A.B. Mathematics

MAT 100 Calculus Foundations Fall QR Introduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting. T. Howard, J. Johnson

MAT 102 Survey of Calculus Not offered this year QR One semester survey of the major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus including approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisites: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT103 and MAT102. Provides adequate preparation for MAT175. Three classes. Staff

MAT 103 Calculus I Fall/Spring QR First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. The fall offering will emphasize applications to physics and engineering in preparation for MAT104; the spring offering will emphasize applications to economics and life sciences, in preparation for MAT175. Prerequisite: MAT100 or equivalent. Three classes. Staff

MAT 104 Calculus II Fall/Spring QR Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. Prerequisite: MAT103 or equivalent. Three classes. Staff

MAT 175 Mathematics for Economics/Life Sciences Fall/Spring QR Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers. Students preparing for math track econometrics and finance courses need MAT201/202 instead. Students who complete 175 can continue in 202 if they wish. Staff

MAT 191 An Integrated Introduction to Engineering, Mathematics, Physics (See EGR 191)

MAT 192 An Integrated Introduction to Engineering, Mathematics, Physics (See EGR 192)

MAT 199 Math Alive (See APC 199)

MAT 201 Multivariable Calculus Fall/Spring QR Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields, and Stokes's theorem. Prerequisite: 104 or equivalent. Three classes. Staff

MAT 202 Linear Algebra with Applications Fall/Spring QR Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems.Three classes. Staff

MAT 203 Advanced Vector Calculus Fall QR Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Three classes. H. Diao

MAT 204 Advanced Linear Algebra with Applications Spring QR Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent. Three classes. C. Taylor

MAT 214 Numbers, Equations, and Proofs Fall QR An 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 from more advanced or more applied number theory such as p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the mathematics department and for non-majors interested in exposure to higher mathematics. C. Skinner

MAT 215 Honors Analysis (Single Variable) Fall/Spring QR An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. 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. C. Fefferman, J. Gómez-Serrano

MAT 217 Honors Linear Algebra Spring QR A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. Staff

MAT 218 Accelerated Honors Analysis II Spring QR Continuation of MAT216, Accelerated Analysis I from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. See the department website for details: (link is external). R. Gunning

MAT 305 Mathematical Logic Not offered this year QR A development of logic from the mathematical viewpoint, including propositional and predicate calculus, consequence and deduction, truth and satisfaction, the Goedel 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. Staff

MAT 306 Advanced Logic (See PHI 323)

MAT 320 Introduction to Real Analysis Fall QR 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. M. Ignatova

MAT 323 Topics in Mathematical Modeling (also

APC 323

MAT 325 Analysis I: Fourier Series and Partial Differential Equations Spring QR Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Fast Fourier Transforms, Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Prerequisites: 215, 218, or permission of instructor. A. Ionescu

MAT 330 Complex Analysis with Applications Spring QR The theory of functions of one complex variable, covering power series expansions, residues, contour integration, and conformal mapping. Although the theory will be given adequate treatment, the emphasis of this course is the use of complex analysis as a tool for solving problems. Prerequisite: MAT201 and MAT202 or equivalent. M. Aizenman

MAT 335 Analysis II: Complex Analysis Fall QR 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. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters. A. Naor

MAT 345 Algebra I Fall QR This course will cover 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, rings and modules. P. Sarnak

MAT 346 Algebra II Spring QR Continuation of MAT345. Further develop knowledge of algebraic structures by exploring examples that connect to higher mathematics. There will be opportunities for a student to explore an advanced topic in great depth, possibly for a junior project. Staff

MAT 355 Introduction to Differential Geometry Spring QR Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet formula, uniformization of surfaces, elementary notions of contact geometry. Prerequisite: MAT218 or 350 or equivalent. Staff

MAT 365 Topology Fall QR Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent. Z. Szabó

MAT 375 Introduction to Graph Theory (also

COS 342

MAT 377 Combinatorial Mathematics (also

APC 377

MAT 378 Theory of Games Spring QR 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. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended. J. Fickenscher

MAT 380 Probability and Stochastic Systems (See ORF 309)

MAT 385 Probability Theory Fall QR Sequence 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 the 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. Prerequisite: MAT203 or 218 or equivalent. Staff

MAT 391 Mathematics in Engineering I (See MAE 305)

MAT 392 Mathematics in Engineering II (See MAE 306)

MAT 393 Mathematical Programming Not offered this year QR Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming. Staff

MAT 407 Theory of Computation (See COS 487)

MAT 419 Topics in Number Theory QR Topics introducing various aspects of number theory, including analytic and algebraic number theory, L-functions, and modular forms. See Course Offerings (link is external) listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent. F. Castella

MAT 425 Analysis III: Integration Theory and Hilbert Spaces Fall QR 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. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters. Prerequisites: MAT215 or 218 or equivalent. M. Dafermos

MAT 427 Ordinary Differential Equations Not offered this year QR Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include explicit solutions of some non-linear equations in two variables by separation of variables and integrating factors, explicit solution of simultaneous linear equations with constant coefficients, explicit solution of some linear equations with variable forcing term by Laplace transform methods, geometric methods (description of the phase portrait), and the fundamental existence and uniqueness theorem. Staff

MAT 429 Topics in Analysis QR Introduction to incompressible fluid dynamics. The course will give an introduction to the mathematical theory of the Euler equations, the fundamental partial differential equation arising in the study of incompressible fluids. We will discuss several topics in analysis that emerge in the study of these equations: Lebesgue and Sobolev spaces, distribution theory, elliptic PDEs, singular integrals, and Fourier analysis. Content varies from year to year. See Course Offerings (link is external) listing for topic details. Staff

MAT 449 Topics in Algebra Fall QR Topics in algebra selected from areas such as representation theory of finite groups and the theory of Lie algebras. Three classes. Prerequisite: MAT 345 or MAT 346. . S