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

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

A mathematics degree from HMC 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, actuarial science, banking, 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 incooperation with the other Claremont Colleges, including graduate courses at the Claremont Graduate University.

The academic experience of our majors is extended by the Mathematics Clinic Program. An educational innovation of Harvey Mudd College, the Clinic Program brings together teams of students to work on a research problem sponsored by business, industry or government. Teams work closely witha 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 have many opportunities to work individually with faculty to do mathematical research. Active areas of mathematical research at HMC and The Claremont Colleges include real and complex analysis, algebra, topology, differential and hyperbolic geometry, probability theory, dynamical systems, partial differential equations, asymptotics, numerical analysis, combinatorics, mathematical biology, graph theory and mathematical modeling. HMC students may also do their senior thesis with mathematicians at the other 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 mathematicsis required of all mathematics majors, selected from the following list: Math 164 (Scientific Computing), Math 165 (Numerical Analysis), Math 167 (Theory of Computation), Math 168 (Algorithms), 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 go onto 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 oftheir 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 have excellent facilities to support educational and research programs. The Department of Mathematics Scientific Computing Laboratory has over a dozen UNIX workstations and servers; these machines are available to any student who is enrolled in advanced mathematics courses or who is involved in mathematical research. The laboratory supports a wide variety of mathematical software packages including Mathematica, Maple and Matlab. The Differential Equations Laboratory adds several more work stations to the departmental network, devoted mainly to student project work in differential equations and applied mathematics. The department has two parallel computers, one a beowulf class distributed memory cluster, and the other a 16-processor shared memory supercomputer; these computers support student project work in numerical analysis and algorithms as well as research in applied mathematics, parallel computing and scientific computing.

Other Mathematical Activities at HMC and in Claremont
There are many opportunities outside of course work to enjoy andparticipate 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. Control algorithm project
  
• 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
  
• 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
  

COURSES
(Includes mathematics courses frequently taken by HMC studentsat the other Claremont Colleges)

11. Calculus of One Real or Complex Variable. Benjamin, Jacobsen, Karp, Orrison. Complex numbers, limits, formal epsilon-delta limitdefinition, derivatives and differentiation rules; proofs by contradiction and induction; infinite series; integration; applications of the calculus; introduction to calculus of complex-valued functions. Prerequisite: One year ofcalculus at the high school level. 2 credit hours. (First half of Fall.)

12. Introduction to Linear Algebra and Discrete Dynamical Systems. de Pillis, Orrison, Pippenger, Su, Tucker. Matrix representation of systems of equations, matrix operations, determinants; linear independence and dependence, bases; inner products, eigenvalues and eigenvectors; examples of discrete dynamical systems, fixed points, chaos, stability, bifurcations. Prerequisite: Mathematics 11 or the equivalent. 2 credit hours. (Fall, both halves)

13. Differential Equations I. Bernoff, Castro, de Pillis, Jacobsen, Levy, 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: Math 11 or the equivalent. 1.5 credit hours. (Fall, second half, and Spring, first half.)

14. Multivariable Calculus I. Castro, Gu, Karp, Orrison, Su, Yong. Vectors, dot and cross products; vector descriptions of lines and planes; partial derivatives and differentiability; gradients and directional derivatives; chain rule; higher order derivatives and Taylor approximations; double and triple integrals in rectangular and other coordinate systems; line integrals; vector fields, curl and divergence; introduction to Green's theorem, divergence theorem and Stoke's theorem. Prerequisite: Math 11. 1.5 credit hours. (Spring, both halves.)

55. Discrete Mathematics. 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. Prerequisite: Math 12 or permission of the instructor. 3 credit hours. (Fall and Spring.)

61. Multivariable Calculus II. Bernoff, Castro, Gu, Su, Yong. Review of basic multivariable calculus; optimization and the Second Derivative Test; constrained optimization using Lagrange multipliers; conservative and nonconservative vector fields; Green's theorem; parametrized surfaces and surface integrals; divergence theorem, outline of proof and applications; Stoke's theorem, outline of proof and applications; unification of major vector theorems. Prerequisite: Math 14. 1.5 credit hours. (First half of Fall.)

62. Introduction to Probability and Statistics. Martonosi, 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; chi-square goodness of fit; simple linear regression; introduction to analysis of variance; applications to analyzing real data sets. Prerequisite: Math 11. 1.5 credit hours. (Second half of Fall.)

63. Linear Algebra II. Benjamin, de Pillis, Gu, Martonosi, Orrison, Pippenger, Williams. Review of basic linear algebra; vector spaces; row and column spaces of matrices, rank-nullity theorem; orthogonal bases and Gram-Schmidt procedure; orthogonal expansion and Fourier coefficients; linear transformations; change of basis and similarity; eigen values, eigen vectors and characteristic polynomials; diagonalization of symmetric matrices; applications of eigen values to systems of ordinary differential equations. Prerequisite: Math 12. 1.5 credit hours. (First half of Spring.)

64. Differential Equations II. Bernoff, Castro, de Pillis, Jacobsen, Levy, Martonosi, Yong. Review of basic ordinary differential equations, especially systems; undriven linear systems; orbital portraits; stability and conservative systems; Lyapunov functions; cycles and long-term behavior of solutions; Sturm-Liouville problems; series solutions near ordinary and regular singular points; Bessel functions; chaos. Prerequisite: Math 13 and Math 63. 1.5 credit hours. (Second half of Spring.)

104. Graph Theory. Benjamin, Martonosi, Orrison, Tucker. An introduction to graph theory with applications. Theory and applications of trees, matchings, graph coloring, planarity, graph algorithms and other topics. Prerequisite: Math 12 and 55. 3 credit hours. (Offered alternate years.)

106. Combinatorics. Benjamin, Orrison, Pippenger. An introduction tothe techniques and ideas of combinatorics, including counting methods, Stirling numbers, Catalan numbers, generating functions, Ramsey theory and partially ordered sets (formerly Math 103). Prerequisite: Mathematics 55 or permission of instructor. 3 credit hours. (Offered alternate years.)

108. History of Mathematics. J. 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. Prerequisite: Math 11. 3 credit hours. (Offered alternate years.)

109. Introduction to the Mathematics of Finance. Aksoy (CMC).This course emphasizes the mathematics 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: Math 63. (Offered alternate years.)

115. Fourier Series and Boundary Value Problems. Bernoff, Castro, Cumberbatch (CGU), 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.) Prerequisite: Mathematics 64. 3 credit hours. (Fall.)

118. Mathematical Biology I (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: Math 64, Bio 52, or permission of instructor. 2 credit hours. (First half of Spring.)

119. Mathematical Biology II (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: Math 64, Bio 52, or permission of instructor. 2 credit hours. (Second half of Spring.)

120. Chirality. Flapan (Pomona). A structure ischiral 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. Prerequisite: Math 12. 2 credit hours. (Offered alternate years in Spring.)

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

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

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

136. Complex Variables and Integral Transforms. Jacobsen, 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. Prerequisite: Mathematics 64. 3 credit hours. (Fall.)

137. Graduate Analysis I (Also listed as Math 331). Castro, Grabiner (Pomona), O'Neill (CMC). Abstract Measures, Lebesgue measure and Lebesgue-Stieljes measures on R; Lebesgue integral and limit theorems; product measures and the Fubini theorem; additional topics. 3 credit hours. Prerequisite: Math 132. (Fall.)

138. Graduate Analysis II (Also listed as Math 332). Castro, Grabiner (Pomona), O'Neill (CMC). Banach and Hilbert spaces; L^p spaces;complex measures and the Radon-Nilodym theorem. Prerequisite: Math 137/331. 3 credit hours. (Spring.)

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

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

147. Topology. 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. Prerequisite: 131 or permission of instructor. 3 credit hours. (Offered jointly with Pomona; Spring.)

148. Knot Theory. Hoste (Pitzer). An introduction totheory 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. Prerequisite: Math 147 or 171 or permission of instructor. 3 credit hours. (Offered alternate years.)

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

156. Stochastic Processes. Krieger (Emeritus), Martonosi, Williams. 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: Math 63 and Math 151 (or Math 157) or permission of the instructor. 3 credit hours. (Offered jointly; Fall at HMC.)

157. Intermediate Probability. 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's theorem and central limit theorem (formerly Math 151). Prerequisite: Math 62 or permission of instructor. 2 credit hours. (First half of Spring.)

158. Statistical Data Analysis. Martonosi, Hardin (Pomona), Williams. 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 on both methods and on applications to data using statistical software. Prerequisite: Math 62 or AP Statistics or permission of instructor. 2 credit hours. (Spring).

159. Experimental Design. Martonosi, Angus (CGU). Many statistics courses focus on how to analyze data that have already been collected in an experiment. However, when planning the experiment, the researcher must take steps to ensure that the data to be collected can be analyzed using statistically sound methods and can yield conclusive results. This is the design of experiments. The course explores the theory and application of the design and analysis of experiments: ANOVA, randomized block designs, factorial designs, central composite designs, response surface methods, random factorsand robust design. Students will analyze real data, design a real experiment, and discuss some of the theory underlying these methods. 2 credit hours. Prerequisite: Math 62 or AP Statistics or permission of instructor. (Spring).

164. Scientific Computing (Jointly listed as Computer Science 144). de Pillis, 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: Math 64 and CS 60 or permission of instructor. 3 credit hours. (Spring.)

165. Numerical Analysis. Bernoff, de Pillis, Pippenger, 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: Math 64 or permission of instructor. 3 credit hours. (Fall.)

167. Complexity Theory (Jointly listed as Computer Science 142). Pippenger, Libeskind-Hadas (Computer Science). Brief review of computability theory through Rice's Theorem and the Recursion Theorem followed by a rigoroust reatment of complexity theory. The complexity classes P, NP and the Cook-LevinTheorem. Approximability of NP-complete problems. The polynomial hierarchy, PSPACE-completeness, L and NL-completeness, #P-completeness. IP and Zero-knowledge proofs.Randomized and parallel complexity classes. The speedup, hierarchy and gap theorems. Prerequisite: Computer Science 81. 3 credit hours. (Fall.)

168. Algorithms (Jointly 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: Math 55 and CS 60. Corequisite: Math 131. 3 credit hours. (Fall and Spring.)

171. Abstract Algebra I: Groups and Rings. Benjamin, Karp, Orrison. The study of some basic structures that appear throughout mathematics, including groups and rings. Topics in group theory will include isomorphism theorems, orbits and stabilizers, and coset partitions. Topics in ring theory will include ideals, quotient rings, and prime and maximal ideals. Ring and field extensions may also be introduced as time remains. Prerequisites: Math 12 and Math 55, or permission of instructor. 3 credit hours. (Offered jointly; Fall at HMC and Pomona, Spring at CMC or Scripps.) 

172. Abstract Algebra II: Galois Theory. Karp, Orrison, Su. The topics covered will include polynomial rings, field extensions, classical constructions, splitting fields, algebraic closure, separability, Fundamental Theorem of Galois Theory, Galois groups of polynomials and solvability. Prerequisite: Math 171. 3 credit hours. This course is independent from Math 174 and may be taken by students who have taken Math 174. (Offered jointly; Spring by HMC and Pomona Colleges.)



173. Advanced Linear Algebra. de Pillis, 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 eigen values of matrices; stability of systems of linear differential equations and Lyapunov's Theorem; iterative solutions of large systems of linear algebraic equations. Prerequisite: Math 131 or equivalent. 3 credit hours. (Offered jointly; alternate years.)

174. Abstract Algebra II: Representation Theory. 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. Benjamin, Towse (Scripps). Properties of integers, congruences, Diophantine problems, quadratic reciprocity, number theoretic functions, primes. Prerequisite: Math 55 or permission of instructor. 3 credit hours. (Spring.) (Offered jointly; Fall at Scripps.)

180. Applied Analysis. Bernoff, Castro, Jacobsen, Levy. Selected topics from Fourier series, Fourier and Laplace transforms, ordinary and partial differential equations. Prerequisite: Math 131. 3 credit hours. (Students may not receive credit for both Mathematics 180 and 115.) (Fall.)

181. Dynamical Systems. Bernoff, Goroff, Jacobsen, 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. Prerequisite: Math 115 or 180. 3 credit hours. (Offered jointly; Fall at Pomona, Spring at HMC in alternate years.)

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

185. Introduction to Wavelets and their Applications. Yong. An introduction tothe mathematical theory of wavelets, with applications to signal processing, data compression and other areas of science and engineering. Prerequisite: Fourier Series (Math 115 or Math 180), or permission of instructor. 2 credit hours.

187. Operations Research. Benjamin, Martonosi. Linear, integer, non-linear and dynamic programming, classical optimization problems and network theory. Prerequisite: Math 12. 3 credit hours. (Offered jointly; Fall at CMC.)

188. Social Choice and Decision Making (Also listed as IE198). Su. Basic concepts of game theory and social choice theory, representations of games, Nashequilibria, utility theory, non-cooperative games, cooperative games, voting games, paradoxes, Arrow's impossibility theorem, Shapley value, power indices, "fair division" problems and applications. Prerequisite: Math 63 and (recommended) Math 55 or permission of instructor. 3 credit hours. (Offered alternate years; Spring.)

189. Special Topics in Mathematics. Staff. A course devoted to exploring topics of current interest to faculty or students. Recent topics have included: Algebraic Topology, Complex Dynamics, Fluid Dynamics, Games and Gambling, Advanced Partial Differential Equations, Nonlinear Functional Analysis, Advanced Discrete Mathematics, and Variational Methods. Prerequisites: Permission of instructor. 3 credit hours.

191. Putnam Seminar. 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. 1 credit hour. (Fall.)

192. Problem Solving Seminar. 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. 1 credit hour. (Spring.)

193. Mathematics Clinic. Castro 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. 3 credit hours per semester. (Fall and Spring.)

196. Independent Study. Staff. Readings in special topics. Prerequisite: permission of department. 1-5 credit hours per semester. (Fall and Spring.)

197. Senior Thesis. Bernoff, de Pillis, Pippenger. 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. Prerequisite: permission of department. 3 credit hours. (Fall and Spring.)

198. Math Forum. Castro, Jacobsen, Levy, Orrison. 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. Required for all math and joint CS-Math majors; recommended for mathematical biology majors, typically in the junior year. 1 credit hour. (Fall and Spring.)

199. Math Colloquium. Staff. Students will attend a 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. Not for credit. (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