The Senior Research Capstone Experience

From 2008 HMC Catalog: Two semesters of Mathematics Clinic (Math 193) or two semesters of Senior Thesis (Math 197) are required and normally taken in the senior year. Clinic and thesis are important capstone experiences for each mathematics major; they represent sustained efforts to solve a complex problem from industry or mathematical research. To do a senior thesis, students must prepare a senior research proposal with the help of their thesis adviser. The proposal will describe the intended senior research project and must be submitted to the Department of Mathematics for approval before the end of the junior year. Clinic teams will be formed in the fall according to the requirements of the projects and student preferences. Students who do Clinic must work on the same Clinic project both semesters.

Instructions to students: On this wiki, you'll find descriptions of various projects for Clinic and Thesis to help you decide which route to pursue. Please keep in mind that completing original mathematical research, while highly desirable, is not a requirement of a senior thesis project. Also, if you have an idea for a senior thesis project but do not see it listed below, you should talk to your adviser and other faculty members around the Claremont Colleges who may share your interests. The HMC senior thesis archives may be a source of ideas. The "Common Application" for both Clinic and Thesis is Capstone Application.

Listing of Projects by Faculty Members

Professor Andrew Bernoff: Modelling of Physical and Biological Systems, Applications of Dynamical Systems, Fluid Mechanics, Self-Similarity and Scaling

Project: Models of Discrete and Continuous Swarming

Description: Swarming of organisms is a ubiquitous phenomena in the world around us; flocks of birds, schools of fish and locust swarms are examples of systems we have considered. There are several projects available that fit in as part of an ongoing effort to understand the dynamics of increasingly detailed models of these social structures.
Figure 1: From left to right: (a) Fish school (Parrish and Keshet, Science, 1999). (b) Army ant mill (Schnierla, Army Ants: A Study in Social Organization, 1971). (c) Marching "hopper band" of locusts (Uvarov, Grasshoppers and Locusts, 1966). (d) Advancing herd of wildebeest (Sinclair, The African Buffalo: A Study of Resource Limitation of Populations, 1977). Background Needed: Some PDE’s, comfort with numerical methods and computing. Skills Learned: Modeling of biological systems, familiarity with energy and variational methods, bifurcation theory, integral equations, numerical and asymptotic methods.
If you are interested send me an e-mail ( ajb@hmc.edu ).

Project: Pattern Formation in Variational Systems Driven by Pairwise Interaction Forces

Description: Many physical systems are driven by pairwise particle interactions; for example in thin liquid and solid layers, pairs of molecules interact electromagnetically. In swarming, the gathering of insects is often modeled via a pairwise social force. These systems can exhibit elaborate changes of morphologies as system parameters change. An example of this is the circle-dogbone-labyrinth transition seen in Langmuir films as the domain size increases (see Figure). In this project we will investigate the relationship between theses morphology changes and the form of the pairwise potential via energy methods and bifurcation theory. While many examples of these transitions have been observed for specific systems, characterizing these transitions for generic potentials remains a fertile area for study.
Figure 1: Morphologies of a Langmuir layer domain. A Langmuir layer is a molecularly thin fluid layer of a liquid crystal, lipid or other active substance on a quiescent 3D subfluid. When the Langmuir layer has a significant dipole moment, as the domain size increases a transition is seen from circular to dogbone shapes and then more elaborately contorted labyrinth patterns. Courtesy of Elizabeth Mann (Kent State) Background Needed: Some PDE’s, comfort with numerical methods and computing. Skills Learned: Familiarity with energy and variational methods, bifurcation theory, integral equations, numerical and asymptotic methods.
If you are interested send me an e-mail ( ajb@hmc.edu ).

Professor Alfonso Castro: Solvability of ordinary and partial differential equations using both elementary integration methods and functional analytic tools. The intermediate value theorem and its generalization to several variables, the contraction mapping principle, variational methods, and the implicit function theorem are examples of techniques that are used in these studies.

Professor Lisette de Pillis: Mathematical Biology, including tumor modeling, immunology modeling, optimal control, HIV/AIDS modeling, epidemiology modeling; numerical linear algebra

Professor Weiqing Gu: Differential geometry, Grassmann manifolds

Professor Jon Jacobsen: PDEs & nonlinear analysis; mathematical biology/ecology. Some areas of interest for thesis topics include pedagogy (e.g., building physical demos to illustrate mathematical concepts from dynamical systems; building GUIs and other animations or ipad/software products to illustrate key ideas from calculus/analysis/dynamical systems/etc. to promote learning), math history (esp. expository thesis on the history of weak solutions in PDE (i.e., solutions which are not as differentiable as the PDE seems to require)), numerical/computational analysis of population models with growth and random dispersal, and topics in fractals/dynamical systems.

Professor Dagan Karp: Algebraic Geometry. Dagan Karp's research is focused on combinatorial algebraic geometry and geometry inspired by theoretical physics. Possible thesis research areas include (1) toric geometry in Gromov-Witten theory; this is a combinatorial approach to an area of mathematics born from interaction with theoretical physics, and (2) tropical geometry, which is a beautiful new combinatorial approach to algebraic geometry. Prof. Karp also welcomes students with interests in related subjects.

Professor Henry Krieger: Measure-theoretic probability, Toeplitz operators, stochastic processes

Professor Rachel Levy: Analytical and numerical solutions of ordinary and partial differential equations modeling fluid flow; projects on Marangoni and gravity driven thin liquid films, ocean surface waves, coordination and control of underwater robotics; projects usually have experimental component.

Professor Susan Martonosi: Operations research, applied probability, homeland security, network optimization, humanitarian logistics.

Lately, I've been developing resource allocation models for the sequential distribution of anti-malaria interventions across a geographic area. These models use optimization techniques, such as Markov decision problems and integer programming, to determine the optimal allocation on an annual basis. The model takes as input data resulting from an SIR (Susceptible, Infected, Recovered) population dynamics model. Research on this project involves a mix of developing and refining the SIR model, developing and refining the optimization model, and researching socioeconomic and health statistics on malaria.

I'm also interested in network disruption methods. Research in this area can involve graph theory, algorithm design, network optimization, operations research, etc.

Professor Michael Orrison: Harmonic analysis on finite groups, and applications of the representation theory of finite groups.

Professor Nicholas Pippenger: Discrete mathematics, probability, and applications to communications and computation

Professor Francis Su: (On sabbatical 2013-2014, so not able to take students) Geometric and topological combinatorics, especially as applied to problems in mathematical economics, fair division, voting.

Professor Talithia Williams: Statistical disease modeling, Environmental statistics, Space - time data analysis

Description

Prevalence and incidence are two important measures of the impact of a disease. For many diseases, incidence is the most useful measure for response planning. We have developed a model to estimate incidence of progressive diseases, with an application to cataract disease in Africa. Initial results suggest different behavior of unilateral and bilateral incidence that might teach us something about the normal course of cataract disease. For example, how long before people with unilateral cataract usually develop bilateral cataract? Does this time depend on age, geographic region, gender or other factors that influence unilateral incidence? There are several natural extensions to this current body of work that would make for an exciting thesis project.
- .
. Figure 1: a) An eye clouded by a cataract. b) The cataract incidence rate by age group of northern (red) and southern (blue) countries in Africa.

Possible thesis projects include:

Developing a data clustering methodology.
Modeling age-dependent mortality due to cataract.
Applying the current model to other progressive diseases, such as cancer.
Developing a model that incorporates factors that influence cataracts (i.e. diabetes in the population)

Professor Darryl Yong: Mathematics education, applied mathematics, perturbation theory, partial differential equations

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