# Mathematics Major

A mathematics degree from Harvey Mudd College will prepare students for a variety of careers in business, industry or academics. Mathematical methods are increasingly employed in fields as diverse as finance, biomedical research, management science, the computer industry and most technical and scientific disciplines. To support the academic and professional goals of our majors, we offer a wide selection of courses in both pure and applied mathematics. This selection is enhanced by courses offered in cooperation with the other Claremont Colleges, including graduate courses at the Claremont Graduate University.

Students will have many opportunities to do mathematical research with faculty through independent study, a summer research experience, or their senior capstone experience. Active areas of mathematical research at HMC and The Claremont Colleges include algebra, algebraic geometry, algorithms and computational complexity, combinatorics, differential geometry, dynamical systems, fluid mechanics, graph theory, number theory, numerical analysis, mathematical biology, mathematics education, operations research, partial differential equations, real and complex analysis, statistical methods and analysis, and topology.

The culmination of the degree is the senior capstone research experience: every student experiences a taste of the life of a professional mathematician as part of a team in the Mathematics Clinic Program or by working individually on a Senior Thesis.

The Mathematics Clinic program extends the academic experience of our majors. An educational innovation of HMC, our Clinic Program brings together teams of students to work on a research problem sponsored by business, industry or government. Teams work closely with a faculty advisor and a liaison provided by the sponsoring organization to solve complex real-world problems using mathematical and computational methods. Clinic teams present their results in bound final reports to the sponsors and give several formal presentations on the progress of the work during the academic year.

Our Senior Thesis program provides students with the opportunity to work independently on a problem of their choosing. Advisors and readers may be chosen from the HMC faculty and the other mathematicians at The Claremont Colleges, providing students with a wealth of research opportunities. As with Clinic, the end product of a thesis is a bound volume as well as presentations made at a professional conference or other venue, during the college-wide Presentations Days and throughout the year.

## Course Requirements

Every HMC student, regardless of major, takes three semesters of mathematics (covering the topics of calculus, multivariable calculus, linear algebra, probability & statistics, and differential equations) as part of the general core curriculum.

Beyond the Core courses, the course of study for a mathematics degree has five components: the Major Core, Computational Mathematics, the Senior Capstone Experience (Clinic or Thesis), Math Forum and Math Colloquium, and the Elective Program. Each of these components to the major program is described below.

## The Major Core

A set of core courses is required of each major. These courses cover a range of fundamental fields of mathematics, and position the student to pursue any one of a variety of elective programs to finish the degree.

- Mathematics 55: Discrete Mathematics
- Mathematics 70: Intermediate Linear Algebra
- Mathematics 80: Intermediate Differential Equations
- Mathematics 131: Mathematical Analysis I
- Mathematics 157: Intermediate Probability
- Mathematics 171: Abstract Algebra I
- Mathematics 180: Introduction to Partial Differential Equations

## Computational Mathematics

Computational techniques are essential to many fields of modern mathematics, and to most mathematical applications in business and industry. *One course in computational mathematics is required* of all majors, selected from the following list:

- Mathematics 164: Scientific Computing
- Mathematics 165: Numerical Analysis
- Mathematics 167: Complexity Theory
- Mathematics 168: Algorithms
- CS 81: Computability and Logic

## Senior Capstone Research Experience

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 major: they represent sustained efforts to solve a complex problem from industry or work on independent mathematical research.

Clinic teams are formed in the fall semester according to the requirements of the projects and student preferences. Students who choose Clinic must work on the same project for both semesters.

For thesis, you must prepare a senior-research proposal with the help of your thesis advisor. Your proposal will describe the intended senior-research project and must be submitted to the department’s Curriculum Committee for approval before the end of your junior year.

See the individual sites for each program for additional details about their requirements and procedures.

## Math Forum and Math Colloquium

All mathematics majors must take one semester of Undergraduate Mathematics Forum (Mathematics 198) and one semester of Math Colloquium (Mathematics 199) generally in your junior year.

In the math forum, students prepare and present talks on mathematical topics taken from the literature. As a requirement for the math forum, students must submit a tentative description of their proposed elective program to the department by the end of the fall semester of the junior year.

The Claremont Colleges Mathematics Colloquia features weekly speakers from the colleges as well as visitors from other institutions discussing their research or giving talks of general mathematical interest. All speakers are encouraged to pitch their talks to undergraduate mathematics majors.

## The Elective Program

To complete the degree, *three elective mathematics courses (at least 7 units)* are required. The student, in consultation with his or her advisor, will design their elective program. To assist students in this process, the Department has prepared a variety of sample programs. These sample programs are courses of study supporting a wide range of career goals in academics, business or industry. Several sample elective programs are listed below; course numbers in bold are strongly recommended (CS = Computer Science, CGU = Claremont Graduate University). Sample programs are advisory; students may follow a sample program, or design a program of their own.

### Pure Mathematics

and at least one course from

- Mathematics 104: Graph Theory
- Mathematics 106: Combinatorics
- Mathematics 136: Complex Variables and Integral Transforms
- Mathematics 142: Differential Geometry
- Mathematics 143: Seminar in Differential Geometry
- Mathematics 147: Topology
- Mathematics 173: Advanced Linear Algebra
- Mathematics 174: Abstract Algebra II: Representation Theory
- Mathematics 175: Number Theory
- CGU Math 331: Real Analysis I
- CGU Math 332: Real and Functional Analysis II
- CGU Math 334: Complex Analysis II

### Applied Mathematics

and at least one course from

- Mathematical and Computational Biology 118A: Introduction to Mathematical Biology
- Mathematical and Computational Biology 118B: Introduction to Computational Biology
- Mathematics 119: Advanced Mathematical Biology
- Mathematics 132: Mathematical Analysis II
- Mathematics 164: Scientific Computing
- Mathematics 165: Numerical Analysis
- Mathematics 173: Advanced Linear Algebra
- Mathematics 184: Graduate Partial Differential Equations
- Mathematics 187: Operations Research
- CGU Math 362: Numerical Methods for Partial Differential Equations
- CGU Math 368: Advanced Numerical Analysis
- CGU Math 382: Perturbation and Asymptotic Analysis

### Probability and Statistics

and at least one course from

- Mathematics 106: Combinatorics
- Mathematics 132: Mathematical Analysis II
- Mathematics 152: Statistical Theory
- Mathematics 153: Bayesian Statistics
- Mathematics 155: Time Series
- Mathematics 158: Statistical Linear Models
- Mathematics 173: Advanced Linear Algebra
- Mathematics 187: Operations Research
- CGU Math 351: Time Series Analysis
- CGU Math 355: Linear Statistical Analysis

### Operations Research

and at least one course from

- Mathematics 104: Graph Theory
- Mathematics 106: Combinatorics
- Mathematics 132: Mathematical Analysis II
- Mathematics 152: Statistical Theory
- Mathematics 158: Statistical Linear Models
- Mathematics 165: Numerical Analysis
- Mathematics 168: Algorithms
- Mathematics 188: Social Choice and Decision Making

### Actuarial or Financial Mathematics

and at least one course from

- Mathematics 152: Statistical Theory
- Mathematics 155: Time Series
- Mathematics 158: Statistical Linear Models
- Mathematics 165: Numerical Analysis
- Mathematics 187: Operations Research
- CGU Math 355: Linear Statistical Analysis
- Econometrics
- CMC Econ 125: Econometrics
- CMC Econ 126: Microeconometrics
- CGU Econ 382: Econometrics I
- CGU Econ 383: Econometrics II
- CGU Econ 384: Econometrics III
- Pomona Econ 167: Econometrics

### Scientific Computing

and at least one course from

- Mathematical and Computational Biology 118A: Introduction to Mathematical Biology
- Mathematical and Computational Biology 118B: Introduction to Computational Biology
- Mathematics 119: Advanced Mathematical Biology
- Mathematics 136: Complex Variables and Integral Transforms
- Mathematics 168: Algorithms
- Mathematics 173: Advanced Linear Algebra
- Mathematics 181: Dynamical Systems
- Mathematics 184: Graduate Partial Differential Equations
- CS 156: Parallel and Real-Time Computation
- CGU Math 362: Numerical Methods for Partial Differential Equations
- CGU Math 368: Advanced Numerical Analysis
- CGU Math 382: Perturbation and Asymptotic Analysis

### Theoretical Computer Science

and at least one course from

- Mathematics 104: Graph Theory
- Mathematics 106: Combinatorics
- Mathematics 165: Numerical Analysis
- Mathematics 167: Complexity Theory
- Mathematics 172: Abstract Algebra II: Galois Theory
- Mathematics 175: Number Theory
- CS 151: Artificial Intelligence
- CS 152: Neural Networks
- CS 156: Parallel and Real-Time Computation

Students interested in this elective program may wish to consider the Joint Major in Computer Science and Mathematics.

### Mathematical Biology

- Mathematical and Computational Biology 118A: Introduction to Mathematical Biology
- Mathematical and Computational Biology 118B: Introduction to Computational Biology
- Mathematics 119: Advanced Mathematical Biology

and at least one course from

- Mathematics 152: Statistical Theory
- Mathematics 156: Stochastic Processes
- Mathematics 158: Statistical Linear Models
- Mathematics 164: Scientific Computing
- Mathematics 168: Algorithms
- Mathematics 173: Advanced Linear Algebra
- Mathematics 181: Dynamical Systems
- Mathematics 184: Graduate Partial Differential Equations
- Mathematics 187: Operations Research

Students interested in this elective program may wish to consider the Joint Major in Mathematical and Computational Biology.

## Comments

Through the Major Core requirement, every major will have a foundation course in several important areas: discrete mathematics, analysis, algebra, differential equations and probability. In addition, every major will have a course relating to computational aspects of mathematics. The Major Core positions each student to move in any of several directions in the design of their elective program. There is a wide range of options to finish the major, supporting a variety of career goals and interests. It is expected that most students will take Math 55 and 131 by the end of the sophomore year, Math 157, 171, 180, 198, and 199 by the end of the junior year, and Math 193 or 197 during the senior year.

Two semesters of Clinic or thesis are required of each major. All students must declare their intentions by the end of their junior year. Students who wish to take Clinic should inform the Mathematics Clinic Director, and preregister for Math 193. Students choosing thesis must arrange to have a thesis advisor by the end of the spring semester of their junior year. In consultation with their advisor, the student must prepare a research proposal describing a suitable thesis problem, and submit the proposal to the mathematics department for approval. We expect that students will begin work on their theses immediately in the fall of the senior year. Thesis students will meet weekly as a group, to discuss their progress, make presentations, and exchange ideas. Students enrolled in Clinic who also wish to do thesis will be able to do a one-semester thesis, if desired. They may arrange their thesis in the fall of the senior year.

The faculty in the mathematics department works closely with each student to develop a coherent program of elective courses that meets the student’s professional and academic goals. The entire department meets once each term to discuss and compare all student programs and to discuss student progress.