(See also Joint Major in Computer Science and Mathematics, and Mathematical Biology)
Professors Bernoff (Chair), Benjamin, Borrelli (emeritus), Castro, Coleman (emeritus), Davis (2010-2012), de Pillis, Gu, Jacobsen, Karp, Krieger (emeritus), Levy, Martonosi, Orrison, Pippenger, Su, Tucker (2007-2011), 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.
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.
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 and during college-wide events, including Presentations Days.
The course of study for a mathematics degree has five components: The Major Core, Computational Mathematics, Clinic or Thesis, Mathematics Forum and Mathematics 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 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) and Mathematics 180 (Applied Analysis).
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: Mathematics 164 (Scientific Computing), Mathematics 165 (Numerical Analysis), Mathematics 167 (Complexity Theory), Mathematics 168 (Algorithms), Biology 188 (Computational Biology) or Computer Science 81 (Computability and Logic).
Clinic or Thesis
Two semesters of Mathematics Clinic (Mathematics 193) or two semesters of Senior Thesis (Mathematics 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 advisor. 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.
Math Forum and Math Colloquium
All mathematics majors must take one semester of Math Forum (Mathematics 198) and one semester of Mathematics Colloquium (Mathematics 199), generally in the junior year. In the mathematics forum, students prepare and present talks on mathematical topics taken from the literature. As a requirement for the mathematics 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 totaling at least seven credit hours are required. The elective program will be designed by the student in consultation with his or her advisor. 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. We emphasize that sample elective programs are advisory. Students may follow a sample program or design one of their own.
(CS = Computer Science, CMC = Claremont McKenna College, CGU = Claremont Graduate University, PO = Pomona College)
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.
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.
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 50 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. The HMC team has placed in the top 10 teams five times in the last 10 years; usually, HMC is the highest ranked undergraduate institution in the nation. 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.
- Michael E. Moody Lecture Series. The Moody evening lecture series brings speakers to the College who illuminate the joy, wonder and applications of mathematics, attracting hundreds of students and other members of the Claremont Colleges community.
- Weekly Mathematics Colloquium. The Claremont Colleges Mathematics Colloquium meets once per week. Most 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, operations research, statistics, financial mathematics, population dynamics and topology. Faculty, CGU graduate students and advanced undergraduate students attend the seminars.
- Mathematical Competition in Modeling/Interdisciplinary Competition in Modeling (MCM/ICM). TThe MCM/ICM contests are sponsored by the Consortium for Mathematics and its Applications and the Society for Industrial and Applied Mathematics. Each year, the MCM/ICM contests propose 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 eight teams competing in the MCM and ICM, out of over 2,500 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.
- CareFusion: Modeling Fluid Transport in Subcutaneous Tissue
- Chicago Trading Company: Building a Multi-Agent Artificial Stock Market
- Laserfiche: Automated Dewarping Algorithms for Enhancing Camera-Based Document Acquisition
- Los Alamos National Laboratories. Mathematical and Computational Modeling of Tumor Development
- 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:
- A Fast Fourier Transform for the Symmetric Group
- Connections Between Voting Theory and Graph Theory
- Foraging Fruit Flies: Lagrangian and Eulerian Descriptions of Insect Swarming
- Improving Cataract Surgery Rates Through Incidence Estimation
- Kolmogorov Complexity of Graphs
- Mathematical AIDS Epidemic Model: Preferential Anti-Retroviral Therapy Distribution in Resource Constrained Countries
- The Negs and Regs of Continued Fractions
- Turing Pattern Dynamics for Spatio-Temporal Models with Growth and Curvature
MATHEMATICS COURSES (Credit hours follow course title)
(Includes mathematics courses frequently taken by HMC students at the other Claremont Colleges)
15. Application and Art of Calculus (1)
Whitcher, Williams. This course is a fun and casual problem solving experience in single variable calculus. We will help the students strengthen mathematical skills essential to excel in the HMC Core. Students work in groups and solve calculus problems with an emphasis on applications to the sciences. Prerequisites: permission of department only. (Fall)
25B/25G. Calculus and Linear Algebra (3)
Benjamin, de Pillis, Karp, Levy, Orrison, Su. Theory and techniques of differential and integral calculus of a single real or complex variable; infinite series, including Taylor series and convergence tests. Theory and applications of vectors and matrices, including systems of linear equations; linear transformations in Euclidean space; determinants, eigenvalues, eigenvectors, and diagonalization. An introduction to multivariable calculus, including partial derivatives, double and triple integrals. The topics covered in 25B are the same as those covered in 25G, but 25B digs deeper into the theory and applications of the materials. Prerequisites: Mastery of single-variable calculus—entry into 25B by department placement only. (Fall)
35/62. 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, 25B or 25G. (Mathematics 62 first half, Fall 2010; replaced by Mathematics 35 starting Spring 2011)
45. Introduction to Differential Equations (1.5)
Bernoff, Castro, de Pillis, Jacobsen, Levy, Su, Whitcher, 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. Prerequisites: Mathematics 25B or 25G. (Spring)
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 Mathematics 25B or 25G; or permission of the instructor. (Fall and Spring)
60. Multivariable Calculus (1.5)
Bernoff, Castro, Gu, Karp, Levy, 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. Prerequisites: Mathematics 25B or 25G. (Fall, 2011 on)
64A/65. Differential Equations/Linear Algebra II (1.5)
Bernoff, Castro, Jacobsen, Levy, Martonosi. General vector spaces and linear transformations; change of basis and similarity; generalized eigenvectors; Jordan canonical forms. Applications to linear systems of ordinary differential equations, matrix exponential; Nonlinear systems of differential equations; equilibrium points and their stability. Prerequisites: Mathematics 64A or Mathematics 12 and 13; Mathematics 46, 60; or permission from the instructor/department. (Mathematics 64A, second half, Fall semester; replaced by Mathematics 65 starting Fall 2011)
70. Intermediate Linear Algebra (1.5)
de Pillis, Gu, Orrison. This half course is a continuation of Mathematics 65 and is designed to increase the depth and breadth of students' knowledge of linear algebra. Topics include: Vector spaces, linear transformations, eigenvalues, eigenvectors, inner-product spaces, spectral theorems, Jordan Canonical Form, singular value decomposition, and others as time permits. Prerequisites: Mathematics 64A or 65; or the equivalent. (First half, Spring semester)
80. Intermediate Differential Equations (1.5)
Bernoff, Castro, de Pillis, Jacobsen, Levy. This half course is a continuation of Math 65 and is designed to increase the depth and breadth of students' knowledge of differential equations. Topics include: nonlinear systems (stability for autonomous systems, Lyapunov functions, periodic solutions, limit cycles, chaos), the Laplace Transform, power-series methods, and others as time permits. Prerequisites: Mathematics 64A or 65; or the equivalent. (Second half, Spring semester)
104. Graph Theory (3)
Martonosi, 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 25B or 25G and Mathematics 55. (Offered alternate years)
106. Combinatorics (3)
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 or Mathematics 25B or 25G. (Offered alternate years)
108. History of Mathematics (3)
Grabiner (Pitzer). A survey of the history of mathematics from anitquity 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 or Mathematics 25B or 25G. (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 iequation), mathematical modeling of financial derivatives (calls and puts), and numerical methods for solving differential equations; Black-Scholes Model. Prerequisites: Mathematics 64A or 65 or (Mathematics 63 and 64) or permission of instructor. (Offered alternate years)
110. Applied Mathematics for Engineering (1.5)
(Also cross-listed as Engineering 72)
Levy, Yong, Bassman (Engineering). Applications of differential equations, linear algebra, and probability to engineering problems in multiple disciplines. Mathematical modeling, dimensional analysis, scale, approximation, model validation. Prerequisites: Mathematics 62 and Mathematics 64A or 65; or the equivalent. (Spring)
115. Fourier Series and Boundary Value Problems (3)
Bernoff, Levy, Yong. Complex variables and residue calculus; Laplace transforms; Fourier series and the Fourier transform; Partial Differential Equations including the heat equation, wave equation, and Laplace’s equation; Separation of variables; Sturm-Liouville theory and orthogonal expansions; Bessel functions. (May not be included in a mathematics major program. Students may not receive credit for both Mathematics 115 and 180.) Prerequisites: Mathematics 64A or 65; or Mathematics 63 and 64; or the equivalent. (Spring)
118. Mathematical Biology I (2)
(Also listed as Biology 118.) de Pillis, Jacobsen, Levy, 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 64A or 65; or Mathematics 63 and 64; Biology 52; or permission of instructor. (First half of Spring semester)
119. Mathematical Biology II (2)
(Also listed as Biology 119.) de Pillis, Jacobsen, Levy, 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 64A or 65; or Mathematics 63 and 64; Biology 52; or permission of instructor. (Second half, 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 or Mathematics 25B or 25G. (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 or Mathematics 25B or 25G. (Offered jointly at Pomona in alternate years)
131. Mathematical Analysis I (3)
Castro, Karp, Su. 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) or (Mathematics 25B or 25G) and Mathematics 60). (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, Spring 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 64A or 65; or Mathematics 63 and 64. (Fall)
137. Graduate Analysis I (3)
(Also listed as Mathematics 331 CG)
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. (Fall)
138. Graduate Analysis II (3) (Also listed as Mathematics 332 CG)
Castro, Krieger, Grabiner (Pomona), O'Neill (CMC). Banach and Hilbert spaces; Lp spaces; complex measures and the Radon-Nikodym theorem. Prerequisites: Mathematics 137 or 331 CG. (Spring)
142. Differential Geometry (3)
Gu, Karp, 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). Prerequisites: Mathematics 64A or 65; or Mathematics 63 and 64. (Fall)
IE 142. Seminar on Math and Science Education (Integrative Experience) (3)
Levy, Yong, Dodds (Computer Science). Students will learn about and contribute to math and science education in our community. Over the course of the semester, students observe math and science classrooms and reach out to integrate with our readings and discussions, which will be centered around questions such as, "What is effective math and science teaching?", "What is effective math and science education?", and, "How does math and science education impact our society?" Prerequisites: none. (Fall or Spring)
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 142; recommended Mathematics 147; or permission of instructor. (Spring)
IE 144. Mathematics, Music, Art: Cosmic Harmony (Integrative Experience) (3)
Orrison, Alves (Humanities, Social Sciences, and the Arts). A seminar exploring some of the many intersections between mathematics and music within our own and non-Western cultures, including proportion in art, tuning systems, algorithmic composition, artificial intelligence and creativity, and music synthesis. The class will also examine the ethical, aesthetic, and cultural ramifications of compression technology, sampling, downloading, and the effects of technology on music and vice-versa. Prerequisites: none. (Fall or 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)
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 171; or permission of instructor. (Offered alternate years)
152. Statistical Theory (3)
Martonosi, Williams, Hardin (Pomona), Huber (CMC). An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals and tests of hypotheses. Prerequisites: Mathematics 151 or 157; or permission of instructor. (Offered jointly; Spring at Pomona and CMC)
156. Stochastic Processes (3)
Benjamin, Martonosi, Huber (CMC). Continuation of Mathematics 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 64A or 65; or Mathematics 63 and 64; and Mathemathics 151 or 157; or permission of instructor. (Offered jointly; Fall 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. (Formerly Mathematics 151.) Prerequisites: Mathematics 62 or 35; or permission of instructor. (Offered first half of Spring)
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 35; or AP Statistics or permission of instructor. (Offered second half of Spring, 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 35; or the equivalent. (Offered in alternate years, second-half 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 64A or 65; or Mathematics 63 and 64; and Computer Science 60 or permission of instructor. (Spring)
165. Numerical Analysis (3)
Bernoff, Castro, de Pillis, Levy, 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: Mathematics 64A or 65; or Mathematics 63 and 64; or permission of instructor. (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 60 and Mathematics 55. (Fall)
168. Algorithms (3)
(Also listed as Computer Science 140.)
Pippenger, Libeskind-Hadas (Computer Science), Sweedyk (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. (Fall and Spring)
171. Abstract Algebra I (3)
Benjamin, Davis, Karp, Orrison, Sarkis (Pomona), Shahriari (Pomona). Groups and isomorphism theorems. Rings and other structures. Prerequisites: (Mathematics 12 or Mathematics 25B or 25G) 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)
Davis, Karp, Orrison, Su, Shahriari (Pomona), Sarkis (Pomona). 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. Prerequisites: Mathematics 171. (Offered jointly; Spring semester at HMC and Pomona)
173. Advanced Linear Algebra (3)
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 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 the equivalent. (Offered jointly in alternate years)
174. Abstract Algebra II: Representation Theory (3)
Davis, 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. Prerequisites: none. (Offered jointly; Spring semester at HMC and Pomona)
175. Number Theory (3)
Benjamin, Pippenger, Towse (Scripps). Properties of integers, congruences, Diophantine problems, quadratic reciprocity, number theoretic functions, primes. Prerequisites: Mathematics 55; or permission of instructor. (Spring; offered jointly Fall semester at Scripps)
180. Applied Analysis (3)
Bernoff, Castro, de Pillis, Jacobsen, Levy. Partial Differential Equations (PDEs) including the heat equation, wave equation, and Laplace’s equation; existence and uniqueness of solutions to PDEs via the maximum principle and energy methods; method of characteristics; Fourier series; Fourier transforms and Green’s functions; separation of variables; Sturm-Liouville theory and orthogonal expansions; Bessel functions. Prerequisites: Mathematics 64A or 65; or Mathematics 63 and 64; and Mathematics 131; or permission of instructor. (Fall)
181. Dynamical Systems (3)
Bernoff, de Pillis, 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 semster at HMC in alternate years)
182. Partial Differential Equations (3)
Bernoff, Castro, Jacobsen. 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. (Spring; alternate years)
185. Introduction to Wavelets and their Applications (2)
Staff. 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 180; or permission of instructor.
187. Operations Research (3)
Benjamin, Krieger, Martonosi, Huber (CMC), Shahriari (Pomona). Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory. Prerequisites: Mathematics 12 or 25B or 25G. (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 or 65; recommended Mathematics 55; or permission of instructor. (Offered alternate years, Spring)
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 Riemann Zeta Functions. Prerequisites: permission of instructor.
190. Mathematical Contest in Modeling/Interdisciplinary Contest in Modeling Seminar (1)
Martonosi. This seminar meets one evening per week during which students solve and present solutions to challenging mathematical problems in preparation for the Mathematical Contest in Modeling (MCM) and Interdisciplinary Contest in Modeling (ICM), an international undergraduate mathematics competition. Prerequisites: none. (Fall)
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. Prerequisites: none. (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. Prerequisites: none. (Spring)
193. Mathematics Clinic (3)
Bernoff, Castro, de Pillis, Gu, Levy, Martonosi, Williams. 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 advisor and a liaison provided by the sponsoring organization to solve complex real-world problems using mathematical and computational methods. Students are expected to present their work orally and to produce a final report conforming to the publication standards of a professional mathematician. (Fall and Spring)
196. Independent Study (1-5)
Staff. Readings in special topics. Prerequisites: permission of department. (Fall and Spring)
197. Senior Thesis (3)
Staff. Senior thesis offers the student, guided by the faculty advisor, a chance to experience a taste of the life of a professional research mathematician. The work is largely independent with guidance from the research advisor. The principal objective of the senior thesis program is to help you develop intellectually and improve your written and verbal communication skills. Students are expected to present their work orally and to produce a thesis conforming to the publication standards of a professional mathematician. Prerequisites: permission of department. (Fall and Spring)
198. Undergraduate Mathematics Forum (1)
Castro, Jacobsen, Levy, 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 computer science and mathematics majors and mathematical biology majors, typically in the junior year. (Fall and Spring)
199. Mathematics Colloquium (0.5)
Benjamin, Jacobsen. Students will attend weekly Claremont Mathematics 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. No more than 2.0 units of credit can be earned for colloquia. Pass/No Credit grading. Prerequisites: none. (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