Mathematics Major

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

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

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

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

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

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:

  • Credits: 3

    Instructors: Benjamin, Bernoff, Orrison, Pippenger

    Offered: Fall and Spring

    Description: 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.

    Corequisites: MATH040 HM or MATH073 HM 

  • Credits: 1.5

    Instructors: de Pillis, Omar, Orrison

    Offered: Spring, first half

    Description: This half course is a continuation of MATH065 HM 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: MATH065 HM 

  • Credits: 1.5

    Instructors: Bernoff, Castro, de Pillis, Jacobsen

    Offered: Spring, second half

    Description: This half course is a continuation of MATH065 HM 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: MATH065 HM 

  • Credits: 3

    Instructors: Castro, Karp, Omar, Su

    Offered: Jointly; Fall semester at HMC and Pomona, Spring semester at HMC and CMC

    Description: 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: MATH055 HM 

  • Credits: 1.5

    Instructors: Benjamin, Martonosi, Pippenger, Su, Williams

    Offered: Fall and Spring

    Description: 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. 

    Prerequisites: MATH019 HM or MATH060 HM  

  • Credits: 3

    Instructors: Benjamin, Karp, Omar, Orrison, Staff (CMC), Staff (Pomona)

    Offered: Jointly; Fall semester at HMC and CMC, Spring semester at HMC and Pomona

    Description: 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: (MATH040 HM or MATH073 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Jacobsen

    Offered: Fall

    Description: 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: MATH080 HM and MATH131 HM 

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.

Computational Mathematics

One course in computational mathematics is required of all mathematics majors, selected from the following list:

  • Credits: 3

    Instructors: Keller, Stone

    Offered: Fall and Spring

    Description: An introduction to some of the mathematical foundations of computer science, particularly logic, automata, and computability theory. Develops skill in constructing and writing proofs, and demonstrates the applications of the aforementioned areas to problems of practical significance.

    Prerequisites: MATH055 HM and (CSCI060 HM or CSCI042 HM)

  • Credits: 3

    Instructors: Bernoff, de Pillis, Yong

    Offered: Spring

    Description: 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and CSCI060 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: Pippenger, Libeskind-Hadas (Computer Science), Staff (Pomona)

    Offered: Fall

    Description: Brief review of computability theory, followed by a rigorous treatment of complexity theory. The complexity classes P, NP, and the Cook-Levin Theorem. 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.

    Prerequisites: (CSCI060 HM or CSCI042 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science)

    Offered: Fall and Spring

    Description: 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: (CSCI070 HM and CSCI081 HM) or ((CSCI060 HM or CSCI042 HM) and MATH131 HM))

Computational techniques are essential to many fields of modern mathematics and to most mathematical applications in business and industry.

Clinic or Thesis

Two semesters of Mathematics Clinic or two semesters of Senior Thesis are required and normally taken in the senior year:

  • (taken twice)

    Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Gu, Martonosi, Williams

    Offered: Fall and Spring

    Description: 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. Students are expected to take the two semesters of Clinic within a single academic year.

or

  • (taken twice)

    Credits: 3

    Instructor: Staff

    Offered: Fall and Spring

    Description: 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

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.

Mathematics Forum and Mathematics Colloquium

All mathematics majors must take:

One semester of Mathematics Forum:

  • (generally in the junior year)

    Credit: 1

    Instructors: Castro, Jacobsen, Orrison, Yong

    Offered: Fall and Spring

    Description: 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 majors; recommended for all joint CS-math majors and mathematical biology majors, typically in the junior year.

and

One semester of Mathematics Colloquium:

  • (generally in the junior year)

    Credit: 0.5

    Instructors: Benjamin, Jacobsen, Su

    Offered: Fall and Spring

    Description: 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. No more than 2.0 credits can be earned for departmental seminars/col­loquia. 

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 credits are required.

The elective program will be designed by the student in consultation with her or his 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.

Pure Mathematics

  • Credits: 3

    Instructors: Castro, Omar, Su, Staff (Pomona)

    Offered: Jointly; Fall semester at HMC, Spring semester at Pomona

    Description: A rigorous study of calculus in Euclidean spaces including multiple Riemann integrals, derivatives of transformations, and the inverse function theorem.

    Prerequisites: MATH131 HM 

  • Credits: 3

    Instructors: Karp, Omar, Orrison, Su, Staff (Pomona)

    Offered: Jointly; Spring semester at HMC and Pomona

    Description: 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: MATH171 HM 

and at least one elective from:

  • Credits: 3

    Instructors: Martonosi, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: An introduction to graph theory with applications. Theory and applications of trees, matchings, graph coloring, planarity, graph algorithms, and other topics.

    Prerequisites: (MATH040 HM or MATH073 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Benjamin, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: 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: MATH055 HM 

  • Credits: 3

    Instructors: Gu, Jacobsen, Karp, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM

  • Credits: 3

    Instructors: Gu, Karp, Staff (Pitzer)

    Offered: Fall

    Description: Curves and surfaces, Gauss curvature; isometries, tensor analy­sis, covariant differentiation with application to physics and geometry (intended for majors in physics or mathematics).

    Prerequisites: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructor: Gu

    Offered: Spring

    Description: Selected topics in Riemannian geometry, low dimensional manifold theory, elementary Lie groups and Lie algebra, and contemporary applications in mathematics and physics.

    Prerequisites: MATH131 HM and MATH142 HMMATH147 HM recommended

  • Credits: 3

    Instructors: Karp, Pippenger, Su, Staff (Pomona)

    Offered: Jointly with Pomona; Spring semester

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: de Pillis, Gu, Orrison

    Offered: Jointly in alternate years

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: Karp, Omar, Orrison, Su

    Offered: Jointly; Spring semester at HMC and Pomona

    Description: The topics covered will include group rings, characters, orthogonality relations, induced representations, applications of representation theory, and other select topics from module theory.

    Prerequisites: MATH171 HM 

  • Credits: 3

    Instructors: Benjamin, Omar, Pippenger, Staff (Scripps)

    Offered: Spring; offered jointly Fall semester at Scripps

    Description: Properties of integers, congruences, Diophantine problems, quadratic reciprocity, number theoretic functions, primes.

    Prerequisites: MATH055 HM 

Applied Mathematics

  • Credits: 3

    Instructors: Gu, Jacobsen, Karp, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM

  • Credits: 3

    Instructors: Bernoff, de Pillis, Jacobsen, Staff (Pomona)

    Offered: Jointly; Fall semester at Pomona, Spring semester at HMC in alternate years

    Description: 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: MATH115 HM or MATH180 HM 

and at least one elective from:

  • Credits: 3

    Instructors: de Pillis, Jacobsen, Adolph (Biology)

    Description: 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: MCBI118A HM and MCBI118B HM 

  • Credits: 3

    Instructors: Castro, Omar, Su, Staff (Pomona)

    Offered: Jointly; Fall semester at HMC, Spring semester at Pomona

    Description: A rigorous study of calculus in Euclidean spaces including multiple Riemann integrals, derivatives of transformations, and the inverse function theorem.

    Prerequisites: MATH131 HM 

  • Credits: 3

    Instructors: Bernoff, de Pillis, Yong

    Offered: Spring

    Description: 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and CSCI060 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: de Pillis, Gu, Orrison

    Offered: Jointly in alternate years

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, Jacobsen

    Offered: Spring, alternate years

    Description: 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: (MATH115 HM and MATH131 HM) or MATH180 HM; recommended MATH132 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC), Staff (Pomona)

    Offered: Fall

    Description: Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory.

    Prerequisites: MATH040 HM or MATH073 HM 

  • MATH362 CG - Numerical Methods for Differential Equations
  • MATH368 CG - Numerical Methods for Matrix Computations
  • MATH382 CG - Perturbation and Asymptotic Analysis
  • Credits: 1.5

    Instructors: Adolph (Biology), de Pillis (Mathematics), Jacobsen (Mathematics)

    Offered: Spring

    Description: An introduction to the field of mathematical 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and BIOL052 HM 

and

  • Credits: 1.5

    Instructors: Bush (Biology), Donaldson-Matasci (Biology), Libeskind-Hadas (Computer Science), Wu (Computer Science)

    Offered: Spring

    Description: An introduction to the field of computational biology. Algorithms for phylogenetic inference and computational methods for solving problems in molecular evolution and population genetics.

    Prerequisites: (CSCI005 HM or CSCI005GR HM) and BIOL052 HM 

Probability and Statistics

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona), Staff (CMC)

    Offered: Jointly; Spring semester at Pomona and CMC

    Description: An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals, and tests of hypotheses.

    Prerequisites: MATH 157 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC)

    Offered: Jointly; Fall, alternate years at HMC

    Description: 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: (MATH040 HM or MATH073 HM) and MATH157 HM 

and at least one elective from:

  • Credits: 3

    Instructors: Benjamin, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: 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: MATH055 HM 

  • Credits: 3

    Instructors: Castro, Omar, Su, Staff (Pomona)

    Offered: Jointly; Fall semester at HMC, Spring semester at Pomona

    Description: A rigorous study of calculus in Euclidean spaces including multiple Riemann integrals, derivatives of transformations, and the inverse function theorem.

    Prerequisites: MATH131 HM 

  • Credits: 3

    Instructor: Williams

    Offered: Spring, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructor: Williams

    Offered: Spring, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona)

    Offered: Fall, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: de Pillis, Gu, Orrison

    Offered: Jointly in alternate years

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC), Staff (Pomona)

    Offered: Fall

    Description: Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory.

    Prerequisites: MATH040 HM or MATH073 HM 

  • MATH351 CG - Time Series Analysis
  • MATH355 CG - Linear Statistical Models

Operations Research

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC)

    Offered: Jointly; Fall, alternate years at HMC

    Description: 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: (MATH040 HM or MATH073 HM) and MATH157 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC), Staff (Pomona)

    Offered: Fall

    Description: Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory.

    Prerequisites: MATH040 HM or MATH073 HM 

and at least one elective from:

  • Credits: 3

    Instructors: Martonosi, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: An introduction to graph theory with applications. Theory and applications of trees, matchings, graph coloring, planarity, graph algorithms, and other topics.

    Prerequisites: (MATH040 HM or MATH073 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Benjamin, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: 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: MATH055 HM 

  • Credits: 3

    Instructors: Castro, Omar, Su, Staff (Pomona)

    Offered: Jointly; Fall semester at HMC, Spring semester at Pomona

    Description: A rigorous study of calculus in Euclidean spaces including multiple Riemann integrals, derivatives of transformations, and the inverse function theorem.

    Prerequisites: MATH131 HM 

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona), Staff (CMC)

    Offered: Jointly; Spring semester at Pomona and CMC

    Description: An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals, and tests of hypotheses.

    Prerequisites: MATH 157 HM 

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona)

    Offered: Fall, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science)

    Offered: Fall and Spring

    Description: 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: (CSCI070 HM and CSCI081 HM) or ((CSCI060 HM or CSCI042 HM) and MATH131 HM))

  • Credits: 3

    Instructor: Su

    Offered: Spring, alternate years

    Description: 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.

    Corequisites: MATH030B HM or MATH030G HMMATH055 HM recommended

Actuarial or Financial Mathematics

  • Credits: 3

    Instructor: Staff (CMC)

    Offered: Alternate years

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC)

    Offered: Jointly; Fall, alternate years at HMC

    Description: 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: (MATH040 HM or MATH073 HM) and MATH157 HM 

and at least one elective from:

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona), Staff (CMC)

    Offered: Jointly; Spring semester at Pomona and CMC

    Description: An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals, and tests of hypotheses.

    Prerequisites: MATH 157 HM 

  • Credits: 3

    Instructor: Williams

    Offered: Spring, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona)

    Offered: Fall, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC), Staff (Pomona)

    Offered: Fall

    Description: Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory.

    Prerequisites: MATH040 HM or MATH073 HM 

  • MATH355 CG - Linear Statistical Models
  • ECON125 CM - Econometrics
  • ECON126 CM - Microeconometrics
  • ECON167 PO - Econometrics
  • ECON382 CG - Econometrics I
  • ECON383 CG - Econometrics II
  • ECON384 CG - Econometrics III

Scientific Computing

  • Credits: 3

    Instructors: Bernoff, de Pillis, Yong

    Offered: Spring

    Description: 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and CSCI060 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

and at least one elective from:

  • Credits: 3

    Instructor: Keller

    Offered: Spring, alternate years

    Description: Characteristics and applications for parallel and real-time systems. Specification techniques, algorithms, architectures, languages, design, and implementation.

    Prerequisites: CSCI105 HM and CSCI140 HM; CSCI131 HM recommended

  • Credits: 3

    Instructors: de Pillis, Jacobsen, Adolph (Biology)

    Description: 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: MCBI118A HM and MCBI118B HM 

  • Credits: 3

    Instructors: Gu, Jacobsen, Karp, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM

  • Credits: 3

    Instructors: Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science)

    Offered: Fall and Spring

    Description: 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: (CSCI070 HM and CSCI081 HM) or ((CSCI060 HM or CSCI042 HM) and MATH131 HM))

  • Credits: 3

    Instructors: de Pillis, Gu, Orrison

    Offered: Jointly in alternate years

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: Bernoff, de Pillis, Jacobsen, Staff (Pomona)

    Offered: Jointly; Fall semester at Pomona, Spring semester at HMC in alternate years

    Description: 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: MATH115 HM or MATH180 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, Jacobsen

    Offered: Spring, alternate years

    Description: 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: (MATH115 HM and MATH131 HM) or MATH180 HM; recommended MATH132 HM 

  • MATH362 CG - Numerical Methods for Differential Equations
  • MATH368 CG - Numerical Methods for Matrix Computations
  • MATH382 CG - Perturbation and Asymptotic Analysis
  • Credits: 1.5

    Instructors: Adolph (Biology), de Pillis (Mathematics), Jacobsen (Mathematics)

    Offered: Spring

    Description: An introduction to the field of mathematical 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and BIOL052 HM 

and

  • Credits: 1.5

    Instructors: Bush (Biology), Donaldson-Matasci (Biology), Libeskind-Hadas (Computer Science), Wu (Computer Science)

    Offered: Spring

    Description: An introduction to the field of computational biology. Algorithms for phylogenetic inference and computational methods for solving problems in molecular evolution and population genetics.

    Prerequisites: (CSCI005 HM or CSCI005GR HM) and BIOL052 HM 

Theoretical Computer Science

  • Credits: 3

    Instructors: Keller, Stone

    Offered: Fall and Spring

    Description: An introduction to some of the mathematical foundations of computer science, particularly logic, automata, and computability theory. Develops skill in constructing and writing proofs, and demonstrates the applications of the aforementioned areas to problems of practical significance.

    Prerequisites: MATH055 HM and (CSCI060 HM or CSCI042 HM)

  • Credits: 3

    Instructors: Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science)

    Offered: Fall and Spring

    Description: 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: (CSCI070 HM and CSCI081 HM) or ((CSCI060 HM or CSCI042 HM) and MATH131 HM))

and at least one elective from:

  • Credits: 3

    Instructor: Boerkoel

    Offered: Fall and Spring

    Description: This course presents a general introduction to the field of Artificial Intelligence. It examines the question: What does (will) it take for computers to perform human tasks? It presents a broad introduction to topics such as knowledge representation, search, learning and reasoning under uncertainty. For each topic, it examines real-world applications of core techniques to problems which may include game playing, text classification and visual pattern recognition.

    Prerequisites: MATH035 HM and CSCI070 HM 

  • Credits: 3

    Instructor: Keller

    Offered: Fall

    Description: Modeling, simulation, and analysis of artificial neural networks and their relation to biological networks. Design and optimization of discrete and continuous neural networks. Back propagation and other gradient descent methods. Hopfield and Boltzmann networks. Unsupervised learning. Self-organizing feature maps. Applications chosen from function approximation, signal processing, control, computer graphics, pattern recognition, time-series analysis. Relationship to fuzzy logic, genetic algorithms, and artificial life.

    Prerequisites: (CSCI060 HM or CSCI042 HM) and MATH065 HM 

  • Credits: 3

    Instructor: Keller

    Offered: Spring, alternate years

    Description: Characteristics and applications for parallel and real-time systems. Specification techniques, algorithms, architectures, languages, design, and implementation.

    Prerequisites: CSCI105 HM and CSCI140 HM; CSCI131 HM recommended

  • Credits: 3

    Instructors: Martonosi, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: An introduction to graph theory with applications. Theory and applications of trees, matchings, graph coloring, planarity, graph algorithms, and other topics.

    Prerequisites: (MATH040 HM or MATH073 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Benjamin, Omar, Orrison, Pippenger

    Offered: Alternate years

    Description: 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: MATH055 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, de Pillis, Pippenger, Yong

    Offered: Fall

    Description: 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: MATH065 HM or (MATH073 HM and MATH082 HM)

  • Credits: 3

    Instructors: Pippenger, Libeskind-Hadas (Computer Science), Staff (Pomona)

    Offered: Fall

    Description: Brief review of computability theory, followed by a rigorous treatment of complexity theory. The complexity classes P, NP, and the Cook-Levin Theorem. 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.

    Prerequisites: (CSCI060 HM or CSCI042 HM) and MATH055 HM 

  • Credits: 3

    Instructors: Karp, Omar, Orrison, Su, Staff (Pomona)

    Offered: Jointly; Spring semester at HMC and Pomona

    Description: 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: MATH171 HM 

  • Credits: 3

    Instructors: Benjamin, Omar, Pippenger, Staff (Scripps)

    Offered: Spring; offered jointly Fall semester at Scripps

    Description: Properties of integers, congruences, Diophantine problems, quadratic reciprocity, number theoretic functions, primes.

    Prerequisites: MATH055 HM 

Mathematical Biology

  • Credits: 1.5

    Instructors: Adolph (Biology), de Pillis (Mathematics), Jacobsen (Mathematics)

    Offered: Spring

    Description: An introduction to the field of mathematical 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and BIOL052 HM 

and

  • Credits: 1.5

    Instructors: Bush (Biology), Donaldson-Matasci (Biology), Libeskind-Hadas (Computer Science), Wu (Computer Science)

    Offered: Spring

    Description: An introduction to the field of computational biology. Algorithms for phylogenetic inference and computational methods for solving problems in molecular evolution and population genetics.

    Prerequisites: (CSCI005 HM or CSCI005GR HM) and BIOL052 HM 

  • Credits: 3

    Instructors: de Pillis, Jacobsen, Adolph (Biology)

    Description: 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: MCBI118A HM and MCBI118B HM 

and at least one elective from:

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona), Staff (CMC)

    Offered: Jointly; Spring semester at Pomona and CMC

    Description: An introduction to the general theory of statistical inference, including estimation of parameters, confidence intervals, and tests of hypotheses.

    Prerequisites: MATH 157 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC)

    Offered: Jointly; Fall, alternate years at HMC

    Description: 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: (MATH040 HM or MATH073 HM) and MATH157 HM 

  • Credits: 3

    Instructors: Martonosi, Williams, Staff (Pomona)

    Offered: Fall, alternate years

    Description: 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: MATH035 HM 

  • Credits: 3

    Instructors: Bernoff, de Pillis, Yong

    Offered: Spring

    Description: 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: (MATH065 HM or (MATH073 HM and MATH082 HM)) and CSCI060 HM 

  • Credits: 3

    Instructors: Pippenger, Sweedyk (Computer Science), Libeskind-Hadas (Computer Science)

    Offered: Fall and Spring

    Description: 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: (CSCI070 HM and CSCI081 HM) or ((CSCI060 HM or CSCI042 HM) and MATH131 HM))

  • Credits: 3

    Instructors: de Pillis, Gu, Orrison

    Offered: Jointly in alternate years

    Description: 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: MATH131 HM 

  • Credits: 3

    Instructors: Bernoff, de Pillis, Jacobsen, Staff (Pomona)

    Offered: Jointly; Fall semester at Pomona, Spring semester at HMC in alternate years

    Description: 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: MATH115 HM or MATH180 HM 

  • Credits: 3

    Instructors: Bernoff, Castro, Jacobsen

    Offered: Spring, alternate years

    Description: 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: (MATH115 HM and MATH131 HM) or MATH180 HM; recommended MATH132 HM 

  • Credits: 3

    Instructors: Benjamin, Martonosi, Staff (CMC), Staff (Pomona)

    Offered: Fall

    Description: Linear, integer, non-linear and dynamic programming, classical optimization problems, and network theory.

    Prerequisites: MATH040 HM or MATH073 HM 

Comments

Through the Major Core requirement, every major will have a foundation course in several important areas: discrete mathematics, analysis, algebra, differential equations and probability. In addition, every major will have a course relating to computational aspects of mathematics. The Major Core positions each student to move in any of several directions in the design of their elective program. There is a wide range of options to finish the major, supporting a variety of career goals and interests. It is expected that most students will take MATH055 HM – Discrete Mathematics and MATH131 HM – Mathematical Analysis I by the end of the sophomore year, MATH157 HM – Intermediate Probability, MATH171 HM – Abstract Algebra I, MATH180 HM – Introduction to Partial Differential Equations, MATH198 HM – Undergraduate Mathematics Forum, and MATH199 HM – Mathematics Colloquium by the end of the junior year, and MATH193 HM – Mathematics Clinic or MATH197 HM – Senior Thesis in Mathematics during the senior year.

Two semesters of Clinic or thesis are required of each major. All students must declare their intentions by the end of their junior year. Students who wish to take Clinic should inform the Mathematics Clinic Director, and preregister for MATH193 HM – Mathematics Clinic. Students choosing thesis must arrange to have a thesis advisor by the end of the spring semester of their junior year. In consultation with their advisor, the student must prepare a research proposal describing a suitable thesis problem, and submit the proposal to the mathematics department for approval. We expect that students will begin work on their theses immediately in the fall of the senior year. Thesis students will meet weekly as a group, to discuss their progress, make presentations, and exchange ideas. Students enrolled in Clinic who also wish to do thesis will be able to do a one-semester thesis, if desired. They may arrange their thesis in the fall of the senior year.

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