Mathematics

Continuation of MAT345. Further develop knowledge of algebraic structures by exploring examples that connect to higher mathematics.

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 MAT300, or MAT203 or equivalent.

Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent.

The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or 204 or 217 or equivalent.

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.

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.

An introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains. Prerequisite: MAT 201, 203, 216, or instructor's permission. Three lectures, one precept.

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.

An introduction to ordinary differential equations. Use of numerical methods. Equations of a single variable and systems of linear equations. Method of undermined coefficients and method of variation of parameters. Series solutions. Use of eigenvalues and eigenvectors. Laplace transforms. Nonlinear equations and stability; phase portraits. Partial differential equations via separation of variables. Sturm-Liouville theory. Three lectures. Prerequisites: MAT 201 or 203, and MAT 202 or 204.

Solution of partial differential equations. Complex variable methods. Characteristics, orthogonal functions, and integral transforms. Cauchy-Riemann conditions and analytic functions, mapping, the Cauchy integral theorem, and the method of residues with application to inversion of transforms. Applications to diffusion, wave and Laplace equations in fluid mechanics and electrostatics. Three lectures, one preceptorial. Prerequisite: 305, MAT 301 or equivalent.

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.

Studies the limits of computation by identifying tasks that are either inherently impossible to compute, or impossible to compute within the resources available. Introduces students to computability and decidability, Godel's incompleteness theorem, computational complexity, NP-completeness, and other notions of intractability. This course also surveys the status of the P versus NP question. Additional topics may include: interactive proofs, hardness of computing approximate solutions, cryptography, and quantum computation. Two lectures, one precept. Prerequisite: 240 or 341, or instructor's permission.

Topics introducing various aspects of number theory, including analytic and algebraic number theory, L-functions, and modular forms. See Course Offerings listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent.

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.

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.

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 listing for topic details.

Topics in algebra selected from areas such as representation theory of finite groups and the theory of Lie algebras. Prerequisite: MAT 345 or MAT 346.

Elements of smooth manifold theory, tensors, and differential forms, Riemannian metrics, connection and curvature; selected applications to Hodge theory, curvature in topology and general relativity.

An introduction to modern cryptography with an emphasis on fundamental ideas. The course will survey both the basic information and complexity-theoretic concepts as well as their (often surprising and counter-intuitive) applications. Among the topics covered will be private key and public key encryption schemes, digital signatures, pseudorandom generators and functions, chosen ciphertext security; and time permitting, some advanced topics such as zero knowledge proofs, secret sharing, private information retrieval, and quantum cryptography. Prerequisites: 226 or permission of instructor. Two 90-minute lectures.

Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles.

This course will cover topics in Extremal Combinatorics including ones motivated by questions in other areas like Computer Science, Information Theory, Number Theory and Geometry. The subjects that will be covered include Graph powers, the Shannon capacity and the Witsenhausen rate of graphs, Szemeredi's Regularity Lemma and its applications in graph property testing and in the study of sets with no 3 term arithmetic progressions, the Combinatorial Nullstellensatz and its applications, the capset problem, Containers and list coloring, and related topics as time permits.

Wiener measure. Stochastic differential equations. Markov diffusion processes. Linear theory of stationary processes. Ergodicity, mixing, central limit theorem for stationary processes. If time permits, the theory of products of random matrices and PDE with random coefficients will be discussed. Prerequisite: MAT385.

Mathematical methods and techniques that are essential for modern theoretical physics. Topics such as group theory, Lie algebras, and differential geometry are discussed and applied to concrete physical problems. Special attention will be given to mathematical techniques that originated in physics, such as functional integration and current algebras. Three classes. Prerequisite: MAT 330 or instructor's permission.

This course is for second-year graduate students to help them develop their writing and speaking skills for communicating mathematics in a wide variety of settings, including teaching, grant applications, teaching statement, research statement, talks aimed at a general mathematical audience, and seminars, etc. In addition, responsible conduct in research (RCR) training is an integral part of this course.

This course will describes abelian extensions of number fields and function fields of curves over finite fields. One example is the celebrated Kronecker-Weber theorem stating that any abelain extension of Q is contained in a field generated by roots of unity. Another example is Kronecker's Jugendtraum stating that all abelian extensions of imaginary quadratic fields can be obtained analogously using torsion points of elliptic curves with complex multiplications. Prerequisites: Galois Theory (such as MAT 322) and MAT 419.