# Mathematics (MATH)

MATH 103 Introduction to Calculus

Introduction to concepts and methods of calculus for students with little or no previous calculus experience. Polynomial and elementary transcendental functions and their applications, derivatives, extremum problems, curve-sketching, approximations; integrals and the fundamental theorem of calculus.

For BA Students: Formal Reasoning and Analysis

One-term course offered either term

Activity: Recitation

1 Course Unit

Notes: This is a Formal Reasoning course.

MATH 104 Calculus, Part I

Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, infinite series, Taylor's theorem, and first order ordinary differential equations. Use of symbolic manipulation and graphics software in calculus.

One-term course offered either term

Prerequisites: None. Engineering students are encouraged to register for the 006 lecture and group 6 recitations.

Activity: Recitation

1 Course Unit

Notes: This is a Formal Reasoning course.

MATH 110 Calculus for Wharton Students

Differential calculus, integral calculus, series, differential equations and elements of multivariable calculus, with an emphasis on applications.

One-term course offered either term

Activity: Lecture

1 Course Unit

MATH 114 Calculus, Part II

Functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors and vector calculus, first order ordinary differential equations. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus.

One-term course offered either term

Prerequisite: Math 104

Activity: Recitation

1 Course Unit

Notes: This is a Formal Reasoning course.

MATH 115 Calculus, Part II with Probability and Matrices

Functions of several variables, partial derivatives, multiple integrals, differential equations; introduction to linear algebra and matrices with applications to linear programming and Markov processes. Elements of probability and statistics. Applications to social and biological sciences. Use of symbolic manipulation and graphics software in calculus.

One-term course offered either term

Prerequisite: Math 104

Activity: Recitation

1 Course Unit

Notes: This is a Formal Reasoning course.

MATH 116 Honors Calculus

Students who are interested in math or science might also want to consider a more challenging Honors version of Calculus II and III, Math 116 and Math 260 (the analogues of Math 114 and Math 240, respectively). These courses will cover essentially the same material as 114 and 240, but more in depth and involve discussion of the underlying theory as well as computations.

One-term course offered either term

Activity: Recitation

1 Course Unit

Notes: This is a Formal Reasoning course.

MATH 123 Community Math Teaching Project

This course allows Penn students to teach a series of hands-on activities to students in math classes at University City High School. The semester starts with an introduction to successful approaches for teaching math in urban high schools. The rest of the semester will be devoted to a series of weekly hands-on activities designed to teach fundamental aspects of geometry. The first class meeting of each week, Penn faculty teach Penn students the relevant mathematical background and techniques for a hands-on activity. During the second session of each week, Penn students will teach the hands-on activity to a small group of UCHS students. The Penn students will also have an opportunity to develop their own activity and to implement it with the UCHS students.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 170 Ideas in Mathematics

Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.

For BA Students: Natural Science and Math Sector

One-term course offered either term

Activity: Lecture

1 Course Unit

Notes: May also be counted toward the General Requirement in Natural Science Mathematics

MATH 180 Analytical Methods in Economics, Law, and Medicine

Elementary applications of decision analysis, game theory, probability and statistics to issues in accounting, contracting, finance, law, and medicine, amongst others.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 202 Proving Things: Analysis

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying real and complex numbers, sequences, series, continuity, differentiability and integrability. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

One-term course offered either term

Corequisites: Math 104, 114 or 240

Activity: Seminar

1 Course Unit

MATH 203 Proving things: Algebra

This course 105uses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying arithmetic, algebra, linear algebra, groups, rings and fields. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

One-term course offered either term

Corequisites: Math 104, 114 or 240

Activity: Lecture

1 Course Unit

MATH 210 Mathematics in the Age of Information

This course counts as a regular elective for both the Mathematics Major and Minor. This is a course about mathematical reasoning and the media. Embedded in many stories one finds in the media are mathematical questions as well as implicit mathematical models for how the world behaves. We will discuss ways to recognize such questions and models, and how to think about them from a mathematical perspective. A key part of the course will be about what constitutes a mathematical proof, and what passes for proof in various media contexts. The course will cover a variety of topics in logic, probability and statistics as well as how these subjects can be used and abused.

One-term course offered either term

Prerequisites: Math 114, Math 115 or equivalent

Activity: Lecture

1 Course Unit

MATH 220 Proof in Math, Phil, Law

Proofs are vital to many parts of life. They arise typically in formal logic, mathematics, the testing of medication, and convincing a jury. How do you prove that the earth is essentially a sphere (in particular, not flat)? In reality, proofs arise anywhere one attempts to convince others. However, the nature of what constitutes a proof varies wildly depending on the situation -- and on whom you are attempting to convince. Convincing your math teacher or a judge is entirely different from convincing your mother or a jury. The course will present diverse views of Proof. On occasion there may be guest lecturers.

Also Offered As: PHIL 220

Activity: Seminar

1 Course Unit

MATH 240 Calculus, Part III

Linear algebra: vectors, matrices, systems of linear equations, vector spaces, subspaces, spans, bases, and dimension, eigenvalues, and eigenvectors, maxtrix exponentials. Ordinary differential equations: higher-order homogeneous and inhomogeneous ODEs and linear systems of ODEs, phase plane analysis, non-linear systems.

One-term course offered either term

Prerequisite: Calculus II

Activity: Recitation

1 Course Unit

MATH 241 Calculus, Part IV

Partial differential equations and their solutions, including solutions of the wave, heat and Laplace equations, and Sturm-Liouville problems. Introduction to Fourier series and Fourier transforms. Computation of solutions, modeling using PDE's, geometric intuition, and qualitative understanding of the evolution of systems according to the type of partial differential operator.

One-term course offered either term

Prerequisite: MATH 240

Activity: Recitation

1 Course Unit

MATH 260 Honors Calculus, Part II

This is an honors version of Math 240 which explores the same topics but with greater mathematical rigor.

Course not offered every year

Prerequisite: Calculus II

Activity: Recitation

1 Course Unit

MATH 312 Linear Algebra

Linear transformations, Gauss Jordan elimination, eigenvalues and eigenvectors, theory and applications. Mathematics majors are advised that MATH 312 cannot be taken to satisfy the major requirements.

Course not offered every year

Also Offered As: MATH 412

Prerequisite: MATH 240

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

MATH 313 Computational Linear Algebra

Many important problems in a wide range of disciplines within computer science and throughout science are solved using techniques from linear algebra. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used. Some specific topics: the solution of systems of linear equations by Gaussian elimination, dimension of a linear space, inner product, cross product, change of basis, affine and rigid motions, eigenvalues and eigenvectors, diagonalization of both symmetric and non-symmetric matrices, quadratic polynomials, and least squares optimazation.

Also Offered As: MATH 513

Prerequisites: Math 240, and some programming experience.

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313. Students can receive credit for at most one of Math 312 and Math 313.

Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra.

Taught by: Staff.

Also Offered As: AMCS 514, MATH 514

Prerequisites: Math 114 or 115.

Activity: Lecture

1 Course Unit

Notes: Math 314/514 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 312 or Math 314.

MATH 320 Computer Methods in Mathematical Science I

Students will use symbolic manipulation software and write programs to solve problems in numerical quadrature, equation-solving, linear algebra and differential equations. Theoretical and computational aspects of the methods will be discussed along with error analysis and a critical comparison of methods.

Course usually offered in fall term

Prerequisites: MATH 240 or concurrent and ability to program a computer, or permission of instructor

Activity: Lecture

1 Course Unit

MATH 321 Computer Methods in Mathematical Sciences II

Continuation of MATH 320.

Course not offered every year

Prerequisite: MATH 320

Activity: Lecture

1 Course Unit

MATH 340 Discrete Mathematics I

Topics will be drawn from some subjects in combinatorial analysis with applications to many other branches of math and science: graphs and networks, generating functions, permutations, posets, asymptotics.

Course not offered every year

Also Offered As: LGIC 210

Prerequisites: MATH 114 or Math 115 or permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 341 Discrete Mathematics II

Topics will be drawn from some subjects useful in the analysis of information and computation: logic, set theory, theory of computation, number theory, probability, and basic cryptography.

Also Offered As: LGIC 220

Prerequisites: Math 340/Logic 210 or permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 350 Number Theory

Congruences, Diophantine equations, continued fractions, nonlinear congruences, and quadratic residues.

Course not offered every year

Activity: Lecture

1 Course Unit

Syllabus for MATH 360-361: a study of the foundations of the differential and integral calculus, including the real numbers and elementary topology, continuous and differentiable functions, uniform convergence of series of functions, and inverse and implicit function theorems. MATH 508-509 is a masters level version of this course.

One-term course offered either term

Prerequisite: MATH 240

Activity: Lecture

1 Course Unit

Continuation of MATH 360.

One-term course offered either term

Prerequisite: MATH 360

Activity: Lecture

1 Course Unit

MATH 370 Algebra

Syllabus for MATH 370-371: an introduction to the basic concepts of modern algebra. Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings and fields. MATH 502-503 is a masters level version of this course.

One-term course offered either term

Prerequisite: MATH 240

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

MATH 371 Algebra

Continuation of MATH 370.

One-term course offered either term

Prerequisite: MATH 370

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

MATH 410 Complex Analysis

Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.

One-term course offered either term

Also Offered As: AMCS 510, MATH 299

Prerequisite: MATH 240 or permission of instructor

Activity: Lecture

1 Course Unit

MATH 420 Ordinary Differential Equations

After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.

One-term course offered either term

Also Offered As: AMCS 520

Prerequisite: MATH 240 permission of instructor

Activity: Lecture

1 Course Unit

MATH 425 Partial Differential Equations

Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.

Course usually offered in fall term

Also Offered As: AMCS 525

Prerequisites: MATH 240 or permission of instructor. Knowledge of PHYS 150-151 will be helpful.

Activity: Lecture

1 Course Unit

MATH 430 Introduction to Probability

Random variables, events, special distributions, expectations, independence, law of large numbers, introduction to the central limit theorem, and applications.

Course not offered every year

Also Offered As: AMCS 530

Prerequisite: MATH 240

Activity: Lecture

1 Course Unit

MATH 432 Game Theory.

A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.

One-term course offered either term

Also Offered As: AMCS 532

Prerequisites: Math 240.

Activity: Lecture

1 Course Unit

MATH 460 Topology

Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

Course not offered every year

Also Offered As: MATH 500

Prerequisite: MATH 240/241

Activity: Lecture

1 Course Unit

MATH 465 Differential Geometry

Differential geometry of curves in the plane and in 3-space;n gauge theories Surfaces in 3-space; The geometry of the Gauss map;ons. The language of Intrinsic geometry of surfaces; Geodesics; Moving frames; of vector bundles, The Gauss-Bonnet Theorem; Assorted additional topics.

Course not offered every year

Also Offered As: MATH 501

Prerequisites: MATH 361 or Permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 480 Topics in Modern Math

Mathematics 480 will open with a review of the basics of real analysis (brief or extended background requires). The review will include: introduction of the real numbers through Dedekind cuts, continuity of real-valued functions on the real line; Cantor nested-interval principle, basic results for continuous functions, Maximum and Intermediate Value theorems, Heine-Borel Theorem, Uniform Continuity on closed intervals; metric spaces, convergence of sequences, Cauchy sequences, completeness, more general uniform continuity and intermediate value theorems; general topology, separation, compactness, product spaces, Tychonoff's Theorem. Special topics in analysis: Weierstrass Polynomial Approximation Theorem, Bernstein polynomials and simultaneous approxfunctions and derivatives, topics from divergent series, summation methods; r measure theory, the Lebesgue integral, Lp spaces, Holder, Minkowski, and and Cauchy-Schwarz inequalities; basics of Functional Analysis, normed spaces, Banach spaces and Hilbert space, with examples (Lp spaces, continuous-functions spaces), Banach spaces and spectral theory, groups and Fourier transforms, Tauberian theorems; approximation theory, again, through the prism of functional analysis; extension of the polynomial approximation theorem (Stone-Weierstrass theorem), Muntz approximation theorem (by polynomials with preassigned powers), compact operators, the Spectral theorem, Stone's theorem (resprsentations of the additive group of real numbers); Peter-Weyl theory (represenations of compact groups). A selection from these topics as time and class preparation allow.

Course not offered every year

Prerequisites: A year of analysis at the 300 level or above (for example, Mathematics 360-361,508-509); a semester of linear algebra at the 300 level or above (for example, Mathematics 370).

Activity: Lecture

1 Course Unit

MATH 499 Supervised Study

Study under the direction of a faculty member. Intended for a limited number of mathematics majors.

One-term course offered either term

Activity: Independent Study

1 Course Unit

Notes: Hours and credit to be arranged

MATH 500 Geometry-Topology, Differential Geometry

Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

Course not offered every year

Also Offered As: MATH 460

Prerequisites: Math 240/241.

Activity: Lecture

1 Course Unit

MATH 501 Geometry-Topology, Differential Geometry

The course moves from a study of extrinsic geometry (curves and surfaces in n-space) to the intrinsic geometry of manifolds. After a review of vector calculus and a section on tensor algebra, we study manifolds and their intrinsic geometry, including metrics, connections, geodesics, and the Riemann curvature tensor. Topics include Eulerian curvature and Euler's theorems, the Gauss map and first/second fundamental forms, the Theorema Egregium, minimal surfaces in n-space; other topics as time permits.

Course not offered every year

Also Offered As: MATH 465

Prerequisites: Math 240, Math 312 (or 370), and Math 361 (or 508); or Math 500; or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 502 Abstract Algebra

An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra -- a more theoretical course than Math 370.

Course usually offered in fall term

Prerequisite: Math 240

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

MATH 503 Abstract Algebra

Continuation of Math 502.

Course usually offered in spring term

Prerequisites: Math 502 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

Notes: Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

MATH 504 Graduate Proseminar in Mathematics

This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.

Course usually offered in fall term

Activity: Lecture

1 Course Unit

MATH 505 Graduate Proseminar in Mathematics

This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.

Course usually offered in spring term

Activity: Lecture

1 Course Unit

Construction of real numbers, the topology of the real line and the foundations of single variable calculus. Notions of convergence for sequences of functions. Basic approximation theorems for continuous functions and rigorous treatment of elementary transcendental functions. The course is intended to teach students how to read and construct rigorous formal proofs. A more theoretical course than Math 360.

Course usually offered in fall term

Prerequisites: Math 240/241. Math 200/201 also recommended.

Activity: Lecture

1 Course Unit

Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods.

Course usually offered in spring term

Prerequisites: Math 508 or with the permission of the instructor. Linear algebra is also helpful.

Activity: Lecture

1 Course Unit

Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra.

Prerequisites: Math 114 or 115.

Activity: Lecture

1 Course Unit

Notes: Math 512 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 312 and Math 512.

MATH 513 Computational Linear Algebra

A number of important and interesting problems in a wide range of disciplines within computer science are solved by recourse to techniques from linear algebra. The goal of this course will be to introduce students to some of the most important and widely used algorithms in matrix computation and to illustrate how they are actually used in various settings. Motivating applications will include: the solution of systems of linear equations, applications matrix computations to modeling geometric transformations in graphics, applications of the Discrete Fourier Transform and related techniques in digital signal processing, the solution of linear least squares optimization problems and the analysis of systems of linear differential equations. The course will cover the theoretical underpinnings of these problems and the numerical algorithms that are used to perform important matrixcomputations such as Gaussian Elimination, LU Decomposition and Singular Value Decomposition.

Also Offered As: MATH 313

Activity: Lecture

1 Course Unit

Also Offered As: AMCS 514, MATH 314

Activity: Lecture

1 Course Unit

MATH 520 Selections from Algebra

Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.

Course not offered every year

Also Offered As: MATH 702

Corequisite: Math 502 or permission of the instructor

Activity: Lecture

1 Course Unit

MATH 521 Selections from Algebra

Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.

Course not offered every year

Corequisite: Math 502 or permission of the instructor

Activity: Lecture

1 Course Unit

MATH 524 Topics in Modern Applied Algebra

Topics such as automata, finite state languages, Boolean algebra, computers and logical design will be discussed.

Course not offered every year

Prerequisite: Math 371 or Math 503

Activity: Lecture

1 Course Unit

MATH 525 Topics in Modern Applied Algebra

Topics such as automata, finite state languages, Boolean algebra, computers and logical design will be discussed.

Course not offered every year

Prerequisite: Math 371 or Math 503

Activity: Lecture

1 Course Unit

MATH 530 Mathematics of Finance

This course presents the basic mathematical tools to model financial markets and to make calculations about financial products, especially financial derivatives. Mathematical topics covered: stochastic processes, partial differential equations and their relationship. No background in finance is assumed.

Course not offered every year

Prerequisites: Math 240, Stat 430

Activity: Lecture

1 Course Unit

MATH 540 Selections from Classical and Functional Analysis

Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.

Course not offered every year

Also Offered As: MATH 730

Corequisite: Math 508 or permission of the instructor

Activity: Lecture

1 Course Unit

MATH 546 Probability Theory

The required background is (1) enough math background to understand proof techniques in real analysis (closed sets, uniform covergence, fourier series, etc.) and (2) some exposure to probability theory at an intuitive level (a course at the level of Ross's probability text or some exposure to probability in a statistics class). After a summary of the necessary results from measure theory, we will learn the probabist's lexicon (random variables, independence, etc.). We will then develop the necessary techniques (Borel Cantelli lemmas, estimates on sums of independent random variables and truncation techniques) to prove the classical laws of large numbers. Next come Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks.

Course usually offered in fall term

Also Offered As: STAT 930

Activity: Lecture

1 Course Unit

MATH 547 Stochastic Processes

Markov chains, Markov processes, and their limit theory. Renewal theory. Martingales and optimal stopping. Stable laws and processes with independent increments. Brownian motion and the theory of weak convergence. Point processes.

Course not offered every year

Also Offered As: STAT 931

Prerequisites: Math 546/Stat 530 or the equivalent.

Activity: Lecture

1 Course Unit

MATH 548 Topics in Analysis

Topics may vary but typically will include an introduction to topological linear spaces and Banach spaces, and to Hilbert space and the spectral theorem. More advanced topics may include Banach algebras, Fourier analysis, differential equations and nonlinear functional analysis.

Course not offered every year

Prerequisites: Math 360/361 and Math 370; or Math 508/509 and Math 502

Activity: Lecture

1 Course Unit

MATH 549 Topics in Analysis

Continuation of Math 548.

Course not offered every year

Prerequisites: Math 548 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 560 Selections from Geometry and Topology

Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.

Course not offered every year

Also Offered As: MATH 760

Corequisite: Math 500 or permission of the instructor

Activity: Lecture

1 Course Unit

MATH 561 Selections from Geometry and Topology

Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.

Course not offered every year

Corequisite: Math 500 or permission of the instructor

Activity: Lecture

1 Course Unit

MATH 570 Introduction to Logic and Computability

Propositional logic: semantics, formal deductions, resolution method. First order logic: validity, models, formal deductions; Godel's completeness theorem, Lowenheim-Skolem theorem: cut-elimination, Herbrand's theorem, resolution method.Computability: finite automata, Turing machines, Godel's incompleteness theorems. Algorithmically unsolvable problems in mathematics.

Course not offered every year

Also Offered As: LGIC 310, PHIL 410

Prerequisite: Math 371 or Math 503

Activity: Lecture

1 Course Unit

MATH 571 Topics in Logic

Continuation of Math 570.

Course not offered every year

Also Offered As: CIS 518, LGIC 320, MATH 671, PHIL 412

Prerequisites: Math 570 or with the permission of the instructor.

Activity: Seminar

1 Course Unit

MATH 575 Mathematical Theory of Computation

Continuation of Math 574.

Course not offered every year

Prerequisites: Math 574 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 580 Combinatorial Analysis and Graph Theory

Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry.

Course not offered every year

Prerequisite: Permission of the instructor

Activity: Lecture

1 Course Unit

MATH 581 Combinatorial Analysis and Graph Theory

Continuation of Math 580.

Course not offered every year

Prerequisites: Math 580 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 582 Applied Mathematics and Computation

This course offers first-hand experience of coupling mathematics with computing and applications. Topics include: Random walks, randomized algorithms, information theory, coding theory, cryptography, combinatorial optimization, linear programming, permutation networks and parallel computing. Lectures will be supplemented by informal talks by guest speakers from industry about examples and their experience of using mathematics in the real world.

Course not offered every year

Also Offered As: AMCS 701

Prerequisites: Math 240-241. Math 312, Math 360. Knowledge of Math 412 and Math 508 is recommended.

Activity: Lecture

1 Course Unit

MATH 583 Applied Mathematics and Computation

Continuation of Math 582.

Course not offered every year

Prerequisites: Math 582 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 584 The Mathematics of Medical Imaging and Measurement

In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly.The data which is accesible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT).In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

Taught by: Staff.

Course not offered every year

Also Offered As: BE 584

Prerequisites: Math 314, Math 508 and 509 or with permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 585 The Mathematics of Medical Imaging and Measurement

Continuation of Math 584.

Course not offered every year

Prerequisites: Math 584 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 586 Topics in Mathematical Biology.

This course will cover various mathematical models and tools that are used to study modern biological problems. The specific emphasis will vary from year to year, but typically will include an introduction to stochastic processes and computational methods that arise in evolutionary biology and population genetics. No prior knowledge of biology is needed to take this course, but a strong background in probability and familiarity with algorithms and combinatorics will be assumed.

Also Offered As: BIOL 586

Prerequisites: Math 430 or equivalent is required. Math 241 and 340 are recommended.

Activity: Lecture

1 Course Unit

MATH 594 Advanced Methods in Applied Mathematics

Introduction to mathematics used in physics and engineering, with the goal of developing facility in classical techniques. Vector spaces, linear algebra, computation of eigenvalues and eigenvectors, boundary value problems, spectral theory of second order equations, asymptotic expansions, partial differential equations, differential operators and Green's functions, orthogonal functions, generating functions, contour integration, Fourier and Laplace transforms and an introduction to representation theory of SU(2) and SO(3). The course will draw on examples in continuum mechanics, electrostatics and transport problems.

Course not offered every year

Also Offered As: PHYS 500

Prerequisites: Math 241 or Permission of Instructor. Physics 151 would be helpful for undergraduates.

Activity: Lecture

1 Course Unit

MATH 599 Independent Study

One-term course offered either term

Activity: Independent Study

1 Course Unit

MATH 600 Topology and Geometric Analysis

Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.

Course usually offered in fall term

Prerequisites: Math 500/501 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 601 Topology and Geometric Analysis

Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

Course usually offered in spring term

Prerequisites: Math 600 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 602 Algebra

Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem, elementary representation theory. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.

Course usually offered in fall term

Prerequisite: Math 370/371 or Math 502/503

Activity: Lecture

1 Course Unit

MATH 603 Algebra

Continuation of Math 602.

Course usually offered in spring term

Prerequisites: Math 602 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 604 First Year Seminar in Mathematics

This is a seminar for first year Mathematics graduate student, supervised by faculty. Students give talks on topics from all areas of mathematics at a level appropriate for first year graduate students. Attendance and preparation will be expected by all participants, and learning how to present mathematics effectively is an important part of the seminar.

Course usually offered in fall term

Prerequisites: Open to first year Mathematics graduate students. Others need permission of the instructor.

Activity: Seminar

1 Course Unit

MATH 605 First Year Seminar in Mathematics

Continuation of Math 604.

Taught by: Staff.

Course usually offered in spring term

Prerequisites: Open to first year Mathematics graduate students. Others need permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 608 Analysis

Complex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. Introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C0 and the Riesz-Markov theorem, Lp-spaces, Fubini Theorem.

One-term course offered either term

Also Offered As: AMCS 608

Prerequisite: Math 508-509

Activity: Lecture

1 Course Unit

MATH 609 Analysis

Real analysis: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L2-theory of the Fourier transform. Functional analysis: normed linear spaces, convexity, the Hahn-Banach theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, Lp-theory for the Fourier transform.

One-term course offered either term

Also Offered As: AMCS 609

Prerequisites: Math 608 or permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 612 Selections from Algebra

Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.

Course not offered every year

Corequisites: MATH 600 and 602

Activity: Lecture

1 Course Unit

MATH 618 Algebraic Topology, Part I

Homotopy groups, Hurewicz theorem, Whitehead theorem, spectral sequences. Classification of vector bundles and fiber bundles. Characteristic classes and obstruction theory.

Course usually offered in fall term

Prerequisite: Math 600/601 or with the permission of the instructor

Activity: Lecture

1 Course Unit

MATH 619 Algebraic Topology, Part I

Rational homotopy theory, cobordism, K-theory, Morse theory and the h-corbodism theorem. Surgery theory.

Course usually offered in spring term

Prerequisites: Math 618 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 620 Algebraic Number Theory

Dedekind domains, local fields, basic ramification theory, product formula, Dirichlet unit theory, finiteness of class numbers, Hensel's Lemma, quadratic and cyclotomic fields, quadratic reciprocity, abelian extensions, zeta and L-functions, functional equations, introduction to local and global class field theory. Other topics may include: Diophantine equations, continued fractions, approximation of irrational numbers by rationals, Poisson summation, Hasse principle for binary quadratic forms, modular functions and forms, theta functions.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 621 Algebraic Number Theory

Continuation of Math 620.

Course not offered every year

Prerequisites: Math 620 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 622 Complex Algebraic Geometry

Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry. Topics include: Complex algebraic varieties, cohomology theories, line bundles, vanishing theorems, Riemann surfaces, Abel's theorem, linear systems, complex tori and abelian varieties, Jacobian varieties, currents, algebraic surfaces, adjunction formula, rational surfaces, residues.

Course not offered every year

Prerequisites: Math 602/603 and Math 609

Activity: Lecture

1 Course Unit

MATH 623 Complex Algebraic Geometry

Continuation of Math 622.

Course not offered every year

Prerequisites: Math 622 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 624 Algebraic Geometry

Algebraic geometry over algebraically closed fields, using ideas from commutative algebra. Topics include: Affine and projective algebraic varieties, morphisms and rational maps, singularities and blowing up, rings of functions, algebraic curves, Riemann Roch theorem, elliptic curves, Jacobian varieties, sheaves, schemes, divisors, line bundles, cohomology of varieties, classification of surfaces.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 625 Algebraic Geometry

Continuation of Math 624.

Course not offered every year

Prerequisites: Math 624 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 626 Commutative Algebra

Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 627 Commutative Algebra

Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 628 Homological Algebra

Complexes and exact sequences, homology, categories, derived functors (especially Ext and Tor). Homology and cohomology arising from complexes in algebra and geometry, e.g. simplicial and singular theories, Cech cohomology, de Rham cohomology, group cohomology, Hochschild cohomology. Projective resolutions, cohomological dimension, derived categories, spectral sequences. Other topics may include: Lie algebra cohomology, Galois and etale cohomology, cyclic cohomology, l-adic cohomology. Algebraic deformation theory, quantum groups, Brauer groups, descent theory.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 629 Homological Algebra

Continuation of Math 628.

Course not offered every year

Prerequisites: Math 628 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 630 Differential Topology

Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory. The instructor may elect to cover other topics such as Morse Theory, h-cobordism theorem, characteristic classes, cobordism theories.

Course not offered every year

Prerequisite: Math 600/601

Activity: Lecture

1 Course Unit

MATH 631 Differential Topology

Continuation of Math 630.

Course not offered every year

Prerequisites: Math 630 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 632 Topological Groups

Fundamentals of topological groups. Haar measure. Representations of compact groups. Peter-Weyl theorem. Pontrjagin duality and structure theory of locally compact abelian groups.

Course not offered every year

Prerequisites: Math 600/601 and Math 602/603

Activity: Lecture

1 Course Unit

MATH 633 Topological Groups

Continuation of Math 632.

Course not offered every year

Prerequisites: Math 632 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 634 Arithmetic Geometry

Activity: Lecture

1 Course Unit

MATH 638 Algebraic Topology, Part II

Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.

One-term course offered either term

Prerequisite: Math 618/619

Activity: Lecture

1 Course Unit

MATH 639 Algebraic Topology, Part II

Continuation of Math 638.

One-term course offered either term

Prerequisites: Math 638 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 640 Ordinary Differential Equations

The general existence and uniqueness theorems for systems of ordinary differential equations and the dependence of solutions on initial conditions and parameters appearing in the equation. The proofs of existence and uniqueness are related to numerical algorithms for finding approximate solutions for systems of ODE's. We consider special properties of constant coefficient and linear systems. We then present the theory of linear equations with analytic coefficients, the theories of singular points, indicial roots and asymptotic solutions. We then turn to boundary value problems for second order equations with an emphasis on the eigenfunction expansions associated with self adjoint boundary conditions and the Sturm comparison theory. The remaining time is devoted to topics; for example: Hamiltonian systems and symplectic geometry, singular boundary value problems, perturbation theory, the Lyapounov-Schmidt theory and the Poincare-Bendixson theorem, the equations of mathematical physics, the calculus of variations, symmetries of ODE's and transformation groups.

Course not offered every year

Prerequisite: Math 508/509

Activity: Lecture

1 Course Unit

MATH 641 Ordinary Differential Equations

Continuation of Math 640

Course not offered every year

Prerequisites: Math 640 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 642 Topics in Partial Differential Equations

Problems in differential geometry, as well as those in physics and engineering, inevitable involve partial derivatives. This course will be an introduction to these problems and techniques. We will use P.D.E. as a tool. Some of the applications will be small, some large. The proof of the Hodge Theorem will be a small application. Discussion of the Yamabe problem and ricci flow (used to prove the Poincare Conjecture) will be larger.

Prerequisites: Math 608, 609. This course will not presume courses in Partial Differential Equations or Differential Geometry. Background will be covered in the course.

Activity: Lecture

1 Course Unit

MATH 644 Partial Differential Equations

Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and non-linear problems. Sobolev spaces and the theory of distributions will be developed as needed.

Course not offered every year

Prerequisites: Math 600/601, Math 608/609

Activity: Lecture

1 Course Unit

MATH 645 Partial Differential Equations

Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and nonlinear problems. Sobolev spaces and the theory of distributions will be developed as needed.

Course not offered every year

Prerequisites: Math 600/601, Math 608/609

Activity: Lecture

1 Course Unit

MATH 646 Several Complex Variables

Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.

Course not offered every year

Prerequisites: Math 600/601, Math 608/609

Activity: Lecture

1 Course Unit

MATH 647 Several Complex Variables

Continuation of Math 646.

Course not offered every year

Prerequisites: Math 646 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 650 Lie Algebras

Connections with Lie groups, universal enveloping algebras, Poincare-Birkhoff-Witt Theorem, Lie and Engels theorems, free Lie algebras, Killing form, semisimple algebras, root systems, Dynkin diagrams, classification of complex simple Lie algebras, representation theory of Lie algebras, cohomology of Lie algebras.

Course not offered every year

Prerequisite: Math 602/603

Activity: Lecture

1 Course Unit

MATH 651 Lie Algebras

Continuation of Math 650.

Course not offered every year

Prerequisites: Math 650 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 652 Operator Theory

Subject matter may include spectral theory of operators in Hilbert space, C*-algebras, von Neumann algebras.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 653 Operator Theory

Subject matter may include spectral theory of operators in Hilbert space, C*-algebras, von Neumann algebras.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 654 Lie Groups

Connection of Lie groups with Lie algebras, Lie subgroups, exponential map. Algebraic Lie groups, compact and complex Lie groups, solvable and nilpotent groups. Other topics may include relations with symplectic geometry, the orbit method, moment map, symplectic reduction, geometric quantization, Poisson-Lie and quantum groups.

Course not offered every year

Prerequisites: Math 600/601, Math 602/603

Activity: Lecture

1 Course Unit

MATH 655 Lie Groups

Continuation of Math 654.

Course not offered every year

Prerequisites: Math 654 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 656 Representation of Continuous Groups

Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 657 Representation of Continuous Groups

Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups.

Course not offered every year

Also Offered As: PHYS 657

Activity: Lecture

1 Course Unit

MATH 660 Differential Geometry

Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions, the soul theorem, Gromov-Hausdorff convergence.

Course not offered every year

Prerequisites: Math 600/601, Math 602/603

Activity: Lecture

1 Course Unit

MATH 661 Differential Geometry

Continuation of Math 660.

Course not offered every year

Prerequisites: Math 660 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 670 Topics in Logic

Course not offered every year

Also Offered As: LGIC 320, PHIL 412

Prerequisite: Math 570/571

Activity: Seminar

1 Course Unit

MATH 671 Topics in Logic

Course not offered every year

Also Offered As: CIS 518, LGIC 320, MATH 571, PHIL 412

Prerequisite: Math 570/571

Activity: Seminar

1 Course Unit

MATH 676 Advanced Geometric Methods in Computer Science

Advanced geometric methods used in geometric modeling,computer graphics, computer vision, and robotics.

Course not offered every year

Also Offered As: CIS 610

Prerequisites: Math 312 or Math 412, or with the permission of the instructor

Activity: Lecture

1 Course Unit

MATH 680 Applied Linear Analysis

Application of techniques from linear algebra to real problems in economics, engineering, physics, etc. and the difficulties involved in their implementation. Particular emphasis is placed on solving equations, the eigenvalue problem for symmetric matrices and the metric geometry of spaces of matrices. Applications to problems such as options pricing, image reconstruction, airplane and ship design, oil prospecting, etc. (these topics will vary from year to year). Analysis of the numerical algorithms available to solve such problems, rates of convergence, accuracy and stability.

Course not offered every year

Prerequisites: Math 241 and one semester of: Math 360/361 or Math 508/509.

Activity: Lecture

1 Course Unit

MATH 681 Applied Linear Analysis

Continuation of Math 680.

Course not offered every year

Prerequisites: Math 680 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 690 Topics in Mathematical Foundations of Program Semantics

This course discusses advanced topics in mathematical foundations of semantics of programming languages and programming structures. "What is to distinguish a digital dollar when it is as easily reproducible as the spoken word? How do we converse privately when every syllable is bounced off a satellite and smeared over an entire continent? How should a bank know that it really is Bill Gates requesting from his laptop in Fiji a transfer of \$100,000,.....,000 to another bank? Fortunately, the mathematics of cryptography can help. Cryptography provides techniques for keeping information secret, for determining that information has not been tampered with, and for determing who authored pieces of information." (From the Foreword by R. Rivest to the "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone.)

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 691 Topics in Mathematical Foundations of Program Semantics

The course discusses advanced topics in mathematical foundations of semantics of programming languages and programming structures.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 692 Numerical Analysis

A study of numerical methods for matrix problems, ordinary and partial differential equations, quadrature and the solution of algebraic or transcendental equations. Emphasis will be on the analysis of those methods which are particularly suited to automatic high-speed computation.

Course not offered every year

Prerequisite: Math 320/321

Activity: Lecture

1 Course Unit

MATH 693 Numerical Analysis

Continuation of Math 692.

Course not offered every year

Prerequisites: Math 692 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 694 Mathematical Foundations of Theoretical Physics

Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.

Course not offered every year

Also Offered As: PHYS 654

Activity: Lecture

1 Course Unit

MATH 695 Mathematical Foundations of Theoretical Physics

Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.

Course not offered every year

Also Offered As: PHYS 655

Activity: Lecture

1 Course Unit

MATH 696 Topics in Mathematical Physics and String Theory

This interdisciplinary course discusses advanced topics in mathematical physics. Topics may include elliptic operators, heat kernels, complexes and the Atiyah-Singer index theorem, Feynman graphs and anomalies, computing Abelian and non-Abelian anomalies, and the relation of anomalies to the index theorem.

Course not offered every year

Also Offered As: PHYS 656

Prerequisites: Math 694 or permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 697 Topics in Mathematical Physics and String Theory

Continuation of Math 696. Topics may include the family index theorem, equivariant cohomology and loop spaces, the homological algebra of BRST invariance and the Wess-Zumino consistency condition, the descent equations, and worldsheet anomalies in string theory.

Course not offered every year

Also Offered As: PHYS 657

Prerequisites: Math 696 or permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 702 Topics in Algebra

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Also Offered As: MATH 520

Activity: Lecture

1 Course Unit

MATH 703 Topics in Algebra

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Activity: Lecture

1 Course Unit

Ramification theory, adeles and ideles, Tate's thesis, group cohomology and Galois cohomology, class field theory in terms of ideles and cohomology, Lubin-Tate formal groups, Artin and Swan conductors, central simple algebras over local and global fields, general Hasse principles. Other topics may include the following: zero-dimensional Arakelov theory, Tate duality, introduction to arithmetic of elliptic curves, local and global epsilon factors in functional equations, p-adic L-functions and Iwasawa theory, modular forms and functions and modular curves.

Course not offered every year

Prerequisite: Math 620/621

Activity: Lecture

1 Course Unit

Continuation of Math 720.

Course not offered every year

Prerequisites: Math 720 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 724 Topics in Algebraic Geometry

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Prerequisite: Either Math 622/623 or Math 624/625

Activity: Lecture

1 Course Unit

MATH 725 Topics in Algebraic Geometry

Topics from the literature. The specific subject will vary from year to year.

Course not offered every year

Prerequisite: Either Math 622/623 or Math 624/625

Activity: Lecture

1 Course Unit

MATH 730 Topics in Algebraic and Differential Topology

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Also Offered As: MATH 560

Prerequisite: Math 618/619

Activity: Lecture

1 Course Unit

MATH 731 Topics in Algebraic and Differential Topology

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Prerequisite: Math 618/619

Activity: Lecture

1 Course Unit

MATH 748 Topics in Classical Analysis

Harmonic analysis in Euclidean space, Riemann surfaces, Discontinuous groups and harmonic analysis in hyperbolic space, Pseudodifferential operators and index theorems, Variational methods in non-linear PDE, Hyperbolic equations and conservation laws, Probability and stochastic processes, Geometric measure theory, Applications of analysis to problems in differential geometry. The specific subjects will vary from year to year.

Course not offered every year

Prerequisites: Math 608 and Math 609 and permission from the instructor

Activity: Lecture

1 Course Unit

MATH 749 Topics in Classical Analysis

Continuation of Math 748.

Course not offered every year

Prerequisites: Math 748 or with the permission of the instructor.

Activity: Lecture

1 Course Unit

MATH 750 Topics in Functional Analysis

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 751 Topics in Functional Analysis

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 752 Topics in Operator theory

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 753 Topics in Operator Theory

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Activity: Lecture

1 Course Unit

MATH 760 Topics in Differential Geometry

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Also Offered As: MATH 560

Prerequisite: Math 660/661

Activity: Lecture

1 Course Unit

MATH 761 Topics in Differential Geometry

Topics from the literature. The specific subjects will vary from year to year.

Course not offered every year

Prerequisite: Math 660/661

Activity: Lecture

1 Course Unit

MATH 794 Physics for Mathematicians

This course is designed to bring mathematicians with no physics background up to speed on the basic theories of physics: mechanics,relativity,quantum mechanics,classical fields,quantum filed theory, the standard model,strings,superstrings,and M-theory.

Course not offered every year

Prerequisite: Math 694

Corequisite: Math 695

Activity: Lecture

1 Course Unit

MATH 820 Algebra Seminar

Seminar on current and recent literature in algebra.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 821 Algebra seminar

Seminar on current and recent literature in algebra.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 824 Seminar in Algebra, Algebraic Geometry, Number Theory

Seminar on current and recent literature in algebra, algebraic geometry, and number theory.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 825 Seminar in Algebra, Algebraic Geometry, Number Theory

Seminar on current and recent literature in algebra, algebraic geometry, and number theory.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 830 Geometry-Topology Seminar

Seminar on current and recent literature in geometry-topology

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 831 Geometry-Topology Seminar

Seminar on current and recent literature in geometry-topology

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 840 Analysis Seminar

Seminar on current and recent literature in analysis.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 841 Analysis Seminar

Seminar on current and recent literature in analysis.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 844 Seminar in Partial Differential Equations

Seminar on current and recent literature in partial differential equations.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 845 Seminar in Partial Differential Equations

Seminar on current and recent literature in partial differential equations.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 850 Seminar in Functional Analysis

Seminar on current and recent literature in functional analysis.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 851 Seminar in Functional Analysis.

Seminar on current and recent literature in functional analysis.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 861 Seminar in Riemannian Geometry

Seminar on current and recent literature in Riemannian geometry.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 870 Logic Seminar

Seminar on current and recent literature in logic.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 871 Logic Seminar

Seminar on current and recent literature in logic.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 872 Seminar in Logic and Computation

Seminar on current and recent literature in logic and computation.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 873 Seminar in Logic and Computation

Seminar on current and recent literature in logic and computation.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 878 Probability and Algorithm Seminar

Seminar on current and recent literature in probability and algorithm.

Taught by: Staff.

Activity: Seminar

1 Course Unit

MATH 879 Seminar on additive combinatorics.

Taught by: Staff.

Activity: Seminar

1 Course Unit

MATH 880 Combinatorics Seminar

Seminar on current and recent literature in combinatorics.

Course not offered every year

Activity: Seminar

1 Course Unit

MATH 881 Combinatorics Seminar

Seminar on current and recent literature in combinatorics.

Course not offered every year

Activity: Seminar

1 Course Unit