﻿ Comps

2013 October 17,

# Comps

Carleton College, Fall 2013 and Spring 2014, Prof. Joshua R. Davis, , Laird 205A, x4473

Note: We meet Wednesday 5A (1:50-3:00) in Laird 205.

## Introduction

Ideally, your mathematics comps project should be driven by your interests and aspirations. It should be aesthetically beautiful and practically useful. It should be challenging but not impossible. It should make use of multiple undergraduate math courses, and should demonstrate your ability to work in a group. Unfortunately, inventing a project to fulfill all of these objectives is not easy.

My current research concerns certain mathematical modeling problems in geology. Despite my background in "pure" mathematics, I find that these "applied" mathematics problems satisfy my aesthetic sensibilities quite well. And they certainly require a variety of tools. For example, Lie groups and the Poincaré-Hopf theorem have both shown up in my recent work. Simply put: There's a lot of cool math out there, and much of it is useful in the sciences.

I propose that we investigate a particular mathematical problem that has arisen in my research, following this plan:

1. We study the Navier-Stokes equations and specialize to the case of Stokes flow. Geologists use these partial differential equations (PDEs) to model rock deformation.
2. We study finite difference methods and, briefly, finite element methods (FEMs). Geologists use these tools to solve Navier-Stokes.
3. Using other researchers' FEM software, we model a particular kind of rock deformation that occurs at spreading ridges.

That plan sounds somewhat like an introductory PDE course with a computational bent. Depending on the students' interests, we could veer off in other directions, such as:

1. We study the Navier-Stokes equations and specialize to the case of Stokes flow.
2. We further specialize to isotropic, linear viscous, incompressible plane Stokes flow. Flows of this type are almost simple enough to be solved by hand.
3. We develop statistical techniques for analyzing geologic data related to such flows.
4. Using our statistics, we model a particular kind of rock deformation that occurs at spreading ridges.

We could veer off in still other directions: measure theory and probability theory on homogeneous spaces, non-parametric statistics, non-linear optimization, PDEs, fluid dynamics, implementation of FEMs, etc. I have many more problems, than I have time to solve them.

## Expectations

The comprehensive exercise is the capstone of your Carleton education. It is your chance to demonstrate that you can pursue a sustained, difficult, open-ended project. The stated expectations of the Mathematics Department are imprecise. Let's be a little clearer:

• Students are not required to work over the summer or winter.
• Students are required to work at least five hours per week, outside of meetings with the instructor. This number is based on the common expectation at Carleton of ten hours outside class per six-credit course. Because comps is supposed to be every student's academic priority, more than five hours should really be expected.
• Students are required to meet as a group at least twice per week, outside of meetings with the instructor.

It should go without saying that a student who is interested in distinction will try to exceed these basic expectations.

Over the summer, the motivated student might want to brush up on multivariable calculus. As far as I know, the divergence theorem is often not taught in Carleton's ten-week multivariable calculus course. So students could read up on partial derivatives, gradients, curl, divergence, surface integrals, and the divergence theorem. A really motivated student might then go on to study the derivation of Navier-Stokes.

## Materials

• 5-slide slideshow that I showed while pitching this comps project.
• A multivariable calculus textbook that covers surface integrals and the divergence theorem.
• Elementary Fluid Dynamics by Acheson has a nice treatment of Navier-Stokes.
• Statistics textbooks on linear regression, and preferably a little non-linear regression.
• Statistics of Directional Data by Mardia.
• Engineering textbooks on numerical methods including FDMs and FEMs.
• Davis and Titus (2011) summarizes many of the data types of structural geology, in the simplified context of steady homogeneous deformation.
• Scott et al. (2013) is a first pass at modeling a ridge-transform intersection.
• Davis et al. (2013) expands Davis and Titus (2011) to handle two more data types.

## Schedule

• Tue May 20 or Thu May 22: comps gala (talk)
• Mon May 26: paper due
• Fri May 30: exit interviews complete