Rigorous and intensive treatment of standard topics from algebra and trigonometry as a preparation for further courses in calculus or statistics. Fulfills the QR requirement. Fall Only.

# All Current Undergraduate courses

You will find below the list of undergraduate courses we offer or have offered.

##### MAT101 Calculus I with Precalculus Review

Last offered Fall 2011. The old MAT101/102 sequence is being replaced by either MAT100/102 or by MAT100/103 starting in the academic year 2012-13.

##### MAT102 Survey of Calculus

One semester survey of major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus such as approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisite: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT102 and MAT103.

##### MAT103 Calculus I

First semester of the standard 3-semester calculus sequence 103/104/201 for science, engineering and finance. Topics include limits, continuity, derivatives and their applications, introduction to the definite integral.
Emphasizes concrete computations over more theoretical considerations. Offered only in the Fall semester.

##### MAT104 Calculus II (One Variable, Continued from 103)

Second semester of the standard 3-semester calculus sequence 103/104/201 for science, engineering and finance. Topics include techniques and applications of integration, convergence of infinite series and improper integrals, Taylor's theorem, introduction to differential equations and complex numbers. Emphasizes concrete computations over more theoretical considerations. Offered both Fall and Spring. Prerequisite: MAT103 or equivalent.

##### MAT175 Calculus & Linear Algebra for Economics & Life Sciences

Replaces MAT200 beginning Fall 2012.
One semester survey of selected topics from linear algebra and calculus chosen with applications to economics and life sciences in mind to give (minimal) preparation for upper division quantitative courses. For students who do not intend to take further mathematics courses at Princeton. Offered Fall and Spring. Prerequisite: MAT103 or equivalent.

##### MAT189 Number, Shape and Symmetry

Calculus, while very important to scientists and engineers, is but one part of modern mathematics, and the technicality of the subject often obscures the underlying mathematical principles. In response, this course is meant to be an alternative to a first semester calculus course for non-scientists, requiring less mathematical background, but of similar depth. In particular, the course will assume only the standard material from high school algebra and geometry. Math 189 attempts to give students an understanding of what mathematics is, what mathematicians do, and the subject's history. The course emphasizes the understanding of ideas and the ability to express them through mathematical arguments. Offered every other Spring, if staffing permits.

##### MAT190 The Magic of Numbers

This course will explore some of the intriguing and beautiful mathematics that underlie the arts, technology, and everyday life. Developed and taught by Professor Manjul Bhargava.

##### MAT198 Useful Fictions: How and why mathematics is developed and then changes the world

This course is geared toward students with no prior university math experience. It aims to provide a view of mathematics as a living, growing, creative human endeavor that classifies as both a science and an art, to give a feeling for, and some mastery of, the mathematical way of thinking (including "doing mathematics") as well as an awareness of some of the many applications of mathematics in today's world. Active class participation is an essential component of the course, required along with participation in Professor Keith Devlin's (free) Stanford MOOC "Introduction to Mathematical Thinking."

##### MAT199 Math Alive

Mathematics has profoundly changed our world, from the way we communicate with each other and
listen to music, to banking and computers. This course is designed for those without college
mathematics who want to understand the mathematical concepts behind important modern applications.
The course consists of individual modules, each focusing on a particular application (e.g., digital
music, sending secure emails, and using statistics to explain, or hide, facts). The emphasis is on
ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set
requirements. Students will learn by doing simple examples.

##### MAT200 Linear Algebra and Multivariable Calculus for Economists

Last offered in Spring 2012. Replaced by MAT175, beginning Fall 2012.
One semester of multivariable mathematics for finance certificate or for math-track economics majors. Covers selected topics from linear algebra and multivariable calculus in order to give minimal preparation for upper division quantitative courses in economics. (Not sufficient preparation for 300-level math courses).

##### MAT201 Calculus III (Multivariable Calculus)

A continuation of MAT103/104, the third semester in the calculus sequence gives a thorough introduction to multivariable calculus. Topics include limits, continuity and differentiability in several variables, extrema, Lagrange multipliers, Taylor's theorem, multiple integrals, integration on curves and surfaces, Green's theorem, Stokes' theorem, divergence theorem. Emphasizes concrete computations over more theoretical considerations. Offered both Fall and Spring. Prerequisite: MAT104 or equivalent.

##### MAT202 Introduction to Linear Algebra

Linear Algebra, mostly in real n-space. Companion course to 201. Main topics are matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvalues, eigenvectors and their applications to quadratic forms and dynamical systems. Offered Fall and Spring.

##### MAT203 Advanced Multivariable Calculus

Advanced multivariable calculus. More theoretical treatment of limits, continuity, differentiation and integration for functions of several variables than that found in MAT201, but less theoretical than MAT218. A course for those with a strong mathematical background and interest. Recommended for physics majors. Offered Fall only.

##### MAT204 Advanced Linear Algebra with Applications

Advanced multivariable calculus. More theoretical treatment of vector spaces and matrices than that found in MAT202, but more concrete than that of MAT217. A course for those with a strong mathematical background and interest. Recommended for physics majors. Offered Spring only.

##### MAT214 Numbers, Equations, and Proofs

Rigorous, proof-based introduction to classical number theory, to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. There will be a topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat's Last Theorem. Suitable both for students preparing to enter the Mathematics Department and for non-majors interested in exposure to higher mathematics. Fall Only.

##### MAT215 Analysis in a Single Variable

Rigorous proof-based introduction to analysis. Topics include the rigorous epsilon-delta treatment of limits; convergence and uniform convergence of sequences and series; continuity, uniform continuity, and differentiability of functions; The Heine-Borel theorem; the Riemann integral; conditions for integrability of functions and term-by-term differentiation and integration of series of functions; Taylor’s theorem.

##### MAT217 Honors Linear Algebra

Rigorous course in linear algebra. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, linear systems of differential equations, the spectral theorem for normal transformations, bilinear and quadratic forms.

##### MAT218 Analysis in Several Variables

Rigorous course in multivariable analysis. Continuation of MAT215/217 (or MAT203/204, with instructor's permission). Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini's theorem, change of variables theorem, and the theorems of Green, Gauss, and Stokes.

##### MAT301 The History of Mathematics

Seminar will examine themes and ideas from the history of mathematics spanning the entirety of human history, from the oldest surviving written texts (numbers on clay tablets) to the present, with a focus on the mathematics of modern Europe. We will examine key debates and turning-points both within mathematics and in approaches to the history of mathematics as we develop a nuanced approach to understanding the historical relationships between mathematical practitioners and their theories, values, cultures, and circumstances.

##### MAT302 Mathematics in Engineering II

This course provides an introduction to partial differential equations, covering PDEs of relevant interest in engineering and science problems.

##### MAT303 Ordinary Differential Equations

Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. (Renumbered as MAT427 beginning AY 2012-13)

##### MAT305 Mathematical Logic

A development of logic from the mathematical viewpoint, including propositional and predicate calculus, consequence and deduction, truth and satisfaction, the Godel completeness and incompleteness theorems. Applications to model theory, recursion theory and set theory, as time permits. Some underclass background in logic or in mathematics is recommended.
(Replaces MAT312 beginning AY 2012-13)

##### MAT306 Introduction to Graph Theory

This course will cover the fundamental theorems and algorithms of graph theory. (Renumbered as MAT375 beginning AY 2012-13)

##### MAT307 Combinatorial Mathematics

This course introduces students to Combinatorics, a fundamental mathematical discipline as well as an essential component of many mathematical areas. (Renumbered as MAT377 beginning in AY 2012-13)

##### MAT308 Theory of Games

The mathematical concept of a game is an abstraction which encompasses conflict-cooperation situations in which strategy (not just chance) plays a role. (Renumbered as MAT378 beginning AY 2012-13)

##### MAT312 Mathematical Logic

Propositional and predicate calculus. Godel completeness theorem. Finitary methods. Godel incompleteness theorem. (Renumbered as MAT305 beginning AY 2012-13)

##### MAT314 Introduction to Real Analysis

Introduction to real analysis, Lebesgue theory of measure and integration on the line and n-dimensional space, introduction to Fourier Series. (Renumbered as MAT320 beginning AY 2012-13)

##### MAT317 Complex Analysis with Applications

Calculus of functions of one complex variable, power series expansions, residues, and conformal mapping. (Renumbered as MAT330 beginning AY 2012-13)

##### MAT320 Introduction to Real Analysis

Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space, and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent.
(Replaces MAT314 beginning AY2012-13)

##### MAT321 Numerical Methods

Introduction to numerical methods with emphasis on algorithms, applications and computer implementation issues. Topics covered include solution of nonlinear equations; numerical differentiation, integration and interpolation; direct and iterative methods for solving linear systems; numerical solutions of differential equations; two-point boundary value problems; and approximation theory. Lectures are supplemented with numerical examples using MATLAB. Prerequisites: MAT201 and MAT202; or MAT203 and MAT204; or equivalent.
(Replaces MAT342/APC342 beginning AY 2012-13)

##### MAT322 Introduction to Differential Equations

An introduction to differential equations, covering both applications and fundamental theory.
(Replaces MAT350/APC350 beginning AY 2012-13)

##### MAT323 Topics in Mathematical Modeling - Mathematical Neuroscience

Draws problems from the sciences & engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics.
This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upperclass undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.
(Replaces MAT351/APC351 beginning AY 2012-13)

##### MAT325 Analysis I: Fourier Series and Partial Differential Equations

Fourier series, Fourier transforms, and applications to the classical partial differential equations.
Prerequisites: MAT215 or MAT218 or consent of instructor.
(Replaces MAT330 beginning AY 2012-13)

##### MAT326 Algebraic Topology

Singular homology, cellular complexes, Poincare duality, Lefschetz fixed point theorem. (Renumbered as MAT560 beginning AY 2012-13)

##### MAT327 Introduction to Differential Geometry

Riemannian geometry of surfaces. Surfaces in Euclidan space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Bonnet Theorem, uniformization of surfaces. (Renumbered as MAT355 beginning AY 2012-13)

##### MAT328 Differential Geometry

Differential geometry is at the basis of modern physical theories, not only of general relativity, Einstein's geometric theory of gravitation, but also of the gauge theories of electromagnetic and nuclear interactions.

##### MAT330 Complex Analysis with Applications

Calculus of functions of one complex variable, power series expansions, residues, and conformal mapping. (Replaces MAT317 beginning AY 2012-13)

##### MAT331 Analysis II: Complex Analysis

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. (Renumbered as MAT335 beginning AY 2012-13)

##### MAT332 Analysis III: Integration Theory and Hilbert Space

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier transforms, and partial differential equations. Introduction to fractals. (Renumbered as MAT425 beginning AY 2012-13)

##### MAT335 Analysis II: Complex Analysis

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. The second course in a four-semester sequence, but may be taken independently.
(Replaces MAT331 beginning AY 2012-13)

##### MAT342 Numerical Methods

Introduction to numerical methods with emphasis on algorithms, applications and computer implementation issues. (Renumbered as MAT321 beginning AY 2012-13)

##### MAT345 Algebra I

Covers the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions and the representation theory of finite groups. Prerequisites: MAT202 or MAT204 or MAT217.
(Replaces MAT322 beginning AY 2012-13)

##### MAT350 Introduction to Differential Equations

An intro to differential equations. Both applications and fundamental theory will be discussed. (Renumbered as MAT322 beginning AY 2012-13)

##### MAT351 Topics in Mathematical Modeling - Mathematical Neuroscience

This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upperclass undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations. (Replaced by MAT323 beginning AY 2012-13)

##### MAT355 Introduction to Differential Geometry

Riemannian geometry of surfaces. Surfaces in Euclidan space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Bonnet Theorem, uniformization of surfaces. Prerequisites: MAT218 or equivalent.
(Replaces MAT327 beginning AY 2012-13)

##### MAT365 Topology

An introduction to point set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisites: MAT202 or MAT204 or MAT218 or equivalent.
(Replaces MAT325 beginning AY 2012-13)

##### MAT375 Introduction to Graph Theory

This course will cover the fundamental theorems and algorithms of graph theory. Topics include: connectivity, mathchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or MAT204 or MAT217, or equivalent.
(Replaces MAT306 beginning AY 2012-13)

##### MAT377 Combinatorial Mathematics

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics, and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations, and hyperplane arrangements. Current work and open problems will also be discussed.

##### MAT378 Theory of Games

The mathematical concept of a game is an abstraction which encompasses conflict-cooperation situations in which strategy (not just chance) plays a role. Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisites: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended.
(Replaces MAT308 beginning AY 2012-13)

##### MAT385 Probability Theory

Sequences of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains, Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisites: MAT203 or MAT218 or equivalent.
(Replaces MAT390 beginning AY 2012-13)

##### MAT391 Random Processes

Wiener measure, Stochastic differential equations, Markov diffusion processes etc. (Renumbered as MAT486 beginning AY 2012-13)

##### MAT407 Mathematical Methods of Physics

Mathematical methods and terminology which are essential for modern theoretical physics.

##### MAT415 Analytic Number Theory

An introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the tight connections between them. Topics covered might include: the prime number theorem, Dirichlet L-functions, zero-free regions, sieve methods, representation by quadratic forms, and Gauss sums. Prerequisites: MAT335 (Complex Analysis) and MAT345 (Algebra I).

##### MAT416 Introduction to Algebraic Geometry

Introduction to Algebraic Geometry; no previous knowledge of the topic is assumed.

##### MAT419 Topics in Number Theory: Algabraic Number Theory

Course on algebraic number theory. Topics covered include number fields and their integer rings, class
groups, zeta and L-functions.

##### MAT425 Analysis III: Integration Theory and Hilbert Space

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier transforms, and partial differential equations. Introduction to fractals. The third semester of a four-semester sequence, but may be taken independently. Prerequisites: MAT215 or MAT218 or equivalent.
(Replaces MAT332 beginning AY 2012-13)

##### MAT427 Ordinary Differential Equations

The course concerns explicit solution of simple differential equations. Methods of proving that one has found all the solutions are discussed. For this purpose, a brief review of foundational concepts in real analysis is provided. The second part concerns explicit solutions of simultaneous linear differential equations with constant coefficients, a topic closely connected with linear algebra (assumed prerequisite knowledge). The third part concerns the proof of the basic existence and uniqueness theorem for ordinary differential equations. Students will do simple proofs.
(Replaces MAT303 beginning AY 2012-13)

##### MAT433 Analysis IV: Special Topics in Analysis

Introductory course in modern Analysis with applications to Partial Differential Equations,Distribution Theory, Maximal Functions, Littlewood/Paley decompositions and applications, Strichartz inequalities, Bilinear Estimates, Concentrated compactness and applications. (Renumbered as MAT520 beginning AY 2012-13)

##### MAT443 Cryptography

An introduction to modern cryptography with an emphasis on the fundamental ideas.

##### MAT449 Topics in Algebra: Representation Theory

An introduction to representation theory of Lie groups and Lie algebras. The goal is to cover roughly the
first half of Knapp's book.

##### MAT451 Advanced Topics in Analysis

The course will cover the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss.

##### MAT453 Advanced Topics in Algebra

Algebraic number theory. (Renumbered as MAT419 beginning AY 2012-13)

##### MAT455 Advanced Topics in Geometry - Lie Theory

Lie algebras and Lie groups are important in many areas of mathematics as well as theoretical physics. The course gives an introduction to the topic.

##### MAT469 Advanced Topology

The course will target the following topics: The definition of knots in the 3-sphere, first invariants; algebraic knots and links in the 3-sphere; classification of algebraic knots, Puiseux pairs, iterated torus knots; fibred links, monodromy, the case of algebraic links; higher dimensional algebraic knots, Milnor theory of complex isolated hypersurface singularities.

##### MAT486 Random Processes

Wiener measure, Stochastic differential equations, Markov diffusion processes, Linear theory of stationary processes, Ergodicity, mixing, central limit theorem for stationary processes, Gibbs random field. If time permits, the theory of products of random matrices and PDE's with random coefficients will be discussed. Prerequisite: MAT390 in the old numbering system or MAT385 in the new system.
(Replaces MAT391 beginning AY 2012-13)

##### MAT520 Functional Analysis

The course is intended as a basic introductory course to the modern methods of Analysis. Specific applications of these methods to problems in other fields, such as Partial Differential Equations, Probability, and Number Theory, will also be presented. Topics will include Lp spaces, tempered distributions and the Fourier transform, the Riesz interpolation theorem, the Hardy–Littlewood maximal function, Calderon–Zygmund theory, the spaces H1 and BMO, oscillatory integrals, almost orthogonality, restriction theorems and applications to dispersive equations, the law of large numbers and ergodic theory. We will also discuss applications of Fourier methods to discrete counting problems, using the Poisson summation formula.

##### MAT522 Introduction to Partial Differential Equations

Introduction to the techniques necessary for the formulation and solution of problems involving partial differential equations in the natural sciences and engineering, with detailed study of the heat and wave equations. Topics include method of eigenfunction expansions, Fourier series, the Fourier transform, inhomogeneous problems, the method of variation of parameters. Prerequisite MAT202 or MAT204 or MAT218.

##### MAT523 Advanced Analysis

The course covers the essentials of the first eleven chapters of the textbook, "Analysis" by Lieb and Loss. Topics include Lebesque integrals, Measure Theory, L^p Spaces, Fourier Transforms, Distributions, Potential Theory, and some illustrative examples of applications of these topics.

##### MAT984 Junior Seminar

Lie-groups, Lie-algebras and their representations - Junior Seminar with Tasho Kaletha