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Our commitment to inclusivity

Dartmouth’s capacity to advance its dual mission of education and research depends upon the full diversity and inclusivity of this community. We must increase diversity among our faculty, students, and staff. As we do so, we must also create a community in which every individual, regardless of gender, gender identity, sexual orientation, race, ethnicity, socio-economic status, disability, nationality, political or religious views, or position within the institution, is respected. On this close-knit and intimate campus, we must ensure that every person knows that they are a valued member of our community.

Information for Majors

*From the ORC*

The major in mathematics is intended both for students who plan careers in mathematics and related fields, and for those who simply find mathematics interesting and wish to continue its study. The content of the major is quite flexible, and courses may be selected largely to reflect student interests. Students who major in mathematics have an opportunity to participate in activities that bring them in close contact with a faculty member—for example, through a small seminar or through an independent research project under the direction of a faculty member. In addition to regular course offerings, a student with specialized interests, not reflected in our current course offerings, often arranges for an independent reading course. Proposals for independent activities should be directed to the Departmental Advisor to Mathematics Majors.

In general, the mathematics major requires the student to pass eight mathematics or computer science courses beyond prerequisites. At least six of the required eight courses must be mathematics, and at least four of these courses must be taken at Dartmouth. In addition, a student must fulfill the College’s requirement for a culminating experience in the major (see below). Additional requirements for honors are described below in a separate section.

Students are encouraged to take MATH 22/ MATH 24 as soon as feasible, since not only is it an explicit prerequisite to many upper-division courses, but also the level of mathematical sophistication developed in MATH 22/ MATH 24 will be presumed in many upper-division courses for which MATH 22/ MATH 24 is not an explicit prerequisite.

*From the ORC*

Statistics has become a ubiquitous tool not only in traditional areas in the natural and social sciences, but in emerging cross-disciplinary fields in data science. The major combines a solid theoretical foundation with application to one or more fields of study.

Students interested in this major should refer to the ORC for details on prerequisites and specific requirements for the major.

Culminating Experience: Majors in Mathematical Data Science must complete a data intensive research project to satisfy the culminating experience. Students may complete this by writing a thesis, completing an independent research project, or completing a course with a significant statistical project. The culminating experience must be approved in advance by the Adviser to Majors.

**Applied math roadmap** — See the vista of applied courses at
Dartmouth and their interdependencies.

**Pure math flowchart** — The (mostly) pure mathematics flowchart.

**Course dependencies** — Dependency graph combining both pure and applied courses.

**Who advises
majors?** — For consistency we have one faculty
member who advises all majors.

**What are the
requirements?** — What are my options in becoming a
major? How much choice in courses can I exercise? Are there
specific recommendations for graduate school, applied
mathematics, teaching careers?

**Research
Opportunities** — Want to work with a professor on an
ongoing research project? Check out the possibilities.

**Jobs** -
Lots of links.

Instructor: The Staff, The Staff, The Staff, The Staff

This course is an introduction to single variable calculus for students who have not taken calculus before. Students who have seen some calculus, but not enough to place out of MATH 3, should take MATH 3. MATH 1 reviews relevant techniques from algebra and pre-calculus, covers the manipulation and analysis of functions, including polynomial, trigonometric, logarithmic, and exponential functions, an introduction to convergence and limits, continuity, rates of change and derivatives, differentiation rules, and applications to approximation. Students wishing to continue their study of calculus after MATH 1 take MATH 3.

Distributive: QDS

Offered: 20F: 10, 11, 12, 2 21F: Arrange

Instructor: Ma, Ma, Lafreniere, Kulkarni, Lin (fall), Lin, Kulkarni, The Staff, The Staff (winter)

This course is an introduction to single variable calculus aimed at students who have seen some calculus before, either before matriculation or in MATH 1. MATH 3 begins by revisiting the core topics in MATH 1 - convergence, limits, and derivatives - in greater depth before moving to applications of differentiation such as related rates, finding extreme values, and optimization. The course then turns to integration theory, introducing the integral via Riemann sums, the fundamental theorem of calculus, and basic techniques of integration.

Distributive: QDS

Offered: 20F: 9L, 10, 11, 12, 2, 21W: 9L, 10, 11, 2, 21F, 21W: Arrange

Instructor: Wallace (Spring)

This course will establish the relevance of calculus to medicine. It will develop mathematical tools extending the techniques of introductory calculus, including some matrix algebra and solution techniques for first order differential equations. These methods will be used to construct simple and elegant models of phenomena such as the mutation of HIV, spread of infectious disease, and biological disposition of drugs and inorganic toxins, enzyme kinetics and population growth.

Prerequisite: MATH 3. Note: This is a sequel to MATH 3, but it does not cover the same material as MATH 8, and does not serve as a prerequisite for MATH 13. There is a version of this course suitable for major credit: see MATH 27.

Distributive: QDS

Offered: 21S: 10A, 22S: Arrange

Language is the medium for politics and political conflict. Candidates debate during elections. Representatives write laws. Nations negotiate peace treaties. Clerics issue Fatwas. Citizens express their opinions about politics on social media sites. These examples, and many others, suggest that to understand what politics is about, we need to know what political actors are saying and writing. This course introduces techniques to collect, analyze, and utilize large collections of text for social science inferences. Students will also have the opportunity to develop their programming abilities.

We will explore a range of datasets from the text of The Federalist Papers to the millions of tweets sent to and from members of Congress.

Prerequisite: GOVT/ECON/PSYC/SOCY/MATH 10 or QSS 15 or COSC 1

Cross-Listed as: GOVT 19.05 QSS 30.02

Distributive: TLA

Offered: Not offered in AY 2020-2021. May be offered in AY 2021-2022.

Instructor: McPeek, Rockmore

Mathematics is the language of science. However, mathematics preparation for most science students typically involves only the study of calculus at the university level. While many scientific problems involve calculus, two other areas of mathematics are equally (if not more) important: linear algebra and probability. For example, linear algebra is fundamental to stoichiometry and the conservation of matter in chemistry, hydrology and atmospheric dynamics in earth sciences, and cell growth and population dynamics in biology. Moreover, most features of the natural world are probabilistic and frequently best described by probability models, such as the firing of neurons in the brain or the timing of earthquakes. Both are also central to all problems in statistics. This course will explore the application of linear algebra and probability to problems across the sciences. We will cover the basics of solving linear algebra and probability problems as well as formulating simple models to describe and analyze natural phenomena from across the sciences.

Cross-Listed as: BIOL 5

Distributive: QDS

Offered: Not offered in AY 2020-2021. May be offered in AY 2021-2022.

Offered: Consult special listing

Instructor: Petkova, Petkova, Petok (fall), Lin, Wang, Petok (winter), Allen, Allen (spring)

This course is a sequel to MATH 3 and is appropriate for students who have successfully completed an AB calculus curriculum (or the equivalent) in secondary school. Roughly half of the course is devoted to topics in one-variable calculus, selected from techniques of integrations, areas, volumes, numerical integration, sequences and series including Taylor series, ordinary differential equations and techniques of their solution. The second half of the course studies scalar valued functions of several variables. It begins with the study of vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength). The balance of the course is devoted to studying differential calculus of functions of several variables. Topics include limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems including the use of Lagrange multipliers.

Prerequisite: MATH 3 or equivalent.

Distributive: QDS

Offered: 20F: 10, 11, 2, 21W: 10, 11, 2 21S: 10, 11, 21F, 22W, 22S: Arrange

Instructor: Schembri, Kaveh

This course includes the multivariable calculus material present in MATH 8 along with a brief introduction to concepts from linear algebra, a topic pervasive throughout mathematics and its applications. The introduction to linear algebra enables a more thorough understanding of multivariable calculus. Topics include vector geometry, equations of lines and planes, matrices and linear transformations, space curves (velocity, acceleration, arclength), functions of several variables (limits and continuity, partial derivatives, the derivative as a linear transformation, tangent planes and linear approximation, the Chain Rule, directional derivatives and applications, and optimization problems including the use of Lagrange multipliers).

First-year students who have successfully completed a BC calculus curriculum in secondary school may complete multivariable calculus either by taking the two-term sequence MATH 9, 13 or by taking the single faster-paced course MATH 11, which covers the second half of Math 8 together with the material from Math 13 in a single term.

Prerequisite: Advanced placement into MATH 9 or MATH 11.

Distributive: QDS

Offered: 20F: 9L, 10 21F: Arrange

Instructor: Zhou, Kaveh, Wong

An introduction to the basic concepts of statistics. Topics include elementary probability theory, descriptive statistics, the binomial and normal distributions, confidence intervals, basic concepts of tests of hypotheses, chi-square tests, nonparametric tests, normal theory t-tests, correlation, and simple regression. Packaged statistical programs will be used. Because of the large overlap in material covered, no student may receive credit for more than one of the courses ECON 10, ENVS 10, GOVT 10, MATH 10, PSYC 10, QSS 15, and SOCY 10, except by special petition to the Committee on Instruction.

Distributive: QDS

Offered: 21S: 10, 11, 2, 22S: Arrange

Instructor: Auel, van Wyk, van Wyk, Wong

This briskly paced course can be viewed as equivalent to MATH 13 in terms of prerequisites, but is designed especially for first-year students who have successfully completed a BC calculus curriculum in secondary school. In particular, as part of its syllabus it includes most of the multivariable calculus material present in MATH 8 together with the material from MATH 13. Topics include vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength), limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems. It continues with multiple integration, vector fields, line integrals, and finishes with a study of Green's and Stokes' theorem.

Students who have successfully completed a BC calculus curriculum in secondary school may complete multivariable calculus either by taking the two term sequence MATH 9 and MATH 13 or by completing the single, faster-paced, MATH 11.

Distributive: QDS

Offered: 20F: 10, 11, 12, 2, 21F: Arrange

Instructor: The Staff (fall), Lord, Lord, Wong, Wong (winter), Webb, Tassy, Tassy (spring)

This course is a sequel to MATH 8 and provides an introduction to calculus of vector-valued functions. Topics include differentiation and integration of parametrically defined functions with interpretations of velocity, acceleration, arclength and curvature. Other topics include iterated, double, triple and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green's theorem, curl and divergence, and Stokes' theorem.

Prerequisite: MATH 8 or Math 9 or equivalent. Note: First-year students who have received two terms of credit from the AP-BC exam generally should take MATH 11 instead. On the other hand, if the student has had substantial exposure to multivariable techniques, they are encouraged to consult with the Frist-Year Advisor for Mathematics during orientation week to determine if placement into MATH 13 is more appropriate.

Distributive: QDS

Offered: 20F: 11, 21W: 10, 11, 12, 2, 21S: 10, 11, 12, 21F, 22W, 22S: Arrange

Instructor: Assaf (winter), Doyle (spring)

Gives prospective Mathematics majors an early opportunity to delve into topics outside the standard calculus sequence. Specific topics will vary from term to term, according to the interests and expertise of the instructor. Designed to be accessible to bright and curious students who have mastered BC Calculus, or its equivalent. This course counts toward the Mathematics major, and is open to all students, but enrollment may be limited, with preference given to first-year students.

Prerequisite: MATH 8

Distributive: QDS

Offered: 21W: 2, 21S, 11, 22S: Arrange

Instructor: Groszek

This course introduces the axioms of set theory, the universe of sets, and set theory as a foundation for mathematics. It touches on historical and philosophical aspects of set theory. Mathematical topics covered include the algebra of sets, ordinals and cardinals, trans\ffinite induction and recursion, and the axiom of choice. Students will learn language and concepts used throughout mathematics, and learn how to write mathematical proofs.

Distributive: QDS

Offered: 21S: 12, 22S: Arrange

Instructor: Zhou, Tassy (fall), The Staff (spring)

Our capacity to fathom the world around us hinges on our ability to understand quantities which are inherently unpredictable. Therefore, in order to gain more accurate mathematical models of the natural world we must incorporate probability into the mix. This course will serve as an introductions to the foundations of probability theory. Topics covered will include some of the following: (discrete and continuous)random variable, random vectors, multivariate distributions, expectations; independence, conditioning, conditional distributions and expectations; strong law of large numbers and the central limit theorem; random walks and Markov chains. There is an honors version of this course: see MATH 60.

Prerequisite: MATH 8.

Distributive: QDS

Offered: 20F: 11, 10A, 21S: 11, 21X, 21F, 22S: Arrange

Instructor: Petok, Williams, Zhou (fall), Orellana, Orellana, Petok, The Staff, Wang (spring)

This course presents the fundamental concepts and applications of linear algebra with emphasis on Euclidean space. Significant goals of the course are that the student develop the ability to perform meaningful computations and to write accurate proofs. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. Applications may be drawn from areas such as optimization, statistics, biology, physics, and signal processing. Students who plan to take either MATH 63 or MATH 71 are strongly encouraged to take MATH 24.

Distributive: QDS

Offered: 20F: 10, 11, 2, 21S: 10, 11, 12, 12, 2, 21X, 21F, 22S: Arrange

Instructor: Gelb, Glaubitz (fall), Ma, The Staff, (winter), Lin, Chernov (spring)

This course is a survey of important types of differential equations, both linear and non-linear. Topics include the study of systems of ordinary differential equations using eigenvectors and eigenvalues, numerical solutions of first and second order equations and of systems, and the solution of elementary partial differential equations using Fourier series.

Prerequisite: MATH 13.

Distributive: QDS

Offered: 20F: 9L, 2, 21W: 11, 12, 21S: 11, 12, 21F, 22W, 22S: Arrange

Instructor: Wang (winter), Trout (spring)

This course is an introduction to the fundamental concepts of linear algebra in abstract vector spaces. The topics and goals of this course are similar to those of MATH 22, but with an additional emphasis on mathematical abstraction and theory. (MATH 24 can be substituted for MATH 22 as a prerequisite for any course or program.)

Prerequisite: MATH 8.

Distributive: QDS

Offered: 21W: 10, 21S: 2, 22W, 22S: Arrange

Instructor: Voight

The great mathematician C. F. Gauss once wrote "Mathematics is the queen of sciences and number theory is the queen of mathematics." Number theory is that part of mathematics dealing with the integers and certain natural generalizations. Topics include modular arithmetic, unique factorization into primes, linear Diophantine equations, and Fermat's Little Theorem. Discretionary topics may include cryptography, primality testing, partition functions, multiplicative functions, the law of quadratic reciprocity, historically interesting problems.

Prerequisite: MATH 8.

Distributive: QDS

Offered: 20F: 12, 21F: Arrange

Instructor: Shepherd

Prerequisite: COSC 1 and COSC 10, or ENGS 20; ENGS 22 or MATH 23, or equivalent.

Cross-Listed as: ENGS 91, COSC 71

Distributive: QDS

Offered: 20F: 12, 21F: Arrange

Instructor: Wallace

This course will prepare students to read the technical literature in mathematical biology, epidemiology, pharmacokinetics, ecological modeling and related areas. Topics include systems of nonlinear ordinary differential equations, equilibria and steady state solutions, phase portraits, bifurcation diagrams, and some aspects of stability analysis. Emphasis is placed on the student's ability to analyze phenomena and create mathematical models. This interdisciplinary course is open to mathematics majors, biology majors, and students preparing for a career in medicine.

Prerequisite: MATH 22. Note: Students without the mathematical prerequisites can take this course as MATH 4: no student may take both MATH 4 and MATH 27 for credit, and only MATH 27 is eligible to count towards the major in mathematics.

Distributive: QDS

Offered: 21S: 10A, 22S: Arrange

Instructor: Orellana

Beginning with techniques for counting-permutations and combinations, inclusion-exclusion, recursions, and generating functions-the course then takes up graphs and directed graphs and ordered sets, and concludes with some examples of maximum-minimum problems of finite sets. Topics in the course have application in the areas of probability, statistics, and computing.

Prerequisite: MATH 8.

Distributive: QDS

Offered: 21W: 11, 22W: Arrange

What does it mean for a function to be computable? This course examines several different mathematical formalizations of the notion of computability, inspired by widely varying viewpoints, and establishes the surprising result that all these formalizations are equivalent. It goes on to demonstrate the existence of noncomputable sets and functions, and to make connections to undecidable problems in other areas of mathematics. The course concludes with an introduction to relative computability. This is a good companion course to COSC 39; the two share only the introduction of Turing machines. Offered in alternate years.

Prerequisite: None, but the student must be willing to learn to work abstractly and to read and write proofs.

Distributive: QDS

Offered: 22S: Arrange

Instructor: Fu

Pioneered by John Maynard Smith and others, evolutionary game theory has become an important approach to studying a wide range of biological and social problems, such as microbial interactions and animal behavior. In evolutionary game dynamics, the fitness of individuals depends on the relative abundance of all individual types in the population, and higher-fitness individual types tend to increase in abundance. This course introduces basic concepts in evolutionary game theory, including evolutionarily stable strategies, replicator dynamics, finite populations, and games on networks, along with applications to social evolution, particularly to understanding human cooperation.

Prerequisite: Math 3 and Math 20

Cross-Listed as: QSS 30.04

Distributive: QDS

Offered: 21S: 12

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography. As a result of the variable syllabus, this course may not serve as an adequate prerequisite for MATH 81. Students who contemplate taking MATH 81 should consider taking MATH 71 instead of this course.

Distributive: QDS

Instructor: Allen

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography. As a result of the variable syllabus, this course may not serve as an adequate prerequisite for MATH 81. Students who contemplate taking MATH 81 should consider taking MATH 71 instead of this course.

Prerequisite: MATH 22.

Distributive: QDS

Offered: 20F: 11, 21X, 21F: Arrange

Topics in intuitive geometry and topology, for example: how to turn a sphere inside out; knots, links, and their invariants; polyhedra In 2, 3, and 4 dimensions; the classification of surfaces; curvature and the Gauss-Bonnet theorem; spherical and hyperbolic geometry; Escher patterns and their quotients; the shape of the universe. Offered in alternate years.

Prerequisite: MATH 22 or MATH 24.

Distributive: QDS

Offered: 22W: Arrange

Instructor: Trout

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Some applications of the theory may be presented. MATH 63 presents similar material, but from a more sophisticated point of view. This course may not serve as an adequate prerequisite for either MATH 73 or 83. Students who contemplate taking one of these two advanced courses should consider taking MATH 63 instead of this course.

Prerequisite: MATH 13 and permission of the instructor, or MATH 22.

Distributive: QDS

Offered: 21W: 11, 22W: Arrange

Instructor: Lord

Disciplines such as anthropology, economics, sociology, psychology, and linguistics all now make extensive use of mathematical models, using the tools of calculus, probability, game theory, network theory, often mixed with a healthy dose of computing. This course introduces students to a range of techniques using current and relevant examples. Students interested in further study of these and related topics are referred to the courses listed in the Mathematics and Social Sciences program.

Prerequisite: MATH 13, MATH 20.

Cross-Listed as: QSS 36

Distributive: TAS

Offered: 20F: 2A, 21F: Arrange

Introduction into cellular automata and agent-based modeling using the Java programming language. Focus of this course will be simulation of stochastic events, model parameterization and calibration, model validation, simulation and result visualization. This is a hands-on course with laboratory sessions and training exercises on individual computers.

Prerequisite: MATH 22 and MATH 24, MATH 23, one of COSC 1, COSC 10, ENGS 20 or equivalent experience.

Distributive: QDS

Offered: W22: OCP

Instructor: Lafreniere

The theory of graphs has roots in both practical and recreational mathematics. Today there are major applications of graph theory in management science (operations research) and computer science. This course is a survey of the theory and applications of graphs. Topics will be chosen from among connectivity, trees, and Hamiltonian and Eulerian paths and cycles; isomorphism and reconstructability; planarity, duality, and genus; independence and coloring problems, including interval graphs, interval orderings and perfect graphs, color-critical graphs and the four-color theorem; matchings; network flows, including applications to matchings, higher connectivity, and transportation problems; matroids and their relationship with optimization.

Prerequisite: MATH 22 (or COSC 55 and permission of the instructor).

Distributive: QDS

Offered: 21S: 12, 22S: Arrange

Instructor: Lee

Introduction to continuous probability and statistical inference for data analysis. Includes the theory of estimation and the theory of hypothesis testing using normal theory t-tests and nonparametric tests for means and medians, tests for variances, chi-square tests, and an introduction to the theory of the analysis of variance and regression analysis. Analysis of explicit data sets and computation are an important part of this hands-on statistics course. *NOTE: Prior to Fall 2014 Math 40 was numbered Math 50.

Prerequisite: MATH 13 and MATH 20, or permission of the instructor.

Distributive: QDS

Offered: 21W: 12, 22W: Arrange

This course will cover curves and surfaces in Euclidean 3-dimensional space. Topics include curvature and torsion of curves, the Frenet-Serret equations, Gaussian and mean curvature of surfaces, geodesics and parallel transport, isometries and Gauss's Theorem Egregium, the Riemann Curvature tensor. One or more of the following topics will be studied if time permits: vector fields, tangent bundles, hypersurfaces, connections, and curvature. Offered in alternate years.

Prerequisite: MATH 22 or permission of the instructor, and MATH 23.

Distributive: QDS

Offered: 22W:

Instructor: van Wyk

This course covers the differential and integral calculus of complex variables including such topics as Cauchy's theorem, Cauchy's integral formula and their consequences; singularities, Laurent's theorem, and the residue calculus; harmonic functions and conformal mapping. Applications will include two-dimensional potential theory, fluid flow, and aspects of Fourier analysis.

Prerequisite: MATH 13.

Distributive: QDS

Offered: 21S: 2, 22S: Arrange

Instructor: Glaubitz

This course introduces a wide variety of mathematical tools and methods used to analyze phenomena in the physical, life, and social sciences. This is an introductory course and is accessible to undergraduate and graduate students in mathematics and other scientific disciplines who have completed the prerequisites. Topics include dimensional analysis and scaling, perturbation analysis, calculus of variations, integral equations, and eigenvalue problems.

Prerequisite: MATH 22 and MATH 23, or permission of the instructor.

Distributive: TAS

Offered: 21S: 2A, 22S: Arrange

Introduction into cancer biology and basic mathematical approaches to simulate cancer dynamics on the subcellular, cellular, and tissue level. Techniques for quantitative modeling are plentiful, and an increasing number of theoretical approaches are successfully applied to cancer biology. Differential equation models and individual-based cell models paved the way into quantitative cancer biology about two decades ago. Herein we will give an introduction on how such models are derived and how they can be utilized to simulate tumor growth and treatment response. We will then discuss a number of different models and discuss their confirmative and predictive power for cancer biology.

Prerequisite: MATH 22 and MATH 24, MATH 23, one of COSC 1, COSC 10, ENGS 20 or equivalent experience.

Distributive: QDS

Offered: W22: OCP

This course provides an introduction to the most common model used in statistical data analysis. Simple linear regression, multiple regression, and analysis of variance are covered, as well as statistical model-building strategies. Regression diagnostics, analysis of complex data sets and scientific writing skills are emphasized. Methods are illustrated with data sets drawn from the health, biological, and social sciences. Computations require the use of a statistical software package such as STATA. Offered in alternate years.

Prerequisite: MATH 10, another elementary statistics course, or permission of the instructor.

Distributive: TAS

Offered: 21F: Arrange

Instructor: Lee

Partial differential equations play critical roles in wide areas of mathematics, science, and engineering. This is an introductory course, accessible to undergraduate and graduate students in mathematics and other scientific disciplines who have completed the prerequisites. Examples will come from both linear and non-linear partial differential equations, including the wave equation, diffusion, boundary value problems, conservation laws, and the Monge-Ampere equations. Offered in alternate years.

Prerequisite: MATH 22 and MATH 23, or permission of the instructor.

Distributive: QDS

Offered: 20F: 10

This course begins with the definitions of topological space, open sets, closed sets, neighborhoods, bases and subbases, closure operator, continuous functions, and homeomorphisms. The course will study constructions of spaces including subspaces, product spaces, and quotient spaces. Special categories of spaces and their interrelations will be covered, including the categories defined by the various separation axioms, first and second countable spaces, compact spaces, and connected spaces. Subspaces of Euclidean and general metric spaces will be among the examples studied in some detail.

Prerequisite: MATH 13 and MATH 22.

Distributive: QDS

Offered: 21X: Arrange

This course introduces computational algorithms solving problems from a variety of scientific disciplines. Mathematical models describing a phenomenon of interest are typically too complex to construct analytical solutions, leading us to numerical methods. Motivated by models from physics, biology, and medicine, students will develop numerical algorithms and mathematically analyze their accuracy, efficiency, and convergence properties. The course will provide external coding resources as students will implement algorithms in MATLAB. Sample topics include matrix decompositions, inverse problems, optimization, data fitting, and differential equations.

Prerequisite: MATH 22 or MATH 24, COSC 1 or ENGS 20, or permission of the instructor.

Distributive: QDS

Offered: 22S: Arrange

This course is a more theoretical introduction to probability theory than MATH 20. In addition to the basic content of MATH 20, the course will include other topics such as continuous probability distributions and their applications. Offered in alternate years.

Prerequisite: MATH 13, or permission of the instructor.

Distributive: QDS

Offered: 22S: Arrange

Instructor: Williams

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Students may not take both MATH 35 and 63 for credit.

Prerequisite: MATH 22 or MATH 24, or MATH 13 and permission of the instructor.

Distributive: QDS

Offered: 21W: 11, 22W: Arrange

This introductory course presents mathematical topics that are relevant to issues in modern physics. It is mainly designed for two audiences: mathematics majors who would like to see modern physics and the historical motivations for theory in their coursework, and physics majors who want to learn mathematics beyond linear algebra and calculus. Possible topics include (but are not limited to) introductory Hilbert space theory, quantum logics, quantum computing, symplectic geometry, Einstein's theory of special relativity, Lie groups in quantum field theory, etc. No background in physics is assumed. Offered in alternate years.

Prerequisite: Math 23 and MATH 24 (or MATH 22 with permission of the instructor).

Distributive: QDS

Offered: 22S: Arrange

This course covers the use of abstract algebra in studying the existence, construction, enumeration, and classification of combinatorial structures. The theory of enumeration, including both Polya Theory and the Incidence Algebra, and culminating in a study of algebras of generating functions, will be a central theme in the course. Other topics that may be included if time permits are the construction of block designs, error-correcting codes, lattice theory, the combinatorial theory of the symmetric group, and incidence matrices of combinatorial structures. Offered in alternate years.

Prerequisite: MATH 28 and MATH 31, or MATH 71, or permission of the instructor.

Distributive: QDS

Offered: 21F: Arrange

Instructor: Groszek

This course begins with a study of relational systems as they occur in mathematics. First-order languages suitable for formalizing such systems are treated in detail, and several important theorems about such languages, including the compactness and Lowenheim-Skolem theorems, are studied. The implications of these theorems for the mathematical theories being formulated are assessed. Emphasis is placed on those problems relating to first-order languages that are of fundamental interest in logic. Offered in alternate years.

Prerequisite: experience with mathematical structures and proofs, as offered by such courses as MATH 71, MATH 54, or MATH 24; or permission of the instructor.

Distributive: QDS

Offered: 21W: 12

Instructor: Demidenko

This course focuses on modern methods of statistical analysis including nonlinear models, data mining, and classification. Students gain a theoretical basis for multivariate statistical analysis, optimal statistical hypothesis testing, and point and interval estimation. The course is grounded in applications and students will gain experience in solving problems in data analysis. Students are required to use the statistical package R.

Prerequisite: MATH 40

Distributive: QDS

Offered: 21S: 12 22S: Arrange

Instructor: Webb

The sequence MATH 71 and 81 is intended as an introduction to abstract algebra. MATH 71 develops basic theorems on groups, rings, fields, and vector spaces.

Prerequisite: MATH 22 or MATH 24.

Distributive: QDS

Offered: 20F: 10, 21F: Arrange

Instructor: Sutton (fall) Webb (spring)

This course develops one or more topics in geometry. Possible topics include hyperbolic geometry; Riemannian geometry; the geometry of special and general relativity; Lie groups and algebras; algebraic geometry; projective geometry. Offered in alternate years.

Prerequisite: MATH 71, or permission of the instructor. Depending on the specific topics covered, MATH 31 may not be an acceptable prerequisite; however, in consultation with the instructor, MATH 31 together with some outside reading should be adequate preparation for the course.

Distributive: QDS

Offered: 20F: 11, 21S: 11

Instructor: Williams

This course is an introduction to graduate level analysis. Divided roughly in half, the first part of the course covers abstract measure theory. The second half of the course covers complex analysis.

Prerequisite: MATH 43 and MATH 63 or a basic course in real analysis and an undergarduate complex analysis course or permission of the instructor.

Distributive: QDS

Offered: 20F: 2 21F: Arrange

Instructor: Chernov

This course provides a foundation in algebraic topology, including both homotopy theory and homology theory. Topics may include: the fundamental group, covering spaces, calculation of the fundamental group, singular homology theory, Eilenberg-Steenrod axioms, Mayer-Vietoris sequence, computations, applications to fixed points and vector fields.

Prerequisite: MATH 31/ MATH 71 and MATH 54 and permission of the instructor or MATH 54 and MATH 101.

Distributive: QDS

Offered: 21S: 11, 21S: Arrange

Provides some applications of number theory and algebra. Specific topics will vary; two possibilities are cryptology and coding theory. The former allows for secure communication and authentication on the Internet, while the latter allows for efficient and error-free electronic communication over noisy channels. Students may take Math 75 for credit more than once. Offered in alternate years.

Distributive: QDS

Offered: 22S: Arrange

Provides some applications of number theory and algebra. Specific topics will vary; two possibilities are cryptology and coding theory. The former allows for secure communication and authentication on the Internet, while the latter allows for efficient and error-free electronic communication over noisy channels. Students may take Math 75 for credit more than once. Offered in alternate years.

Prerequisite: MATH 25 or MATH 22/ MATH 24 or MATH 31/ MATH 71, or permission of the instructor.

Distributive: QDS

Offered: 22S: Arrange

Instructor: Song

The numerical nature of twenty-first century society means that applied mathematics is everywhere: animation studios, search engines, hedge funds and derivatives markets, and drug design. Students will gain an in-depth introduction to an advanced topic in applied mathematics. Possible subjects include digital signal and image processing, quantum chaos, computational biology, cryptography, coding theory, waves in nature, inverse problems, information theory, stochastic processes, machine learning, and mathematical finance.

Distributive: QDS

Offered: 21S: 10

Instructor: Song

The numerical nature of twenty-first century society means that applied mathematics is everywhere: animation studios, search engines, hedge funds and derivatives markets, and drug design. Students will gain an in-depth introduction to an advanced topic in applied mathematics. Possible subjects include digital signal and image processing, quantum chaos, computational biology, cryptography, coding theory, waves in nature, inverse problems, information theory, stochastic processes, machine learning, and mathematical finance.

Prerequisite: MATH 22, MATH 23, or permission of the instructor.

Distributive: QDS

Offered: 21S: 10, 22W: OCP

Instructor: Voight

This course provides a foundation in core areas in the theory of rings and fields. Specifically, it provides an introduction to commutative ring theory with a particular emphasis on polynomial rings and their applications to unique factorization and to finite and algebraic extensions of fields. The study of fields continues with an introduction to Galois Theory, including the fundamental theorem of Galois Theory and numerous applications.

Prerequisite: MATH 71. In general, MATH 31 is not an acceptable prerequisite; however, in consultation with the instructor, MATH 31 together with some outside reading should be adequate preparation for the course.

Cross-Listed as: MATH 111

Distributive: QDS

Offered: 21W: 10, 22W: Arrange

Instructor: van Erp

Financial derivatives can be thought of as insurance against uncertain future financial events. This course will take a mathematically rigorous approach to understanding the Black-Scholes-Merton model and its applications to pricing financial derivatives and risk management. Topics may include: arbitrage-free pricing, binomial tree models, Ito calculus, the Black-Scholes analysis, Monte Carlo simulation, pricing of equities options, and hedging.

Prerequisite: MATH 20 and MATH 40, or MATH 60; MATH 23; and COSC 1 or the equivalent.

Distributive: QDS

Offered: 21S: 11, W22: Arrange

Advanced undergraduates occasionally arrange with a faculty member a reading course in a subject not occurring in the regularly scheduled curriculum.

Offered: All terms: Arrange

A study of selected topics in logic, such as model theory, set theory, recursive function theory, or undecidability and incompleteness. Offered in alternate years.

Prerequisite: MATH 39 or MATH 69.

Distributive: QDS

Open only to students who are officially registered in the Honors Program. Permission of the adviser to majors and thesis adviser required. This course does not serve for major credit nor for distributive credit, and may be taken at most twice.

Offered: All terms: Arrange