(See also Joint Major in Computer Science and Mathematics, and Mathematical Biology)

Professors Bernoff (Chair), Benjamin, Borrelli (emeritus), Castro, Coleman (emeritus), Goroff, Gu, Henriksen (emeritus), Jacobsen, Karp, Krieger (emeritus), Levy, Martonosi, Orrison, Pippenger, Su, Tucker (2007–2010), Whitcher (2009–2011), Williams and Yong.

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

An educational innovation of HMC, the 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 adviser and a liaison provided by the sponsoring organization to solve complex real-world problems using mathematical and computational methods. The 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.

In addition to the Clinic Program, students will have many opportunities to do mathe-matical research with faculty as an independent study, a summer research experience, or as a Senior Thesis. 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. HMC students may do their Senior Thesis with an HMC faculty member or any mathematician at The Claremont Colleges, which considerably expands the available opportunities and areas of research.

The course of study for a mathematics degree has five components: The Major Core, Computational Mathematics, Clinic or Thesis, 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 mathematics 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. The Major Core consists of Math 55 (Discrete Mathematics), Math 131 (Mathematical Analysis I), Math 157 (Intermediate Probability), Math 171 (Abstract Algebra I) and Math 180 (Applied Analysis).

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 mathematics majors, selected from the following list: Math 164 (Scientific Computing), Math 165 (Numerical Analysis), Math 167 (Complexity Theory), Math 168 (Algorithms), Biology 188 (Computational Biology) or CS 81 (Computability and Logic).

Clinic or Thesis
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.

Colloquium
All mathematics majors must take one semester of Math Forum (Math 198) and one semester of Mathematics Colloquium (Math 199), generally in the junior year. In the 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 Elective Program
To complete the degree, three elective mathematics courses totalling at least seven credit hours are required. The elective program will be designed by the student in consultation with his or her adviser. To assist students in designing their elective program, the department has prepared a variety of sample programs. These sample programs list courses that support a wide range of career goals in academics, business or industry. About half of our graduates immediately join the workforce and about half enter graduate school. Several sample elective programs are listed below. In each of these samples, the first two courses are strongly recommended; at least one additional course is to be selected in order to complete the elective program.

(CS = Computer Science, CMC = Claremont McKenna College, CGU = Claremont Graduate University, PO = Pomona College). We emphasize that sample elective programs are advisory. Students may follow a sample program or design one of their own.

Pure Mathematics: 132, 172 and at least one elective from 104, 106, 123, 135, 136, 142, 143, 147, 173, 175, 185, CGU 331, CGU 332, CGU 334.

Applied Mathematics: 136, 181 and at least one elective from 118, 119, 120, 132, 164, 165, 173, 182, 185, 187, 189, CGU 362, CGU 368, CGU 382.

Probability and Statistics: 152, 156 and at least one elective from 106, 132, 158, 159,173, 187, CGU 351, CGU 355.

Operations Research: 156, 187 and at least one elective from 104, 106, 132, 152, 158, 159, 165, 168, 188.

Actuarial or Financial Mathematics: 109, 156 and at least one elective from 152, 158, 159,165, 187, CGU 355, Econometrics (CMC 125, 126; CGU 382, 383, 384; PO 167).

Theoretical Computer Science: CS 81, 168 and at least one elective from 104, 106, 107, 123,165, 167, 172, 175, CS 151, CS 152, CS 156.

Scientific Computing: 164, 165 and at least one elective from 118, 119, 136, 168, 173, 181, 182, 185, CS 156, CGU 362, CGU 368, CGU 382.

Advising
The mathematics faculty works closely with each mathematics major to develop a coherent program of elective courses that meets the student’s professional and academic goals. The department meets once each year to discuss and evaluate student programs and to discuss student progress.

Facilities
The Department of Mathematics and the college provide excellent computational facilities.  The department’s Scientific Computing Laboratory houses workstations supporting classroom activities and student and faculty research in numerical analysis, algorithms, parallel computing, and scientific computing, addressing diverse problems in mathematical modeling (such as problems in fluid mechanics and mathematical biology), operations research and statistical analysis. Additional resources include Beowulf-style distributed parallel-computing clusters and multiprocessor, large-memory, parallel compute servers. The department supports a wide variety of commercial and free/open-source mathematical software packages such as Mathematica, Maple, MATLAB, R, and SAGE.

Other Mathematical Activities at HMC and in Claremont
There are many opportunities outside of course work to enjoy and participate in mathematics. Some of these activities are described below.

• The William Lowell Putnam Examination. The Putnam Exam is a national collegiate mathematics competition. Over 2,500 students from more than 400 institutions take the exam. It is a very challenging, 12-question exam lasting six hours (three hours in the morning and three in the afternoon). The problems on the exam can be solved using elementary methods so that students can take the exam every year they are at college. About 70 HMC students take the exam each year, one of the highest participation rates in the country. The HMC Putnam team has done very well in the competition. In the 2003 Putnam Mathematical Competition, the HMC team placed fifth nationwide, the only undergraduate college in the last 30 years to have done so. The Putnam Seminar (Math 191) meets weekly and is open to all students. This is a one-unit course that will help to prepare students for the competition.

• Weekly Mathematics Colloquium. The Claremont Colleges Mathematics Colloquium meets once per week. Most of the colloquium speakers are mathematicians from around the country who speak about their research or give talks of general mathematical interest. To encourage undergraduates to attend, all speakers are encouraged to design their talk to be accessible to undergraduate mathematics majors.

• Mathematics Seminars. Several weekly seminars on special mathematical topics are offered in Claremont each year. Recent seminars include combinatorics, analysis, applied mathematics, wavelet theory, monte carlo methods, population dynamics and topology. Faculty, CGU graduate students and advanced undergraduate students attend the seminars.

• Mathematical Competition in Modeling (MCM). The MCM competition is sponsored by the Consortium for Mathematics and its Applications and the Society for Industrial and Applied Mathematics. Each year, the MCM contest proposes two challenging open-ended problems in applied mathematics. Competing schools form teams of three students to work on the problems over a long weekend. Teams cannot consult with any person on their solution, but otherwise can use any resource available to them: computers, reference literature from the library or Internet resources. Each year, HMC has between two and six teams competing in the MCM, out of over 400 teams internationally. HMC has earned the highest award of Outstanding more than any other institution in the competition.

Some Recent Clinic Projects
As described above, Clinic teams work together for two semesters to solve an open problem from business, industry or government. Listed below are a few examples of recent Clinic projects and the names of the sponsors.

• Cardinal Health. Modeling Fluid Transport in Subcutaneous Tissue
• Hewlett-Packard Labs. Characterization and Sources of Printer Color Instability
• Los Alamos National Laboratories. Mathematical and Computational Modeling of Tumor Development
• National Renewable Energy Laboratory. Advanced Modeling of Renewable Energy Market Dynamics
• Overture Services, Inc. Improved Relevance Ordering for Web Search
• Sandia National Laboratories. Improving GPS Algorithms
• ViaSat, Inc. Elliptic Curve Cryptography Scheme for Asymmetric Key Generation

Some Recent Senior Theses
Several students write a senior thesis each year. It is common that theses result in papers that are submitted to mathematical journals for publication. Listed below are the titles of several recent senior theses:

• An Investigation of Rupture in Thin Fluid Films
• Mathematical Model of the Chronic Lymphocytic Leukemia Microenvironment
• Mathematical AIDS Epidemic Model: Preferential Anti-Retroviral Therapy Distribution in Resource Constrained Countries
• Connections Between Voting Theory and Graph Theory
• A Fast Fourier Transform for the Symmetric Group
• The Negs and Regs of Continued Fractions
• Turing Pattern Dynamics for Spatio-Temporal Models with Growth and Curvature
• Kolmogorov Complexity of Graphs
• Foraging Fruit Flies: Lagrangian and Eulerian Descriptions of Insect Swarming
• Optimal Control of a Building During an Earthquake
• Mathematical Models of Image Processing

MATHEMATICS COURSES
(Includes mathematics courses frequently taken by HMC students at the other Claremont Colleges)

11. Calculus (2)
Benjamin, Jacobsen, Karp, Levy, Williams. Limits, derivatives and differentiation rules; partial derivatives; gradients and directional derivatives; introduction to calculus of complex-valued functions; infinite series, Taylor series, convergence tests; fundamental theorem of calculus; techniques of integration; double and triple integrals. Prerequisites: One year of calculus at the high school level. (Offered Fall, both halves)

12. Introduction to Linear Algebra (2)
Benjamin, de Pillis, Gu, Karp, Martonosi, Orrison, Pippenger, Su, Tucker, Williams, Yong. Complex numbers; proofs by contradiction and induction; matrix representation of systems of equations, matrices, operations, determinants, row and column spaces; vectors, dot and cross products; vector descriptions of lines and planes; linear independence and dependence, bases; eigenvalues and eigenvectors; examples of discrete dynamical systems. Prerequisites: One year of calculus at the high school level. (Offered Fall, both halves)

13. Differential Equations (1.5)
Bernoff, Castro, de Pillis, Jacobsen, Levy, Su, Yong. Modeling physical systems, first-order ordinary differential equations, existence, uniqueness, and long-term behavior of solutions; bifurcations; approximate solutions; second-order ordinary differential equations and their properties, applications; first-order systems of ordinary differential equations. Prerequisite: Mathematics 11; or the equivalent. (Offered Spring, both halves)

14. Multivariable Calculus I (1.5)
Castro, Gu, Karp, Orrison, Su, Yong. Review of basic multivariable calculus; optimization and the second derivative test; higher order derivatives and Taylor approximations; line integrals; vector fields, curl, and divergence; Green’s theorem, divergence theorem and Stokes’ theorem, outline of proof and applications. Prerequisite: Mathematics 11; or the equivalent. (Offered Spring, both halves)

55. Discrete Mathematics (3)
Benjamin, Bernoff, Orrison, Tucker. Topics include combinatorics (clever ways of counting things), number theory, and graph theory with an emphasis on creative problem solving and learning to read and write rigorous proofs. Possible applications include probability, analysis of algorithms and cryptography. Prerequisites: Mathematics 12; or permission of the instructor. (Offered Fall and Spring)

61. Multivariable Calculus II (1.5)
Bernoff, Gu, Karp, Su, Yong. Constrained optimization using Lagrange multipliers; conservative and nonconservative vector fields; Green’s theorem; parameterized surfaces and surface integrals; divergence theorem, outline of proof and applications; Stokes’ theorem, outline of proof and applications, unification of major vector theorems. Prerequisites: Mathematics 14. (Offered first half of Fall semester)

62. Introduction to Probability and Statistics (1.5)
Benjamin, Martonosi, Orrison, Su, Williams. Sample spaces, events, axioms for probabilities; conditional probabilities and Bayes’ theorem; random variables and their distributions, discrete and continuous; expected values, means and variances; covariance and correlation; law of large numbers and central limit theorem; point and interval estimation; hypothesis testing; simple linear regression; applications to analyzing real data sets. Prerequisites: Mathematics 12. (Offered second half of Fall semester)

63. Linear Algebra II (1.5)
Benjamin, de Pillis, Gu, Martonosi, Orrison. General vector spaces and linear transformations; rank-nullity theorem; orthogonal expansion and Fourier coefficients; change of basis and similarity; generalized eigenvalues and eigenvectors; diagonalization of symmetric matrices; applications of eigenvalues to systems of ordinary differential equations; LU, QR, and singular value decomposition theorems; Jordan canonical forms. Prerequisites: Mathematics 12. (Offered first half of Spring semester)

64. Differential Equations II (1.5)
Bernoff, Castro, de Pillis, Jacobsen, Levy, Martonosi, Yong. Linear systems of homogeneous
ordinary differential equations, matrix exponential and non-homogeneous linear systems; non-linear systems; equilibrium points and their stability; phase portraits. Prerequisites: Mathematics 12 and Mathematics 13. (Offered second half of Spring semester)

104. Graph Theory (3)
Benjamin, Orrison, Pippenger. An introduction to graph theory with applications. Theory and applications of trees, matchings, graph coloring, planarity, graph algorithms and other topics. Prerequisites: Mathematics 12 and Mathematics 55. (Offered alternate years)

106. Combinatorics (2)
Benjamin, Orrison, Pippenger. An introduction to the techniques and ideas of combinatorics, including counting methods, Stirling numbers, Catalan numbers, generating functions, Ramsey theory and partially ordered sets. Prerequisites: Mathematics 55; or permission of instructor. (Offered alternate years)

107. Set Theory (3)
Bull (Pomona). Naive set theory, Zermelo-Fraenkel axioms and the axiom of choice; ordinal and cardinal arithmetic; construction of real numbers. Prerequisites: Mathematics 12. (Offered alternate years)

108. History of Mathematics (3)
Grabiner (Pitzer). A survey of the history of mathematics from antiquity to the present. Topics emphasized will include: the development of the idea of proof, the “analytic method’’ of algebra, the invention of the calculus, the psychology of mathematical discovery, and the interactions between mathematics and philosophy. Prerequisites: Mathematics 11. (Offered alternate years)

109. Introduction to the Mathematics of Finance (3)
Aksoy (CMC). This course emphasizes the math used in the valuation of derivative securities. Topics will include partial differential equations (diffusion equation), mathematical modeling of financial derivatives (calls and puts), and numerical methods for solving differential equations; Black-Scholes Model. Prerequisites: Mathematics 63. (Offered alternate years)

115. Fourier Series and Boundary Value Problems (3)
Bernoff, Castro, Jacobsen, Levy, Yong. Sturm-Liouville theory, orthogonal expansions, convergence properties of Fourier series, separation of variables for partial differential equations, regular singular point theory, Bessel functions and Legendre polynomials. (May not be included in a mathematics major program. Students may not receive credit for both Mathematics 115 and 180.) Prerequisites: Mathematics 64. (Offered Fall)

118. Mathematical Biology I (2) (Also listed as Biology 118)
de Pillis, Jacobsen, Adolph (Biology), Nadim (CGU/KGI). Mathematical models of biological processes emphasizing continuous models. May include models in epidemiology, population dynamics, cancer modeling, and disease treatment modeling. Prerequisites: Mathematics 64, Biology 52; or permission of instructor. (Offered first half of Spring semester)

119. Mathematical Biology II (2) (Also listed as Biology 119)
de Pillis, Jacobsen, Adolph (Biology), Nadim (CGU/KGI). Mathematical models of biological processes emphasizing discrete and continuous models. May include one- and two-locus population genetics, metapopulations, and matrix population models as well as models in physiology and neurobiology. Prerequisites: Mathematics 64, Biology 52; or permission of instructor. (Offered second half of Spring semester)

120. Chirality (2)
Flapan (Pomona). A structure is chiral if it is different from its mirror image. This interdisciplinary course introduces students to topological and geometric symmetry and provides descriptions of chirality in molecular systems. Connections will be made between the chemical and mathematical theories of chirality. Molecules with interesting topological features will be introduced and their structural behavior discussed. Prerequisites: Mathematics 12. (Offered alternate years in Spring semester)

123. Logic (3)
Bull (Pomona). Propositional and first order predicate logic. The completeness, compactness and Lowenheim/Skolem theorems. Decidable theories. Applications to other areas of mathematics, e.g., nonstandard analysis. Prerequisites: Mathematics 12. (Offered jointly at Pomona in alternate years)

131. Mathematical Analysis I (3)
Castro, Karp, Su, S. Grabiner (Pomona), Askoy (CMC). Countable sets, least upper bound, and metric space topology including compactness, completeness, connectivity, and uniform convergence. Related topics as time permits. Prerequisites: Mathematics 12 and Mathematics 14. (Offered jointly; Fall semester at Pomona, Spring semester at HMC and CMC)

132. Mathematical Analysis II (3)
Castro, Su, Radunskaya (Pomona). A rigorous study of calculus in Euclidean spaces including multiple Riemann integrals, derivatives of transformations and the inverse function theorem. Prerequisites: Mathematics 131. (Offered jointly; Fall semester at HMC, second semester at Pomona)

136. Complex Variables and Integral Transforms (3)
Gu, Jacobsen, Karp, Yong. Complex differentiation, Cauchy-Riemann equations, Cauchy integral formulas, residue theory, Taylor and Laurent expansions, conformal mapping, Fourier and Laplace transforms, inversion formulas, other integral transforms, applications to solutions of partial differential equations. Prerequisites: Mathematics 64. (Offered Fall)

137. Graduate Analysis I (3) (Also listed as Mathematics 331)
Castro, Krieger, Grabiner (Pomona), O’Neill (CMC). Abstract Measures, Lebesgue measure, and Lebesgue-Stieltjes measures on R; Lebesgue integral and limit theorems; product measures and the Fubini theorem; additional topics. Prerequisites: Mathematics 132. (Offered Fall)

138. Graduate Analysis II (3) (Also listed as Mathematics 332)
Castro, Krieger, Grabiner (Pomona), O’Neill (CMC). Banach and Hilbert spaces; Lp spaces; complex measures and the Radon-Nikodym theorem. Prerequisites: Mathematics 137 or Mathematics 331. (Offered Spring)

142.  Differential Geometry (3)
Gu, Karp. Curves and surfaces, Gauss curvature; isometries, tensor analysis, covariant differentiation with application to physics and geometry (intended for majors in physics or mathematics). Prerequisite: Mathematics 64. (Offered Fall)

143. Seminar in Differential Geometry (3)
Gu. Selected topics in Riemannian geometry, low dimensional manifold theory, elementary Lie groups and Lie algebra, and contemporary applications in mathematics and physics. Prerequisites: Mathematics 131 or Mathematics 132 or Mathematics 142; recommended Mathematics 147; or permission of instructor. (Offered Spring)

147. Topology (3)
Pippenger, Su, Flapan (Pomona). Topology is the study of properties of objects preserved by continuous deformations (much like geometry is the study of properties preserved by rigid motions). Hence, topology is sometimes called “rubber-sheet” geometry. This course is an introduction to point-set topology with additional topics chosen from geometric and algebraic topology. It will cover topological spaces, metric spaces, product spaces, quotient spaces, Hausdorff spaces, compactness, connectedness and path connectedness. Additional topics will be chosen from metrization theorems, fundamental groups, homotopy of maps, covering spaces, the Jordan curve theorem, classification of surfaces and simplicial homology. Prerequisites: Mathematics 131; or permission of instructor. (Offered jointly with Pomona; Spring semester)

148. Knot Theory (3)
Hoste (Pitzer). An introduction to theory of knots and links from combinatorial, algebraic, and geometric perspectives. Topics will include knot diagrams, p-colorings, Alexander, Jones, and HOMFLY polynomials, Seifert surfaces, genus, Seifert matrices, the fundamental group, representations of knot groups, covering spaces, surgery on knots, and important families of knots. Prerequisites: Mathematics 147 or Mathematics 171; or permission of instructor. (Offered alternate years)

152. Statistical Theory (3)
Williams, Hardin (Pomona). An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals and tests of hypotheses. Prerequisites: Mathematics 157 or Mathematics 151; or permission of instructor. (Offered jointly; Spring semester at Pomona and CMC)

156. Stochastic Processes (3)
Martonosi, Krieger. Continuation of Math 157. This course is particularly well suited for those wanting to see how probability theory can be applied to the study of phenomena in fields such as engineering, management science, the physical and social sciences, and operations research. Topics include conditional expectation, Markov chains, Poisson processes, and queuing theory. Additional applications chosen from such topics as reliability theory, Brownian motion, finance and asset pricing, inventory theory, dynamic programming, and simulation. Prerequisites: Mathematics 63 and Mathematics 157; or permission of the instructor. (Offered jointly; Fall semester at HMC)

157. Intermediate Probability (2)
Benjamin, Martonosi, Pippenger, Su, Williams. Continuous random variables, distribution functions, joint density functions, marginal and conditional distributions, functions of random variables, conditional expectation, covariance and correlation, moment generating functions, law of large numbers, Chebyshev’ theorem and central-limit theorem. Prerequisites: Mathematics 62; or permission of instructor. (Offered first half of Spring semester)

158. Statistical Data Analysis (3)
Martonosi, Williams, Hardin (Pomona). An introduction to analysis of variance (including one-way and two-way fixed effects ANOVA) and linear regression (including simple linear regression, multiple regression, variable selection, stepwise regression and analysis of residual plots). Emphasis will be both on methods and on applications to data using statistical software. Prerequisites: Mathematics 62; or AP Statistics or permission of instructor. (Offered second half of Spring semester, alternate years)

159. Design and Analysis of Experiments (2)
Martonosi, Williams. Prior to conducting an experiment, a scientist or engineer must properly structure the trials in order to draw meaningful conclusions from the data s/he collects. This course addresses, from a statistical perspective, how experiments should be designed so that the effects of the factors being tested can be distinguished from one another and from the variability inherent in the system. We will consider several design types, from practical and mathematical standpoints, such as Randomized Blocks, Latin Squares, Two-Level Factorial and Fractional Factorial designs, Response Surface Methods, Random Factors, and Robust Design. Students will use statistical software to analyze real data and complete a term project. Prerequisites: Mathematics 62; or equivalent. (Offered in alternate years, second-half of spring semester)

164. Scientific Computing (3)
Bernoff, de Pillis, Levy, Yong. Computational techniques applied to problems in the sciences and engineering. Modeling of physical problems, computer implementation, analysis of results; use of mathematical software; numerical methods chosen from: solutions of linear and nonlinear algebraic equations, solutions of ordinary and partial differential equations, finite elements, linear programming, optimization algorithms and fast-Fourier transforms. Prerequisites: Mathematics 64 and Computer Science 60; or permission of instructor. (Offered Spring)

165. Numerical Analysis (3)
Bernoff, Castro, de Pillis, Yong. An introduction to the analysis and computer implementation of basic numerical techniques. Solution of linear equations, eigenvalue problems, local and global methods for non-linear equations, interpolation and approximate integration. Prerequisites: Mathematics 64; or permission of instructor. (Offered Fall)

167. Complexity Theory (3) (Also listed as Computer Science 142)
Pippenger, Libeskind-Hadas (Computer Science), Bull (Pomona). Specific topics include finite automata, pushdown automata, Turing machines, and their corresponding languages and grammars; undecidability; complexity classes, reductions, and hierarchies. Prerequisites: Computer Science 81. (Offered Fall)

168. Algorithms (3) (Also listed as Computer Science 140)
Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science). Algorithm design, computer implementation, and analysis of efficiency. Discrete structures, sorting and searching, time and space complexity, and topics selected from algorithms for arithmetic circuits, sorting networks, parallel algorithms, computational geometry, parsing, and pattern-matching. Prerequisites: Mathematics 55 and Computer Science 60 and Mathematics 131. (Offered Fall and Spring)

171. Abstract Algebra I (3)
Benjamin, Karp, Orrison, Shahriari (Pomona), Sarkis (Pomona). Groups and isomorphism theorems. Rings and other structures. Prerequisites: Mathematics 12 and Mathematics 55; or permission of instructor. (Offered jointly; Fall semester at HMC and CMC, Spring semester at Pomona)

172. Abstract Algebra II: Galois Theory (3)
Orrison, Su, Shahriari (Pomona), Sarkis (Pomona). Selected topics in the theories of rings, modules, groups, and fields, such as Galois theory of equations and the structure of finitely generated modules over Euclidean and/or principal ideal domains with application to linear algebra and finitely generated abelian groups. Prerequisites: Mathematics 171. (Offered jointly; Spring semester at HMC)

173. Advanced Linear Algebra (3)
Gu, Orrison. Topics from among the following: Similarity of matrices and the Jordan form, the Cayley-Hamilton theorem, limits of sequences and series of matrices; the Perron-Frobenius theory of nonnegative matrices, estimating eigenvalues of matrices; stability of systems of linear differential equations and Lyapunov’s Theorem; iterative solutions of large systems of linear algebraic equations. Prerequisites: Mathematics 131; or equivalent. (Offered jointly in alternate years)

174. Abstract Algebra II: Representation Theory (3)
Karp, Orrison, Su. The topics covered will include group rings, characters, orthogonality relations, induced representations, applications of representation theory, and other select topics from module theory. Prerequisite: Math 171. 3 credit hours. This course is independent from Math 172 and may be taken by students who have taken Math 172. (Offered jointly; Spring by HMC and Pomona Colleges)

175. Number Theory (3)
Benjamin, Towse (Scripps). Properties of integers, congruences, Diophantine problems, quadratic reciprocity, number theoretic functions, primes. Prerequisites: Mathematics 55; or permission of instructor. (Offered Spring; offered jointly Fall semester at Scripps)

180. Applied Analysis (3)
Bernoff, Castro, Jacobsen, Levy. Selected topics from Fourier series, Fourier and Laplace transforms, ordinary and partial differential equations. Prerequisites: Mathematics 131. (Offered Fall)

181. Dynamical Systems (3)
Bernoff, Jacobsen, Levy, Radunskaya (Pomona). Existence and uniqueness theorems for systems of differential equations, dependence on data, linear systems, fundamental matrices, asymptotic behavior of solutions, stability theory, and other selected topics, as time permits. Prerequisites: Mathematics 115 or Mathematics 180. (Offered jointly; Fall semester at Pomona, Spring
semester at HMC in alternate years)

182. Partial Differential Equations (3)
Bernoff, Castro, Jacobsen, Levy. Theory and applications of quasi-linear and linear equations of first order, including systems. Theory of higher order linear equations, including classical methods of solutions for the wave, heat, and potential equations. Prerequisites: Mathematics 115 or Mathematics 180. (Offered Spring; offered in alternate years)

185. Introduction to Wavelets and their Applications (2)
Jacobsen, Yong. An introduction to the mathematical theory of wavelets, with applications to signal processing, data compression and other areas of science and engineering. Prerequisites: Mathematics 115 or Mathematics 180; (Fourier series) or permission of instructor.

187. Operations Research (3)
Benjamin, Martonosi, Shahriari (Pomona). Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory. Prerequisites: Mathematics 12. (Offered jointly; Fall semester at HMC/CMC, alternate years)

188. Social Choice and Decision Making (3) (Also listed as Integrative Experience 198)
Su. Basic concepts of game theory and social choice theory, representations of games, Nash equilibria, utility theory, non-cooperative games, cooperative games, voting games, paradoxes, Arrow’s impossibility theorem, Shapley value, power indices, “fair division” problems, and applications. Prerequisites: Mathematics 63; recommended Mathematics 55; or permission of instructor. (Offered alternate years in Spring semester)

189. Special Topics in Mathematics (1–3)
Staff. A course devoted to exploring topics of current interest to faculty or students. Recent topics have included: Algebraic Geometry, Algebraic Topology, Complex Dynamics, Fluid Dynamics, Games and Gambling, Mathematical Toys, and Rieman Zeta Functions. Prerequisites: permission of instructor.

191. Putnam Seminar (1)
Bernoff, Pippenger, Su. This seminar meets one evening per week during which students solve and present solutions to challenging mathematical problems in preparation for the Putnam Examination, a national undergraduate mathematics competition. (Offered Fall)

192. Problem Solving Seminar (1)
Bernoff. This seminar meets one evening per week during which students solve and present solutions to problems posed in mathematics journals, such as the American Mathematical Monthly. Solutions are submitted to these journals for potential publication. (Offered Spring)

193. Mathematics Clinic (3)
Gu and staff. Participation in projects or problems with a substantial mathematical and/or computational content. Students typically work in teams of two to four, with appropriate faculty supervision. Problems vary considerably, depending upon student interest and program of study, but normally require computer implementation and documentation. All work required for completion of Mathematics Clinic must be completed in a form acceptable to the clinic advisor by noon on Monday of the week prior to graduation. (Offered Fall and Spring)

196. Independent Study (1–5)
Staff. Readings in special topics. Prerequisites: permission of department. (Offered Fall and Spring)

197. Senior Thesis (3)
Staff. A research or expository paper based on independent work done under the supervision of a faculty member. The paper must be submitted to the mathematics department in a form suitable for publication in a mathematics journal. Prerequisites: permission of department. (Offered Fall and Spring)

198. Undergraduate Mathematics Forum (1)
Castro, Jacobsen, Levy, Martonosi, Orrison, Yong. The goal of this course is to improve students’ ability to communicate mathematics, both to a general and technical audience. Students will present material on assigned topics and have their presentations evaluated by students and faculty. This format simultaneously exposes students to a broad range of topics from modern and classical mathematics. Prerequisites: Required for all majors; recommended for all joint CS-math majors and mathematical biology majors, typically in the junior year. (Offered Fall and Spring)

199. Math Colloquium (0)
Jacobsen, Martonosi, Orrison. Students will attend weekly Claremont Math Colloquium, offered through the cooperative efforts of the mathematics faculty at the Claremont Colleges. Most of the talks discuss current research in mathematical sciences, and are accessible to undergraduates. (Offered Fall and Spring)

In addition to the courses described above, the graduate program in mathematics at the Claremont Graduate University offers a variety of courses. Graduate courses most often taken by advanced students at HMC include:

331. Measure and Integration
332. Functional Analysis
334. Advanced Complex Analysis
351. Time Series Analysis
362. Numerical Methods for PDEs
382. Perturbation and Asymptotics