An introduction to graph theory, one of the central disciplines of discrete mathematics. Topics include: graphs and subgraphs, trees, bond and cycle spaces, connectivity, Euler tours and Hamiltonian cycles, matchings, independent sets, cliques and networks (if time permits). (Offered jointly with MATH-401*.)

Three term-hours, fall; lectures.

An introduction to two subjects which together with MATH-801* provide an entry into discrete mathematics and its applications. Among the enumeration techniques covered are inclusion-exclusion, recurrence relations, and generating functions. The study of designs includes finite geometries and Latin squares. (Offered jointly with MATH-402*.) Three term-hours, winter; lectures.

Subject matter may vary from year to year. (Offered jointly with MATH-411*.)

Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Time estimates for arithmetic and elementary number theory algorithms (division algorithm, Euclidean algorithm, congruences), modular arithmetic, finite fields, quadratic residues. Design of simple cryptographic systems; public key, RSA systems. Primality and factoring: pseudoprimes, Pollard's rho-method, index calculus. Elliptic curve cryptography. Offered jointly with MATH-418. Three term hours, fall or winter; lectures.

General properties of operators on Banach spaces: adjoints, closed graph theorem, spectrum and spectral radius, and the holomorphic functional calculus. The main part of the course will deal with operators on Hilbert spaces and will include: classification and examples of spectra, examples of operators, and the spectral theorem and its applications. Three term-hours, fall; lectures.

Topics include: global properties of flows and diffeomorphisms; invariant sets and dynamics; bifurcations of fixed and periodic points; stability and chaos. (Offered jointly with MATH-427*.) Three term-hours, fall or winter; lectures.

The classical calculus of variations: the Gateaux variation, necessary conditions, transversality, corner conditions, Euler-Lagrange multiplier theorem; applications to Hamiltonian mechanics and the Maximum Principle. (Offered jointly with MATH-432*.) Three term-hours, winter; lectures.

Problems of maximization and minimization of functions of many variables when the variables are restricted by inequality constraints, with applications to resource allocation, technology, estimation theory, etc.; convex sets and their properties, convex functions; Lagrange multipliers and the Kuhn-Tucker theorem. (Offered jointly with MATH-434*.) Three term-hours, fall; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Subject matter may vary from year to year. Three term-hours, fall or winter; lectures.

Topological equivalence and topological invariants; homotopy and the fundamental group; covering spaces; homotopy type; Brower’s fixed point theorem; triangularization;  homology and cohomology groups.    (Offered jointly with MATH 443*.) Three term-hours, fall or winter; lectures.

Calculus on manifolds; transversality; Sard's Theorem; immersions and submersions; intersection theory; Jordan curve theorem; Lefschetz fixed point theorem; Poincaré-Hopf Theorem. (Offered jointly with MATH-444*.)

Three term-hours, winter; lectures.

Subject matter will vary from year to year. Three term-hours, Fall or winter; seminar or reading course.

A review of probability models and introduction to applied stochastic processes. Topics may include Markov chains, birth and death processes, random walk problems, elementary renewal theory, Markov processes, Brownian motion and Poisson processes, queuing theory. (Offered jointly with STAT-455*.)

Three term-hours, fall or winter; lectures.

This course treats probability topics not included in other courses and is offered occasionally.

Three term-hours, fall or winter; lectures.

An introduction to aspects of computer software consistent with modern professional practice of statistics. Particular attention is given to the use of the statistical packages SAS and S-Plus. (Offered jointly with STAT-462*.)

Three term-hours, fall; lectures.

Autocorrelation and autocovariance, stationarity; ARIMA models; model identification and forecasting; spectral analysis. Applications to biological, physical and economic data. (Offered jointly with STAT-464.) Three term-hours, fall; lectures.

An overview of the tools used in the measurement and improvement of quality in business, government and industry today. About a third of the course will be concerned with the design of experiments, particularly factorial and fractional factorial designs. Includes a project. Three term-hours, fall or winter; lectures.

Analysis of data consisting of measurements of several variables on a sample of individuals. Distribution theory, maximum-likelihood estimation and likelihood-ratio testing, Bayesian inference, discrimination, principal components, other topics. (Offered jointly with STAT-466*.) Three term-hours, winter; lectures.

Introduces the theory and application of survival analysis: survival distributions and their applications, parametric and nonparametric methods, proportional hazards models, counting process and proportional hazards regression, planning and designing clinical trials. (Offered jointly with STAT-486*.) Three term-hours, winter; lectures.