# Math 490

### Seminar

###### Spring 2011

**1. Math 490 seminar**: **Topics in Group Representations and Character Theory**

Instructor: George Rublein (gtrubl@math.wm.edu)

The topic for this section of the senior seminar will be group representations and character theory. Character theory is a basic tool in the study of symmetry. My hope is that we will see some applications to symmetries in the solution of differential equations. We will use a text: Lederman, Introduction to Group Charaters. Students who have taken Math 307 are encouraged to enroll.

**2. Math 490 seminar: Topics in Mathematical
Biology**

Instructor: Sarah Day (sday@math.wm.edu)

Mathematical biology is a rapidly growing field that utilizes a variety of interesting mathematical techniques in studying biological processes. In this seminar, students will choose projects from the field of mathematical biology, make presentations to the class, and write a paper on their work. Possible topics include phylogenetic trees, population dynamics, molecular evolution, genetics, and infectious diseases. Students in the Applied Mathematics track are especially encouraged to enroll.

###### Spring 2009

**1. Math 490 Seminar: Models of reaction-diffusion and mathematical biology**

Instructor: Junping Shi (shij@math.wm.edu)

Reaction-diffusion (R-D) systems are some of the most widely used models in situations where spatial dispersal plays significant role. We will learn some theory of reaction-diffusion equations in the context of models in mathematical biology. Applications to be discussed in class include spatial spread of genes and of diseases, random dispersal of population, random and chemotactic motion of microorganisms, cellular maturation, pattern formations in developmental biology and morphogenesis, and animal coat patterns (why zebra has the stripes......). While introducing many biology models, we will also develop related mathematical theory and methods like, diffusion mechanism, waves, bifurcation theory, Turing's instability mechanism. Computation and simulation of solutions will be used throughout the class. (We are going to use software Maple and Matlab, but no prior knowledge is required.)

Recommended prerequisite is Math 302 (Differential equations) or Math 345 (Mathematical Models in Biology). An interested student who has not take Math 302 or Math 345 can read up on the necessary background material between semesters.

**2. Math 490 Seminar: Math 490-02 (Discrete Optimization)**

Instructor: Rex Kincaid (rrkinc@math.wm.edu)

Discrete optimization problems are those problems with decisions that are logical (yes/no) or countable. Both exact and heuristic methods for discrete optimization models will be presented in the course. Topics include relaxation techniques, constructive heuristics, improving search techniques (simplex method, simulated annealing, tabu search and genetic algorithms), branch and bound schemes, and valid inequalities for branch and cut methods.

Recommended pre-requisite is Math 323. An interested student who has not take Math 323 can read up on the necessary background material between semesters.

###### Fall 2007

**1. Math 490 Seminar: Investigating Linear Algebra**

Instructor: Charles Johnson (crjohnso@math.wm.edu)

With the assistance of the instructor, each student will choose either a research topic (appropriate for the student's background) in elementary linear algebra to be explored, or a well-developed topic in advanced linear algebra to be learned. The instructor will suggest a number of research topics fitting into broader questions, and will offer a list of advanced topics (e.g. linear matrix equations. Lyapunov's theorem, special classes of matrices, etc.) for students in the course. In each case, participants will be guided by the instructor, will write about their progress, and will give two talks to the group as a whole about their work. Enrollment in the course will be limited and students need to consult with the instructor before registering for the course.

**2. Math 490-01 Senior seminar. **

(targeted for but not limited to prospective teachers)

Instructor: Chi-Kwong Li (ckli@math.wm.edu)

In this seminar, we will study various approaches in teaching and learning mathematics in different countries, and different time of history. We will also consider different methods in generating interest in studying mathematics. One focus will be on finding interesting problems in recreative, applied or theoretical settings that may stimulate interest in studying mathematics. Students are required to do one or two projects on the above themes.

###### Spring 2007

1. Senior Seminar

Instructor: Leiba Rodman (lxrodm@math.wm.edu)

Description: Students will study a narrow topic of mathematics, make oral presentations for discussion in class, and write a report. The topic need not be the same for all students. The choice of topic(s) is flexible and may reflect particular interests of each student.

**2. Computational Topology and Dynamics**

Instructor: Sarah Day (sday@math.wm.edu)

Textbook: *Computational Homology*, by T. Kaczynski, K. Mischaikow, and M. Mrozek, Applied Mathematical Sciences 157, Springer 2004.

Homology is used to study a pattern generated from a nonlinear partial differential equation. Another application of computational topology is in numerical studies of dynamical systems. Numerical techniques based on topological tools are now being used to prove the existence of chaos in dynamical systems models in fields from ecology to mechanics. In this course we will study the mathematical theory behind homology, discuss efficient means of computation, and study applications. Related topics include techniques for image processing and the efficient implementation of algorithms on large data sets. This will be a discussion based class and students are encouraged to pursue and present individual topics of interest related to the course material.