiquitous in mathematics, often starring as the main protagonist of fundamental results in arithmetic, combinatorics, complex analysis and even physics.

The mathematician Martin Eichler famously quipped that modular forms are the "fifth" operation of arithmetic (after addition, subtraction, multiplication and division), given their status as one of the most fundamental tools in mathematics. In his work, Patel finds new patterns in the coefficients of a special type of modular form called "theta functions." These functions satisfy transformation laws whose coefficients belong to a representation of the modular group, called the "Weil Representation." Building on the previous work of Yilong Wang, Patel showed that after a suitable change of basis, these coefficients are algebraic integers lying in a quadratic extension of the rational numbers, dramatically improving previous results which did not give a bound on the degree of the extension.

These types of results were first motivated by integral topological quantum field theory and they could certainly open up new areas of exploration in connection with theoretical physics. In addition, Patel shows that theta functions should be thought of as purely (quadratic) integral objects, whereas traditionally at least a few primes are inverted. This new point of view could have interesting ramifications in arithmetic geometry, by giving integral geometric interpretations of the transformation laws of theta functions.

Yatin's excellence in scholarship and teaching has been recognized with the Paul Catlin Prize in 2015, the Maurice Zelonka Endowed Mathematics Scholarship in 2019, the Robert and Nancy Irvan Endowed Scholarship in Mathematics in 2020, the M.F. Janowitz Endowed Scholarship in 2021, the WSU Department of Mathematics Graduate Teaching Award for the year 2020-2021 and a Thomas Rumble Graduate Fellowship for the year 2021-2022.

Congratulations to Yatin for the awards from the Graduate Research Symposium and from the Automorphic Forms Workshop recognizing his excellent work!