The 2019 DOmath program ran from **May 20 to July 12, 2019**. Here are the five projects, along with the teams' summaries of what they accomplished. The links contain more detailed information, including more technical summaries and final reports where available. **Each project's web page now contains a video of students from the team describing their project and experience.**

**Mysterious unramified zeta functions**

led by Professor **Jayce R. Getz**

In our project we studied a certain L-function, which is a generalization of the famous Riemann zeta function. Because functions like these can be written as an infinite product over the prime numbers, we can learn many facts about number theory and the distribution of the prime numbers by studying them. Dr. Getz found a new class of these L-functions and asked us to look for a simplest rational form for one term of this infinite product, which we found using techniques from representation theory and algebraic combinatorics.

**Solving Nahm's equation**led by Professor

**Ákos Nagy**

Nahm's equations are a system of nonlinear ordinary differential equations that come from mathematical physics, in particular, gauge theory and particle physics. They arise out of the Nahm transform, which allow solutions of Nahm's equations to be transformed into solutions of a much more complicated partial differential equation problem, and vice versa. From purely mathematical perspective, the space of solutions to Nahm's equations is interesting because it has a particularly nice geometry. Since the nonlinearity of the system almost precludes a general solution, we utilized the natural structure that the equations are defined on and exploited symmetries of Nahm's equations to construct and study solutions in special cases. We also performed numerical studies of the equations.

**ODEs with random parameters**

led by Professor **James Nolen**

**Polarization of disordered materials**

led by Professor **Alexander Watson **

**Dynamics of floating plates on thin films**

led by Professors **Thomas Witelski** and **Jeffrey Wong**