I'm pleased to announce that I was accepted into the doctoral program in Applied Mathematics at the University of Washington. I've been looking forward to an opportunity like this for some time now and I'm excited to contribute to the fascinating research and work of this excellent department.

As I now end the second quarter of my Master's career in the department, I'm faced with the question "Which topics of research should I pursue and with whom should I pursue this research?" I've narrowed down my choices to working with three professors in the department. I decided upon these three based on my pure-mathematical upbringing, my interest in computational mathematics and algorithm development, and my skills as a programmer.

I thought I would share a little bit about the kind of research these three professors do in the Applied Mathematics department. Below are just some words about their work and what part of their work I find interesting. None of my comments are by any means comprehensive. I just want to offer an introduction to what I'm considering on doing for the next four to five more years.

Option #1: Exact Computational PDEs with Bernard Deconinck

Website: http://www.amath.washington.edu/~bernard/research.html

Bernard's works are quite varied; especially the fields of mathematics he pursues to arrive at his results. One of his many projects I'm interested in working on involves computing exact solutions to the KdV equation: $$\partial_t \phi + \partial_{xxx} \phi - 6 \phi \partial_x \phi = 0$$ This equation is used to describe the behavior of shallow wave propagation. Relatively recently, exact solutions to this differential equation were found in terms of Riemann theta functions. A challenging problem itself is computing values of these theta functions. Part of Bernard's work has been creating efficient algorithms for doing so and, given the mathematics required for such a feat, is part of what draws me to his work.

There is a lot of fascinating theory of Riemann surfaces and other real/complex analytical objects surrounding his work including some mathematical objects I encountered while working on my senior thesis in number theory. Additionally, the computational aspect of this project may possibly involve writing code in Sage. Hopefully my previous experience with Sage would prove useful were I to work with Bernard.

Option #2: Linear Analysis with Anne Greenbaum

Website: http://www.math.washington.edu/~greenbau/

Anne's specialty is in numerical linear algebra and iterative methods. I'm currently taking a course in linear analysis: the primary field of math encompassing her work. I'm particularly drawn to this field for two main reasons: linear analysis draws from many different fields of mathematics, and I find it difficult to choose a single field to focus on; and that there are a wide variety of applications of the theory to "real world" problems.

My first encounter with her work was during a talk she gave for the Mathematics department's current topics course on Crouzeix's Conjecture. The statement of the conjecture is quite compact:

For all polynomials, $p$, $$||p(A)|| \leq 2 \max_{z \in W(A)} |p(A)|$$ where $W(A)$ is the field of values of the matrix $A$.

The point of this investigation is to find our what sort of information about a matrix captures the behavior of a matrix as much as possible. Knowing the eigenvalues alone can only take you so far and in application to numerical methods, like determining the stability of a finite difference scheme using the matrix, they often don't provide the desired information one needs. The field of values $W(A)$, on the other hand, can give us information about a matrix that the eigenvalues can't. Knowing the bound above would assist in knowing something about the rate of convergence of iterative methods such as the GMRES algorithm; another subject where Anne is an expert.

Option #3: Numerical PDE Solvers with Randy LeVeque

Website: http://www.amath.washington.edu/~rjl/

I had the pleasure of working with Randy for the summer quarter on the Clawpack project in which I started designing an effective Sage wrapper. Given that Sage is a Python/C-based piece of software and, in order to perform the more powerful computations, Clawpack is Fortran-based; the task proved to be quite a challenge! So much so that I'm still not done with the job I was given. I hope to finish what I've started by the end of the Spring quarter.

Randy's work, amongst those of the other professors I'm interested in studying under, is the most "computational". He specializes in designing large-scale algorithms for solving hyperbolic partial differential equations. In particular, his focus is on the advection-diffusion equation $$\partial_t q + \partial_x f(q) = \beta \partial^2_{xx} q$$ (This is just one class of advection-diffusion equations with scalar diffusive coefficient. I'm sure he studies much more complicated sets of equations as well.) This equation is often used to describe the propagation of dispersive substance, such as a tracer, through advecting media.

Working with Randy would expose me to the exciting world of large-scale supercomputing and computational mathematics. Once one begins to distribute their algorithms over thousands of nodes very interesting mathematical problems appear. I would have ample opportunity to write code and contribute to an excellent open-source project.

Additional Thoughts

Now, I still have to pass my qualifying exams before I can do any of the aforementioned research projects. Supposing I succeed, however, you can read about my progress on this blog. I'll also keep a general summary of my work on my Research page. Whatever path I end up taking I'm sure that I will be doing exciting and rewarding work. I look forward to discovering more fascinating mathematics with each of these professors in the years to come.