Professors Bernoff (Chair; on leave fall 2012), Benjamin, Borrelli (emeritus), Byrne (2011-2013), Castro, Coleman (emeritus), de Pillis, Dresch (2012-2014), Gu, Jacobsen, Karp, Krieger (emeritus), Levy, Martonosi, Orrison, Pippenger, Su (acting Chair, fall 2012), 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, 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. 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 Presentation Days.
Mathematics Major Requirements
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), 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, 136, 142, 143, 147, 173, 174, 175, CGU 331, CGU 332, CGU 334.
Applied Mathematics: 136, 181 and at least one elective from 118, 119, 132, 164, 165, 173, 182, 187, CGU 362, CGU 368, CGU 382.
Probability and Statistics: 152, 156 and at least one elective from 106, 132, 152, 153, 155, 158, 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, 155, 158, 165, 187, CGU 355, Econometrics (CMC ECON 125, 126; CGU ECON 382, 383, 384; PO ECON 167).
Scientific Computing: 164, 165 and at least one elective from 118, 119, 136, 168, 173, 181, 182, CS 156, CGU 362, CGU 368, CGU 382.
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.
Mathematical Biology: 118, 119 and at least one elective from 152, 156, 158, 164, 168, 173, 181, 182, 187.
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.
- 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 analysis, algebra, applied mathematics, combinatorics, number theory, operations research, statistics, financial mathematics and topology. Faculty, CGU graduate students and advanced undergraduate students attend the seminars.
- The William Lowell Putnam Examination. The Putnam Exam is a national collegiate mathematics competition. Over 4,000 students from more than 500 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.
- Mathematical Competition in Modeling/Interdisciplinary Competition in Modeling (MCM/ICM). The 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 2,700 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
- DYNAR Collaboration with CGU: Cooperative Autonomous Aquatic Vehicles: Mathematics and Robotics
- Laserfiche: Automated Dewarping Algorithms for Enhancing Camera-Based Document Acquisition
- Los Alamos National Laboratories. Mathematical and Computational Modeling of Tumor Development
- Southwest Research Institute: Application of Iterative Blind Deconvolution Algorithms
- Space Systems/Loral: Application of Robust Control to Spacecraft Attitude Stability
Some Recent Senior ThesesSeveral students each year write a senior thesis. 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 Method for Approximating Solutions to Differential Equations via Schauder's Theorem
- Applications of q-Binomial Coefficients to Counting Problems
- Characteristics of Optimal Solutions to the Sensor Location Problem
- Group Actions and Divisors on Abstract Tropical Curves
- Markov Bases for Analysis of Partially Ranked Data
- Mathematical AIDS Epidemic Model: Preferential Anti-Retroviral Therapy Distribution in Resource Constrained Countries
- Modeling Wave Propagation in Viscoelastic Fluids
- A Multistage Incidence Estimation Model for Diseases with Differential Mortality
- A Nonlinear ODE Model of Tumor Growth and Effect of Immunotherapy and Chemotherapy Treatment in Colorectal Cancer
- 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 (0.5)
Byrne, Dresch, Karp, 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, first half)
30G. Calculus (1.5)
Benjamin, de Pillis, Karp, Levy, Orrison, Su. A comprehensive view of the theory and techniques of differential and integral calculus of a single variable; infinite series, including Taylor series and convergence tests. Focus on mathematical reasoning, rigor and proof, including continuity, limits, induction. Introduction to multivariable calculus, including partial derivatives, double and triple integrals. Prerequisites: One year of calculus at the high-school level. (Fall, first half)
30B. Calculus (1.5)
Benjamin, de Pillis, Karp, Levy, Orrison, Su. A comprehensive view of the theory and techniques of differential and integral calculus of a single variable; infinite series, including Taylor series and convergence tests. Focus on mathematical reasoning, rigor and proof, including continuity, limits, induction. Introduction to multivariable calculus, including partial derivatives, double and triple integrals. Placement into Math 30B is by exam and assumes a more thorough background than Math 30G; it allows for a deeper study of selected topics in calculus. Prerequisites: Mastery of single-variable calculus—entry by department placement only. (Fall, first half)
35. Probability and Statistics (1.5)
Benjamin, Dresch, 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 30B or Mathematics 30G. (Fall, second half)
40. Introduction to Linear Algebra (1.5)
Benjamin, de Pillis, Gu, Martonosi, Orrison, Pippenger, Su, Yong. Theory and applications of linearity, including vectors, matrices, systems of linear equations, dot and cross products, determinants, linear transformations in Euclidean space, linear independence, bases, eigenvalues, eigenvectors, and diagonalization. Prerequisites: One year of calculus at the high-school level. (Spring, first half)
45. Introduction to Differential Equations (1.5)
Bernoff, Castro, de Pillis, Dresch, 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. Prerequisites: Mathematics 30B or Mathematics 30G. (Spring, second half)
55. Discrete Mathematics (3)
Benjamin, Bernoff, Orrison, Pippenger. 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 40; or permission of instructor. (Fall and Spring)
60. Multivariable Calculus (1.5)
Bernoff, Castro, Gu, Karp, Levy, Orrison, Su, Yong. Linear approximations, the gradient, directional derivatives and the Jacobian; 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 30B or Mathematics 30G) and Mathematics 40. (Fall, first half, and summer)
65. Differential Equations and Linear Algebra II (1.5)
Bernoff, Castro, Jacobsen, Levy, Martonosi. General vector spaces and linear transformations; change of basis and similarity. Applications to linear systems of ordinary differential equations, matrix exponential; nonlinear systems of differential equations; equilibrium points and their stability. Prerequisites: Mathematics 40 and Mathematics 45; or permission of instructor. (Fall, second half, and summer)
70. Intermediate Linear Algebra (1.5)
de Pillis, Orrison. This half course is a continuation of Math 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 65; or equivalent. (Spring, first half)
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 Existence and Uniqueness, Power Series and Frobenius Series Methods, Laplace Transform, and additional topics as time permits. Prerequisites: Mathematics 65; or equivalent. (Spring, second half)
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 40 and Mathematics 55. (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. (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 30B or Mathematics 30G. (Alternate years)
109. Introduction to the Mathematics of Finance (3)
Aksoy (CMC). This is a first course in Mathematical Finance sequence. This course introduces the concepts of arbitrage and risk-neutral pricing within the context of single- and multi-period financial models. Key elements of stochastic calculus such as Markov processes, martingales, filtration and stopping times will be developed within this context. Pricing by replication is studied in a multi-period binomial model. Within this model, the replicating strategies for European and American options are determined. Prerequisites: Mathematics 65; or equivalent or permission of instructor. (Alternate years)
110. Applied Mathematics for Engineering (1.5) (Also listed as 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, Laplace Transforms. (May not be included in a mathematics major program.) Prerequisites: Mathematics 35 and Mathematics 65; or equivalent. (Spring, first half)
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 65; or equivalent. (Spring)
118. Introduction to Mathematical and Computational Biology (3) (Also listed as Biology 118)
de Pillis, Jacobsen, Levy, Adolph (Biology), Bush (Biology), Libeskind-Hadas (Computer Science). An introduction to the fields of mathematical and computational biology. Continuous and discrete mathematical models of biological processes and their analytical and computational solutions. Examples may include models in epidemiology, ecology, cancer biology, systems biology, molecular evolution, and phylogenetics. Prerequisites: Mathematics 65 and Biology 52; or permission of instructor. (Spring)
119. Advanced Mathematical Biology (2) (Also listed as Biology 119)
de Pillis, Jacobsen, Levy, Adolph (Biology). Further study of mathematical models of biological processes, including discrete and continuous models. Examples are drawn from a variety of areas of biology, which may include physiology, systems biology, cancer biology, epidemiology, ecology, evolution, and spatiotemporal dynamics. Prerequisites: Biology 52, Mathematics 118 or Biology 118; or permission of instructor. (Fall)
131. Mathematical Analysis I (3)
Castro, Karp, Su. This course is a rigorous analysis of the real numbers, and an introduction to writing and communicating mathematics well. Topics include properties of the rational and the real number fields, the least upper bound property, induction, countable sets, metric spaces, limit points, compactness, connectedness, careful treatment of sequences and series, functions, differentiation and the mean value theorem, and an introduction to sequences of functions. Additional topics as time permits. Prerequisites: Mathematics 55 or Mathematics 101. (Jointly; Fall semester at HMC and 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. (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 65; or equivalent. (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. (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. (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 65; or equivalent. (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 and Mathematics 142; recommended Mathematics 147; or permission of instructor. (Spring)
147. Topology (3)
Karp, 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. (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 40; or permission of instructor. (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 Mathematics 157; or permission of instructor. (Jointly; Spring semester at Pomona and CMC)
153. Bayesian Statistics (3)
Williams. An introduction to principles of data analysis and advanced statistical modeling using Bayesian inference. Topics include a combination of Bayesian principles and advanced methods; general, conjugate and noninformative priors, posteriors, credible intervals, Markov Chain Monte Carlo methods, and hierarchical models. The emphasis throughout is on the application of Bayesian thinking to problems in data analysis. Statistical software will be used as a tool to implement many of the techniques. Prerequisites: Mathematics 35; or permission of the instructor. (Spring, alternate years)
155. Time Series (3)
Williams. An introduction to the theory of statistical time series. Topics include decomposition of time series, seasonal models, forecasting models including causal models, trend models, and smoothing models, autoregressive (AR), moving average (MA), and integrated (ARIMA) forecasting models. Time permitting we will also discuss state space models, which include Markov processes and hidden Markov processes, and derive the famous Kalman filter, which is a recursive algorithm to compute predictions. Statistical software will be used as a tool to aid calculations required for many of the techniques. Prerequisites: Mathematics 35; or permission of the instructor. (Spring, alternate years)
156. Stochastic Processes (3)
Benjamin, Martonosi, Huber (CMC). This course is particularly well-suited for those wanting to see how probability theory can be applied to the study of random 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 40 and (Mathematics 151 or Mathematics 157); or permission of instructor. (Jointly; Alternate 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. (Formerly Math 151.) Prerequisites: Mathematics 35; or permission of instructor. (Fall and Spring, first half)
158. Statistical Linear Models (3)
Martonosi, Williams, Hardin (Pomona). An introduction to linear regression including simple linear regression, multiple regression, variable selection, stepwise regression and analysis of residual plots and analysis of variance including one-way and two-way fixed effects ANOVA. Emphasis will be on both methods and applications to data. Statistical software will be used to analyze data. Prerequisites: Mathematics 35; or permission of instructor. (Fall, alternate years)
164. Scientific Computing (3)
(Also listed as Computer Science 144)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 65 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, approximate integration (quadrature), and numerical solutions to ordinary differential equations. Prerequisites: Mathematics 65; or equivalent 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, 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. (Fall and Spring)
171. Abstract Algebra I (3)
Benjamin, Karp, Orrison, Shahriari (Pomona), Sarkis (Pomona). Groups, rings, fields and additional topics. Topics in group theory include groups, subgroups, quotient groups, Lagrange's theorem, symmetry groups, and the isomorphism theorems. Topics in Ring theory include Euclidean domains, PIDs, UFDs, fields, polynomial rings, ideal theory, and the isomorphism theorems. In recent years, additional topics have included the Sylow theorems, group actions, modules, representations, and introductory category theory. Prerequisites: Mathematics 40 and Mathematics 55; or permission of instructor. (Jointly; Fall semester at HMC and CMC, Spring semester at HMC and Pomona)
172. Abstract Algebra II: Galois Theory (3)
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. (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 permission of instructor. (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. Prerequisites: Mathematics 171. (Jointly; Spring by 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)
176. Algebraic Geometry (3)
Karp. Topics include affine and projective varieties, the Nullstellensatz, rational maps and morphisms, birational geometry, tangent spaces, nonsingularity and intersection theory. Additional topics may be included depending on the interest and pace of the class. Prerequisites: Mathematics 171; recommended previous courses in Analysis, Galois Theory, Differential Geometry and Topology are helpful but not required; or permission of the instructor. (Fall, alternate years)
180. Introduction to Partial Differential Equations (3)
Bernoff, Byrne, 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 80 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; or permission of instructor. (Jointly; Fall semester at Pomona, Spring semester at HMC in alternate years)
182. Graduate Partial Differential Equations (3)
Bernoff, Castro, Jacobsen, Levy. Advanced topics in the study of linear and nonlinear partial differential equations. Topics may include the theory of distributions; Hilbert spaces; conservation laws, characteristics and entropy methods; fixed point theory; critical point theory; the calculus of variations and numerical methods. Applications to fluid mechanics, mathematical physics, mathematical biology and related fields. Prerequisites: (Mathematics 115 and Mathematics 131) or Mathematics 180; recommended Mathematics 132. (Spring; offered in 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 Mathematics 180; or permission of instructor.
187. Operations Research (3)
Benjamin, Martonosi, Huber (CMC), Shahriari (Pomona). Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory. Prerequisites: Mathematics 40; or equivalent. (Fall)
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: Prior or concurrent enrollment in Mathematics 30 or equivalent; recommended Mathematics 55; or permission of instructor. (Spring, alternate years)
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: Varies with topics chosen.
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 William Lowell Putnam Mathematics Competition, a national undergraduate mathematics contest. 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. Prerequisites: none. (Fall and Spring)
196. Independent Study (1–5)
Staff.Readings in special topics. Prerequisites: Permission of department or instructor. (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: none; Required for all majors; recommended for all joint CS-math majors and mathematical biology majors, typically in the junior year. (Fall and Spring)
199. Math Colloquium (0.5)
Benjamin, Jacobsen, Su.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. Prerequisites: none. (Fall and Spring)