The 2019 DOmath program will run from May 20 to July 12, 2019, and is open to all Duke undergraduate students. Student applications should be done through mathprograms.org and will be due February 15, 2019, although we will consider late applications until all slots are filled.
There are five teams planned for summer 2019, each of which will involve 2-5 undergraduate student researchers under the supervision of a faculty member(s) and a project manager, typically a graduate student. Here are descriptions of the five projects:
Mysterious unramified zeta functions, led by Professor Jayce R. Getz
Zeta functions are a marriage of algebra and analysis that are of fundamental importance throughout number theory. They can often be viewed as Euler products, in particular they admit an infinite product expansion indexed by the prime numbers (including infinity). Getz has recently uncovered a family of zeta functions that are, for the moment, mysterious. In this DOmath project the team will investigate these functions at "unramified" primes using techniques coming from finite group theory and the combinatorics of symmetric functions. Familiarity with permutations and linear algebra is useful but not necessary.
Solving Nahm's equation, led by Professor Ákos Nagy
Nahm's equation is a system of ordinary differential equations. It was first discovered as the dimensional reduction of the famous self-dual Yang-Mills equation (coming from particle physics). The difficulty in solving Nahm's equation comes from two sources: the equation is nonlinear, and the physically interesting solutions have singularities. The project's goal is to look for and investigate special cases in which Nahm's equation simplifies, and is solvable.
Since Nahm's equation is a system of ordinary differential equations, the team is expected to be comfortable with (or willing to learn) the basics of linear algebra and ordinary differential equations. If time permits, and if there is interest from the team, the connection to the so-called monopole equations can also be explored.
ODEs with random parameters, led by Professor James Nolen
This project deals with Ordinary Differential Equations (ODEs) having random parameters. In many ODE models of biological systems, for example, parameters vary widely across a population or may fluctuate randomly within an individual, so it is important to understand the role of randomness in such models. Some interesting mathematical research directions include: 1) What are the properties of the map from parameters (or from probability measures on the space of parameters) to solution data? 2) Given approximate observations of a solution, what can be inferred about parameters in the model? These questions involve some probability and differential equations, as well as some tools from data analysis and manifold learning. As motivation, we will study some specific mathematical models from the cell-biology literature. Some knowledge of probability and differential equations will be helpful. Also, it will be useful to have some experience with either python or matlab.
Polarization of disordered materials, led by Professor Alexander Watson
In an insulating material (an insulator), electrons cannot flow easily about the material. Nonetheless it may happen that when such a material is placed in an electric field the electrons in the material re-arrange themselves causing an overall dipole moment. Materials which behave this way are known as dielectrics and this dipole moment is known as the dielectric's polarization. In the 1990s it was discovered that accurate computation of polarization (and magnetization, the magnetic counterpart of the polarization) is surprisingly subtle, requiring understanding of Berry phase and other subtle quantum mechanical concepts. This project aims to extend this theory to disordered materials, where the present theory breaks down. Of particular interest are so-called disordered topological insulators: exotic materials whose properties are an active research topic in materials science. The project would be appropriate for students with an interest in physics who are comfortable coding or are willing to learn.
Dynamics of floating plates on thin films, led by Professors Thomas Witelski and Jeffrey Wong
Floating objects on the surface of a liquid layer can disperse or aggregate. Such dynamics are important for understanding the behavior of ice floes and debris on rivers. While the equations that govern fluid flows can be very challenging to study, we can make progress using appropriate modeling assumptions and simplifying the geometry. For thin liquid films, the equations reduce to a single non-linear convection-diffusion equation for the film height. Objects introduced on surfaces modify the local balance of forces, exerting influences on the fluid underneath that couple their motion to the flow.
In this project, we will investigate the effect of simple floating objects on thin film flows through analysis and numerical simulations. Building on basic background in physics, multivariable calculus, and ordinary differential equations, we will develop understanding of interesting behaviors in the fluid dynamics of fluid-structure interactions. We will examine how the presence of the object changes the speed of the flow, and how pairs of objects interact via the flow.