Course Summary: Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem. This is a second or intermediate course in analysis, with a fast review of introductory topics.
Course Summary: Group theory with emphasis on concrete examples and applications. Field theory: ruler and compass constructions, Galois theory, and solvability of equations by radicals.
Additional Information: Class only offered during Summer II of years ending in an even number.
Course Summary: Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green's theorem, Stokes's theorem, divergence theorems, and applications to engineering and the physical sciences.
Course Summary: Matrices, rank and inverse of a matrix, decomposition theorems, eigenvectors, unitary and similarity transformations on matrices.
Additional Information: Not open to students with credit in MATH 35100.
Course Summary: Descriptive statistics; elementary probability; random variables and their distributions; expectation; normal, binomial, Poisson, and hypergeometric distributions; sampling distributions; estimation and testing of hypotheses; one-way analysis of variance; and correlation and regression.
Course Summary: Inference in simple and multiple linear regression, estimation of model parameters, testing and prediction. Residual analysis, diagnostics and remedial measures. Multicollinearity. Model building, stepwise, and other model selection methods. Weighted least squares. Nonlinear regression. Models with qualitative independent variables. One-way analysis of variance. Orthogonal contrasts and multiple comparison tests. Use of existing statistical computing package.
Course Summary: Control charts and acceptance sampling, standard acceptance plans, continuous sampling plans, sequential analysis, and response surface analysis. Use of existing statistical computing packages.
Course Summary: Fundamentals, completely randomized design, and randomized complete blocks. Latin squares, multiclassification, factorial, nested factorial, incomplete blocks, fractional replications, confounding, general mixed factorial, split-plot, and optimum design. Use of existing statistical computing packages.
Course Summary: Real-world projects in biostatistics involving participation in consulting sessions, directed reading in the literature, research ethics, design of experiments, collection of data and applications of biostatistical methods. Detailed written and oral reports required.
Additional Information: May be repeated, up to 6 credits.
Course Summary: Consultation on real-world problems involving statistical analysis under the guidance of a faculty member. A detailed written report and an oral presentation are required. To earn credit for this course the project must have an academic component typically involving some creative application of statistical techniques in an on-going or recently concluded investigation under the supervision of a qualified researcher. To receive approval for this course, the student must write a short proposal describing the scope of the project, methodology to be used and the output to be developed. The student should also clearly state the exact role of all team members involved in the project and give a likely timetable for completion. The project supervisor must send a written consent to supervise the student's project and authorize dissemination of project results to the statistics faculty for their evaluation in determining grade.
Course Summary: A first course in probability intended to serve as a foundation for statistics and other applications. Intuitive background; sample spaces and random variables; joint, conditional, and marginal distributions; special distributions of statistical importance; moments and moment generating functions; statement and application of limit theorems; and introduction to Markov chains.
Course Summary: A basic course in statistical theory covering standard statistical methods and their applications. Includes unbiased, maximum likelihood, and moment estimation; confidence intervals and regions; testing hypotheses for standard distributions and contingency tables; and introduction to nonparametric tests and linear regression.
Course Summary: This course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods.
Additional Information: Course is offered in the Spring of odd numbered years.
Course Summary: See STAT 51900.
Course Summary: Sample spaces and axioms of probability, conditional probability, independence, random variables, distribution functions, moment generating and characteristics functions, special discrete and continuous distributions--univariate and multivariate cases, normal multivariate distributions, distribution of functions of random variables, modes of convergence and limit theorems, including laws of large numbers and central limit theorem.
Course Summary: Sturm-Liouville theory, singular boundary conditions, orthogonal expansions, separation of variables in partial differential equations, and spherical harmonics.
Course Summary: A first course in stationary time series with applications in engineering, economics, and physical sciences. Stationarity, autocovariance function and spectrum; integral representation of a stationary time series and interpretation; linear filtering; transfer function models; estimation of spectrum; and multivariate time series. Use of existing statistical computing packages.
Course Summary: Nonlinear ODEs, critical points, stability and bifurcations, perturbations, averaging, nonlinear oscillations and chaos, and Hamiltonian systems.
Course Summary: Survey designs; simple random, stratified, and systematic samples; systems of sampling; methods of estimation; ratio and regression estimates; and costs. Other related topics as time permits.
Course Summary: Method of characteristics for quasilinear first-order equations, complete integral, Cauchy-Kowalewsky theory, classification of second-order equations in two variables, canonical forms, difference methods of hyperbolic and parabolic equations, and Poisson integral method for elliptic equations.
Course Summary: Models generating binary and categorical response data, two-way classification tables, measures of association and agreement, goodness-of-fit tests, testing independence, large sample properties. General linear models, logistic regression, and probit and extreme value models. Loglinear models in two and higher dimensions; maximum likelihood estimation, testing goodness-of-fit, partitioning chi-square, and models for ordinal data. Model building, selection, and diagnostics. Other related topics as time permits. Computer applications using existing statistical software.
Course Summary: Extension of univariate tests in normal populations to the multivariate case, equality of covariance matrices, multivariate analysis of variance, discriminant analysis and misclassification errors, canonical correlation, principal components, and factor analysis. Strong emphasis on the use of existing computer programs.
Course Summary: Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; and elementary conformal mapping. See Course Web Site for more information.
Course Summary: Generalized linear models, likelihood methods for data analysis, and diagnostic methods for assessing model assumptions. Methods covered include multiple regression, analysis of variance for completely randomized designs, binary and categorical response models, and hierarchical loglinear models for contingency tables.
Course Summary: Ordinary and partial differential equations of physical problems, simplification, dimensional analysis, scaling, regular and singular perturbation theory, variational formulation of physical problems, continuum mechanics, and fluid flow.
Course Summary: Prepares biostatisticians for support of clinical trial projects. Topics: fundamental aspects of the appropriate design and conduct of medical experiments involving human subjects including ethics, design, sample size calculation, randomization, monitoring, data collection analysis and reporting of the results.
Prerequisites: SSTAT 51200, exposure to survival analysis; or consent of instructor.
Course Summary: Divergence theorem, Stokes's Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.
Course Summary: Sufficiency and completeness, the exponential family of distributions, theory of point estimation, Cramer-Rao inequality, Rao-Blackwell Theorem with applications, maximum likelihood estimation, asymptotic distributions of ML estimators, hypothesis testing, Neyman-Pearson Lemma, UMP tests, generalized likelihood ratio test, asymptotic distribution of the GLR test, and sequential probability ratio test.
Course Summary: Foundation of statistical analysis, Bayesian and decision theoretic formulation of problems; construction of utility functions and quantifications of prior information; methods of Bayesian decision and inference, with applications; empirical Bayes; combination of evidence; and game theory and minimax rules, Bayesian design, and sequential analysis. Comparison of statistical paradigms.
Course Summary: Covers a broad range of statistical methods used in many areas of bioinformatics research, including sequence alignment, genome sequencing and gene finding, gene expression microarray analysis, transcriptional regulation and sequence motif finding, comparative genomics, and proteomics.
Prerequisites: STAT 51200, 51900; or consent of instructor.
Course Summary: Complex numbers, holomorphic functions, harmonic functions, and linear transformations. Power series, elementary functions, Riemann surfaces, contour integration, Cauchy's theorem, Taylor and Laurent series, and residues. Maximum and argument principles. Special topics. See Course Web Site for more information.
Course Summary: Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge's theorem, Mittag-Leffler theorem, analytic continuation and Riemann surfaces, and Picard theorems.
Additional Information: Class is only offered during years ending in an even number.
Course Summary: See course listing for STAT 53200.
Course Summary: A basic course in stochastic models including discrete and continuous time processes, Markov chains, and Brownian motion. Introduction to topics such as Gaussian processes, queues and renewal processes, and Poisson processes. Application to economic models, epidemic models, and reliability problems.
Course Summary: Binomial test for dichotomous data, confidence intervals for proportions, order statistics, one-sample signed Wilcoxon rank test, two-sample Wilcoxon test, two-sample rank tests for dispersion, and Kruskal-Wallis test for one-way layout. Runs test and Kendall test for independence, one- and two-sample Kolmogorov-Smirnov tests, and nonparametric regression.
Course Summary: Continuum mechanics deals with the analysis of the motion of materials modeled as a continuous mass rather than as discrete particles. Applications of continuum mechanics are ubiquitous in science and engineering, and are getting more and more popular in medicine too. The goal of this course is to study the basic principles of continuum mechanics for deformable bodies, including conservation laws and constitutive equations, while discussing the mathematical challenges in solving these equations analytically and/or numerically.
Course Summary: Deals with the modern statistical methods for analyzing time-to-event data. Background theory is provided, but the emphasis is on the applications and the interpretations of results. Provides coverage of survivorship functions and censoring patterns; parametric models and likelihood methods, special life-time distributions; nonparametric inference, life-tables, estimation of cumulative hazard functions, and the Kaplan-Meier estimator; one- and two-sample nonparametric tests for censored data; and semiparametric proportional hazards regression Cox Regression, parameters' estimation, stratification, model fitting strategies, and model interpretations. Heavy use of statistical software such as Splus and SAS.
Course Summary: Covers theories, techniques, and applications of partial differential equations, Fourier transforms, and Laplace transforms. Overall emphasis is on applications to physical problems.
Course Summary: Algebras of sets, real number system, Lebesgue measure, measurable functions, Lebesgue integration, differentiation, absolute continuity, Banach spaces, metric spaces, general measure and integration theory, and Riesz representation theorem.
Course Summary: Continues the study of measure theory begun in MATH 54400. See Course Web Site for more information.
Course Summary: Covers modern methods for analysis of repeated measures, correlated outcomes and longitudinal data. Topics: repeated measures ANOVA, random effects and growth curve models, generalized estimating equations (GEE) and generalized linear mixed models (GLMMs). Extensive use of statistical software, e.g. SAS, R.
Course Summary: Banach spaces, Hahn-Banach theorem, uniform boundedness principle, closed graph theorem, open mapping theorem, weak topology, and Hilbert spaces.
Course Summary: Set theory, logic, relations, functions, Cauchy's inequality, metric spaces, neighborhoods, and Cauchy sequence.
Semester(s) Offered: Summer (odd years) with sufficient student interest or as needed. See advisor.
Course Summary: Applications of mathematics to problems in the physical sciences, social sciences, and the arts. Content varies.
Additional Information: May be repeated for credit with the consent of the instructor. Course is offered on an as needed basis.
Course Summary: The first part of the course focuses on numerical integration techniques and methods for ODEs. The second part concentrates on numerical methods for PDEs based on finite difference techniques with brief surveys of finite element and spectral methods.
Prerequisites: MATH 55900 and consent of instructor.
Additional Information: This course is available on an as-needed basis.
Course Summary: Group theory: finite abelian groups, symmetric groups, Sylow theorems, solvable groups, Jordan-H-lder theorem. Ring theory: prime and maximal ideals, unique factorization rings, principal ideal domains, Euclidean rings, and factorization in polynomial and Euclidean rings. Field theory: finite fields, Galois theory, and solvability by radicals.
Course Summary: Review of basics: vector spaces, dimension, linear maps, matrices, determinants, and linear equations. Bilinear forms, inner product spaces, spectral theory, and eigenvalues. Modules over principal ideal domain, finitely generated abelian groups, and Jordan and rational canonical forms for a linear transformation.
Course Summary: This course will broaden the horizon of pure and applied math graduate students in mathematical biosciences. Upon successful completion of this course, students will gain general knowledge of the mathematical methods used in various life-sciences applications. No special knowledge of biology is necessary, but students are expected to be familiar with ODE, PDE and linear algebra.
Course Summary: Computer arithmetic, interpolation methods, methods for nonlinear equations, methods for solving linear systems, special methods for special matrices, linear least square methods, methods for computing eigenvalues, iterative methods for linear systems; methods for systems of nonlinear equations.
Additional Information: Course is offered on an as needed basis.
Course Summary: Projective invariants, Desargues' theorem, cross-ratio, axiomatic foundation, duality, consistency, independence, coordinates, and conics.
Semester(s) Offered: Summer (Even) with sufficient student interest or as needed. See advisor.
Additional Information: Class is available only during Summer II of years ending in an even number and alternates with MATH 56300.
Course Summary: Smooth manifolds, tangent vectors, inverse and implicit function theorems, submanifolds, vector fields, integral curves, differential forms, the exterior derivative, DeRham cohomology groups, surfaces in E3, Gaussian curvature, two-dimensional Riemannian geometry, and Gauss-Bonnet and Poincare theorems on vector fields.
Course Summary: Analysis of axiom systems, finite geometries, critique of Euclid; Hilbert's incidence, betweenness, congruence, and continuity axioms; non-Archimedean geometry, geometry over fields, segment arithmetic and area, non-Euclidean geometry.
Prerequisites: MATH 30000 or consent of instructor.
Additional Information: Class is available only during Summer II of years ending in an even number and alternates with MATH 56100.
Course Summary: Covers the basic notions and theorems of the theory of dynamical systems and their connections with other branches of mathematics. Topics covered include fundamental concepts and examples, one-dimensional systems, symbolic dynamics, topological entropy, hyperbolicity, structural stability, bifurcations, invariant measures, and ergodicity.
Course Summary: Topological spaces, metric spaces, continuity, compactness, connectedness, separation axioms, nets, and function spaces.
Course Summary: Singular homology theory, Ellenberg-Steenrod axioms, simplicial and cell complexes, elementary homotopy theory, and Lefschetz fixed point theorem.
Course Summary: Topics in special functions, representation theory, spectral theory, noncommutative geometry, mathematical foundations of statistical physics. The specific content of the course will vary from year to year, to be determined by the instructor.
Course Summary: Linear systems modeling, mass-spring-damper systems, free and forced vibrations, applications to automobile suspension, accelerometer, seismograph, etc., RLC circuits, passive and active filters, applications to crossover networks and equalizers, nonlinear systems, stability and bifurcation, dynamics of a nonlinear pendulum, van der Pol oscillator, chemical reactor, etc., introduction to chaotic dynamics, identifying chaos, chaos suppression and control, computer simulations, and laboratory experiments.
Course Summary: Logical connectives, rules of sentential inference, quantifiers, bound and free variables, rules of inference, interpretations and validity, theorems in group theory, and introduction to set theory.
Course Summary: A survey and treatment of the content of major developments of mathematics through the eighteenth century, with selected topics from more recent mathematics, including non-Euclidean geometry and the axiomatic method.
Course Summary: Formal theories for propositional and predicate calculus with study of models, completeness, and compactness. Formalization of elementary number theory; Turing machines, halting problem, and the undecidability of arithmetic.
Course Summary: An introduction to set theory, including both so-called "naive" and formal approaches, leading to a careful development using the Zermelo-Fraenkel axioms for set theory and an in-depth discussion of cardinal and ordinal numbers, the Axiom of Choice, and the Continuum Hypothesis. Proofs form a major part of the course and of the homework assignments.
Course Summary: Depending on the interests of the students, the content may vary from year to year. Emphasis will be on mathematical modeling of a variety of physical systems. Topics will be chosen from the volumes Mathematics in Industrial Problems by Avner Friedman. Researchers from local industries will be invited to present real-world applications. Each student will undertake a project in consultation with one of the instructors or an industrial researcher.
Course Summary: Directed study and reports for students who wish to undertake individual reading and study on approved topics.
Course Summary: Introduction to Banach and Hilbert spaces, linear operators, spectral theory of compact linear operators, applications to linear integral equations and to regular Sturm-Liouville problems for ordinary differential equations, boundary value problems, Green's functions, distributions, Fourier transforms, classical PDEs (Laplace, heat, wave equations) of mathematical physics, spectral theory integral equations, and Fredholm Theory.
Course Summary: Continuation of Banach and Hilbert spaces, linear operators, spectral theory of compact linear operators, applications to linear integral equations and to regular Sturm-Liouville problems for ordinary differential equations, boundary value problems, Green's functions, distributions, Fourier transforms, classical PDEs (Laplace, heat, wave equations) of mathematical physics, Fredholm theory.
Course Summary: Probability Theory is the Foundation of statistical methodologies, which is fundamental in the practice of science. From this course students will get a precise mathematical understanding of probabilities and sigma-algebras, random weak convergence, characteristic functions, the central limit theorem, Lobesgue decomposition, conditioning and martingales.
Course Summary: This course covers selected computational techniques useful in advanced statistical applications and statistical research, such as methods for solving linear equations, numerical optimization, numerical integration, Bayesion methods, bootstrap methods and stochastic search algorithms.
Prerequisites: STAT 52100; experience with R/S plus programming.
Course Summary: Nature of applied mathematics, deterministic systems and ordinary differential equations, random processes and partial differential equations, Fourier analysis, dimensional analysis and scaling.
Course Summary: An overview of the drug development process, including the various phases of development from pre-clinical to post-marketing. Topics: statistical issues in design, study monitoring, analysis and reporting. Additional topics may include regulatory and statistical aspects of population pharmacokinetics and real world applications.
Prerequisites: STAT 51200; BIOS S527, S546.
Course Summary: Theories of continuous fields, continuous medium, field equations of continuum mechanics, inviscid fluid flow, viscous flow, turbulence. Additional topics to be dicussed include application of the theory of dynamical systems, methods for analysis of nonlinear ordinary and partial differential equations, and others. This course is an advancement of topics covered in MATH 62600.
Course Summary: Real analysis for inference, statistics and subfields, conditional expectations and probability distributions, UMP tests with applications to normal distributions and confidence sets, invariance, asymptotic theory of estimation and likelihood based inference, U-statistics, Edgeworth expansions, saddle point method.
Course Summary: Discusses the theoretical basis of concepts and methodologies associated with survival data and censoring, nonparametric tests, and competing risk models. Much of the theory is developed using counting processes and martingale methods. Material is drawn from recent literature.
Course Summary: Presents classical and modern approaches to the analysis of multivariate observations, repeated measures, and longitudinal data. Topics include the multivariate normal distribution, Hotelling's T2, MANOVA, the multivariate linear model, random effects and growth curve models, generalized estimating equations, statistical analysis of multivariate categorical outcomes, and estimation with missing data. Discusses computational issues for both traditional and new methodologies.
Course Summary: Advanced topics in functional analysis, varying from year to year at the discretion of the instructor.
Course Summary: Topics in dynamics. Continuation of MATH 56700. See Course Web Site for more information.
Course Summary: Continuation of MATH 57200; cohomology, homotopy groups, fibrations, and further topics.
Course Summary: Continuation of MATH 67200, covering further advanced topics in algebraic and differential topology such as K-theory and characteristic classes.
Course Summary: Topics in inverse spectrum and scattering problems and their applications to the theory of solitons, spectral theory of difference equations and orthogonal polynomials, geometric quantization, and rigorous results in statistical physics, quantum field theory and random processes. The specific content of the course varies from year to year, to be determined by the instructor.
Course Summary: Research topics of current interest in Applied Mathematics to be chosen by the instructor.
Course Summary: Research topics of current interest in Analysis to be chosen by the instructor.
Course Summary: Research topics in Differential Equations related to Physics and Engineering. Topics of current interest will be chosen by the instructor.
Course Summary: Research topics in Topology and their relationships to other branches of Mathematics. Topics of current interest will be chosen by the instructor.
Course Summary: Research required by the graduate students for the sole purpose of writing a Ph.D. Dissertation.