The following is a list of some graduate courses that have been offered over the last five years. In general, courses numbered 600-699 are basic graduate courses preparing students to take the basic part of the qualifying exams, while 700-799 are more advanced courses. The exact topics covered by each of these classes may vary from year to year. Statistics courses are listed separately from mathematics courses.
Courses marked with an asterisk (*) are special topics courses, designed by the instructor to lead graduate students to deeper study of a particular area that might lead to thesis research. The other courses are offered at least every other year, and many are offered every year.
Math 611: Algebra I
Description: This course will focus on rings and modules, including a review of linear algebra, topics in multilinear algebra, and the structure theory of finitely generated modules over a principal ideal domain.
Math 612: Algebra II
Description: This course is a continuation of Math 611. Topics covered will include group theory, Galois Theory, and the basics of representation theory.
Math 621: Complex Analysis
Description: We will cover the basic theory of functions of one complex variable, at a pace that will allow for the inclusion of some non-elementary topics at the end. Basic Theory: Holomorphic functions, conformal mappings, Cauchy's Theorem and consequences, Taylor and Laurent series, singularities, residues, other topics as time permits.
Math 623: Real Analysis I
Description: General theory of measure and integration and its specialization to Euclidean spaces and Lebesgue measure; modes of convergence, $L^{p}$ spaces; product spaces; differentiation of measures and functions, signed measures, Radon-Nikodym theorem.
Math 624: Real Analysis II
Description: Continuation of Math 623. Signed measures and differentiation. Introduction to functional analysis including Hilbert spaces; Banach spaces and elementary operator theory. $L^P$-spaces: duality and approximation theorems. Elementary Fourier analysis and applications.
Math 645: ODE and Dynamical Systems
Description: Consider a set LP (the state space) and a map X (the time evolution). A trajectory of the system is the sequence T:X--> X. For interesting systems the question ``What is the state of the system at time x, T(x), T2(x) = T(T(x)), ...?'' is impractical, too hard, and, in a way irrelevant and we should ask instead ``What is the probability that at timex, T(x), T2(x), ... the system is in some specified subset of the state space?'' Of greatest interest is to study the asymptotics as x, T(x), T2(x), ... tends to infinity. This is the subject of ergodic theory. We will introduce and study the concepts of measure-preserving transformation, ergodicity, mixing, entropy. We will use examples taken from dynamical systems, probability, number theory and physics.
Math 651: Numerical Analysis I
Description: This course covers a broad range of fundamental numerical methods, including: nonlinear equations and systems, direct and iterative methods for solving large linear systems, interpolation and least square, numerical integration and the methods solving the initial value problems of ODE's. There will be regular homework, exam and programs.
Math 652: Numerical Solution of PDE's
Description: The following is a brief outline: Boundary and initial value problems in ordinary differential equations (finite difference and shooting methods), Finite difference methods for partial differential equations (consistency, stability and convergence), Finite Element Methods and Spectral Methods. Applications to various contexts from physics, chemistry, and biology will be discussed. Homework and programming will be assigned and the tools to do this type of programming in MATLAB will be developed.
Math 671: Topology I
Description: Mostly point set topology including product and quotient topologies, continuity, compactness, connectedness, and complete metric spaces. Introduction to algebraic topology, specifically the fundamental group. Grade will be based on homework problems and exams.
Math 691Y: Applied Mathematics Project Seminar
Description: This course is the group project that is required for the MS program in Applied Mathematics. Each academic year we undertake an in-depth study of an applied science problem, combining modeling, theory and computation to understand it. The main goal of the course is to emulate the process of teamwork in problem solving, such as is the norm in industrial applied mathematics. The particular topic to be investigated will be announced at the first class meeting.
Math 697A: Number Theory
Paul Gunnells (2004)
Description: This is a one-semester course on modular forms and their applications. Topics to be covered include the modular group and congruence subgroups, quotients of the upper halfplane, basic definition of modular forms, Fourier expansions of modular forms, cusp forms and Eisenstein series, Hecke operators, Petersson product, and modular curves. Additional possible topics include the connection between modular forms and the arithmetic of elliptic curves, modular symbols, and the Eichler-Selberg trace formula.
Math 697AA: Convex Polytopes and Toric Varieties
Tom Braden (2005)
Description: A toric variety is an algebraic variety built using data from a convex lattice polytope in Euclidean space. Toric varieties form a rather special class of varieties, but they are general enough to contain a lot of interesting examples, and many things such as cohomology and line bundles can be explicitly computed in terms of the geometry of the polytope. On the other hand, it is also possible to work the other way and study convex polytopes using the geometry of toric varieties. We will use Ewald and Fulton as basic references, together with other sources including recent research papers. After we learn the basic definitions, the course will be run as a seminar. There are a lot of different applications of toric geometry, and which ones we spend the most time on will depend partly on student interest.
Math 697AG: Introduction to Algebraic Geometry
Eyal Markman (2007)
Description: This will be the first semester of a year long introductory course in algebraic geometry. The Spring 2008 semester will be taught by Prof. Tevelev. Topics include: 1) Introduction: Algebraic sets, Zariski Topology, Nullstellensatz. Hilbert basis theorem. 2) Affine varieties, projective varieties, morphisms, birational maps, dimension, nonsingularity. 3) Curves: Valuation rings, completions, integral closure, discrete valuation rings. Extending maps of nonsingular curves to projective varieties. 4) Divisors, line bundles, linear systems, and morphisms to projective space. Picard group. Class group. Degree. 5) Riemann-Roch for projective algebraic curves and applications.
Math 697B: Introduction to Riemann Surfaces
Eyal Markman (2006), Tom Weston (2008)
Description: We will start with a gentle introduction to complex affine and projective algebraic curves, with emphasis on plane curves. Highlights include: Bezout's theorem (on the degree of the intersection of two curves in the projective plane), the group law on a plane cubic curve, and the degree-genus formula for a smooth plane curve. We will then proceed to the theory of compact Riemann surfaces. A deep result states, roughly, that any compact Riemann surface is a (projective) algebraic curve. The study of Riemann surfaces uses complex analysis, while the study of algebraic curves uses algebra. We will develop the language of line bundles and sheaves on Riemann surfaces and their sheaf cohomology. Highlights include: elliptic functions, the Riemann-Roch and Serre's Duality Theorems, the Abel-Jacobi Theorem, and their many applications.
Math 697D: Introduction to Algebraic Groups
Eric Sommers (2005)
Description: Algebraic groups sit at the intersection of algebraic geometry and group theory and are a useful tool in representation theory, number theory, and algebraic geometry. In this course we will cover the classification of reductive algebraic groups. We will then survey some applications, including the Borel-Bott-Weil Theorem.
Math 697G: Introduction to Algebraic Geometry
Tom Weston (2006)
Description: This course will be an introduction to algebraic geometry, a subject which is fundamental to much of mathematics. Topics to be covered include: algebraic curves, algebraic varieties, rational functions, dimension, singularities, divisors, differential forms, the Riemann-Roch Theorem, intersection theory on surfaces.
Math 697I: Introduction to Financial Engineering
Mike Sullivan (2006)
Description: This course is an introduction to the theory and applications of stochastic processes. Topics include Markov chains, Poisson processes, continuous-time Markov processes, Brownian motion. In addition, if time permits, selected applications from simulation of Markov processes, hidden Markov models, queuing theory, etc., will be discussed.
Math 697J: Multigrid/Multilevel Methods
Hans Johnston (2006)
Description: The course deals with multigrid/multilevel methods for problems with many variables, especially numerical solution of elliptic partial differential equations. The motivation and applications are from various fields of scientific computing, including image processing and analysis, PDEs, CFD, elasticity, statistical mechanics, QCD, integral equations, etc., The course topics are: basic concepts, local and global processing, discretization of PDEs, 1D model problems and their direct and iterative solution, convergence analysis, 2D model problem, survey of classical relaxation methods, error-smoothing by relaxation, grid-refinement algorithms, two-grid and multigrid algorithm, Fourier analysis of convergence, ellipticity, nonlinear and anisotropic problems, advance techniques, and applications. While rigorous theory is at the heart of any successful numerical method, the emphasis of the course will be the hands-on implementation of the methods. To facilitate this, MATLAB will be used for all examples and homework. It is striking the problems that can be solved with just a few lines of code in MATLAB. Grades will be based on homeworks, midterm and final exam.
Math 697K: Lie Groups
Paul Gunnells (2007)
Description: This course is an introduction to Lie groups, Lie algebras, and their representation theory. Topics will potentially include the following: basics of Lie groups and Lie algebras, structure theory and classification of complex semisimple Lie algebras, and repreentation theory of Lie algebras and Lie groups. Throughout the course emphasis will be placed on explicit examples to illustrate the general theory.
Math 697L: Wave-Packet Analysis and Applications
Andrea Nahmod (2007)
Description: Wave-packet - or time-frequency - analysis is based on the fundamental fact that a function can - up to Heisenberg''s uncertainty principle - be "simultaneously" analyzed both in the physical and frequency variables ({it phase space } ). Roughly, it consists in decomposing complex structures into basic building blocks - which are localized in phase space and easier to understand - via modulated waveforms. Then, piecing them back together in a straightforward manner. The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, wave propagation and other phenomena of nonlocal nature. In this course we will lintroduce and develop a working knowledge of wave-packet analysis and its phase space representation. We will then show how one can use this analysis to obtain results both in Analysis and Partial Differential Equations. {f Some topics will include:} Littlewood-Paley theory and Multiscale analysis. Wavelet and wavelet-packet decompositions; Fourier series vs. wavelet series. Special examples: Haar basis, Shannon''s basis, Walsh wave-packets; Meyer''s wavelets and wavelet-packets; Daubechies'' wavelets. Wavelet description of function space (BV, Sobolev, Besov, etc.). Product and other bilinear estimates and applications (div-curl, fractional Leibnitz rule, null forms of wave equations, etc). The role of oscillations in certain nonlinear evolution equations; the Schr"{o}dinger equation, the Navier-Stokes equations; other nonlinear PDE. The role of modulation invariance: Carleson''s theorem on Fourier convergence. The analysis of singular multipliers and other bilinear pseudo-differential operators. Other related topics will be intertwined as time permits.
Math 697U: Stochastic Processes
Luc Rey-Bellet (2007, 2006)
Description: This course is an introduction to the theory and applications of stochastic processes. Topics include Markov chains, Poisson processes, continuous-time Markov processes, Brownian motion. In addition, if time permits, selected applications from simulation of Markov processes, hidden Markov models, queuing theory, etc., will be discussed.
Math 703: Theory of Manifolds I
Description: An introduction to the basic concepts of Differential Geometry, Differential Topology and Lie Theory. Topics include: A review of differential maps between Euclidean spaces, Inverse and Implicit Function Theorems. Differentiable manifolds, definition and examples. Regular and critical values, Sard's Theorem. submanifolds, immersions and embeddings. Vector bundles, tangent and cotangent bundles. Vector fields, ODE's on manifolds, Lie bracket, integrable distributions, Frobenius Theorem. Differential forms, exterior differential.
Math 704: Theory of Manifolds II
Description: This course is the continuation of Math 703 We will discuss differential forms, integration of manifolds and deRham cohomology, as well as basic Riemannian geometry (connections and curvature). If time permits, we may also do some Morse theory.
Math 713: Introduction to Algebraic Number Theory
Farshid Hajir (2005), Paul Gunnells (2008)
Description: An algebraic number field is a field obtained by adjoining to the rational numbers the roots of an irreducible rational polynomial. Algebraic number theory is the study of properties of such fields. This course will cover the basics of algebraic number theory, with topics to be studied possibly including the following: number fields, rings of integers, factorization in Dedekind domains, class numbers and class groups, units in rings of integers, valuatons and local fields, and zeta-and L-functions.
Math 731: Partial Differential Equations I
Description: This course introduces the modern methods of analysis in partial differential equations. Emphasis is placed on the theory of existence, uniqueness and stability of solutions to boundary-value problems and initial-value problems. This theory is developed in the context of the prototypical PDEs arising in mathematical physics -- The Laplace/Poisson equation, the Heat/Diffusion equation, and the Wave equation. The theory of distributions is presented along with some functional analysis. Elliptic problems are studied in Hilbert-Sobolev spaces, using the variational formulation. Parabolic problems are considered in the setting of the analytic theory of semigroups. Hyperbolic problems are treated with energy methods. The course assumes that the student has some familiarity with the elementary methods of solution of linear ODEs and PDEs. Modern analysis at the first-year graduate level is presumed.
Math 732: Linear/Nonlinear Waves
Description: This course will continue the first semester course, Math 731, which concentrates on prototype linear PDEs. In the second semester we will build on and extend the first semester in two ways. First we will consider the theory of general classes of PDE of elliptic, parabolic and hyperbolic type. Second, we will study some nonlinear problems in PDEs motivated by real applications. An outline of topics follows:
1. Theory of linear hyperbolic systems. Existence, uniqueness and regularity of solutions. Analysis of waves in special systems.
2. Linear and quasi-linear elliptic equations. Variational techniques for nonlinear elliptic boundary-value problems.
3. Parabolic equations and initial value problems. Regularity properties of solutions.
4. Hamilton-Jacobi equations, and viscosity solutions. Applications and motivations in optimal control theory.
5. If time permits, the Navier-Stokes equations of fluid dynamics. Existence and regularity of solutions.
Math 781: Algebraic Topology
Description: Topics to be covered include: Fundamental group, covering spaces, homology of chain complexes and homology theories (singular homology and cohomology, deRham cohomology), homology of CW complexes. Duality theorems, time permitting.
Math 791A: Non-linear Fourier Analysis
Andrea Nahmod (2003)
Description: The purpose of this course is to give a self contained introduction to recent results and techniques concerning nonlinear evolution equations and the behavior of their solutions. We will mainly focus on nonlinear wave and Schrödinger equations. We will start with the classical local existence theory. Then we move to modern techniques involving nonlinear Fourier analysis and geometric consolidation that allow the regularity on the data to be weakened considerably and still get existence and uniqueness results. The course will mix lectures once a week with student's presentations. It will proceed as a working seminar.
Math 797AG: Algebraic Geometry II
Jenia Tevelev (2008)
Description: This will be the second part of an introductory course in Algebraic Geometry. The main goal will be to master cohomological machinery and its applications roughly at the level of Chapter III of Hartshorne's textbook. Some topics not covered in Hartshorne will include Hilbert schemes and more generally Grothendiecks functorial approach to algebraic geometry. Students taking this class willl be encouraged (but not required) to participate in the reading seminar on Algebraic Geometry that meets on Fridays at 2:30. In Spring we will discuss recent important results in Algebraic Geometry on the level accessible to graduate students taking 797AG.
Math 797C: Theory of Large Deviations and Applications
Richard S. Ellis (2005)
Description: The theory of large deviations studies the exponential decay of probabilities in certain random systems. It is an extremely active branch of probability today, having applications to numerous areas including statistics, statistical mechanics, queueing systems, communication networks, information theory, and risk-sensitive control. This course will be an introduction to the theory. It will cover the following topics: Convex analysis and the Cramér theory for sums of i.i.d. random variables; The three levels of large deviations: sample means, empirical measures, empirical processes; Some generalities about large deviations; A new approach to large deviations based on the theory of weak convergence of probability measures; Applications to statistical mechanics and other areas. A central concept in the course is entropy, which arises as a bridge between a microscopic and a macroscopic description of random systems. The course will begin by presenting the basic concepts of probability theory. Although the course is intended for non-experts, it will cover some exciting recent material that could lead to a dissertation.
Math 797D: Topology and Geometry of Singular Spaces
Paul Gunnells (2006)
Description: Singular spaces arise naturally in many contexts, including algebraic geometry and representation theory. Any singular space has a decomposition into manifolds, and so the study of them is a mixture of topology, geometry, and combinatorics. Topics to be studied possibly include manifolds with boundary, manifolds with corners, Whitney stratifications, links, normal slices, the Thom presentation of a stratified space, orbifolds, hypersurface singularities, symmetric spaces and their compactifications.
Math 797E: Homological Algebra
Ivan Mirkovic (2006)
Description: Homological algebra is a general tool useful in various areas of mathematics. One tries to apply it to constructions that morally should contain more information then meets the eye. The homological algebra, if it applies, produces "derived" versions of the construction ("the higher cohomology"), which contain the "hidden " information. The goal is to understand the usefulness of homological ideas in applications and to use this process as an excuse to visit various interesting topics in mathematics.
Math 797F: Spectral Theory
Wei-Min Wang (2006)
Description: We start from introducing basic notions in spectral theory. Some emphasis will be put on matrix operators on $ell^{2}(mathbf{Z}^{d})$, which has the advantage of illuminating the basic ideas without the distraction of some technical issues. We will then study equations coming from quantum mechanics, where there is an underlying dynamical system. This has connections to nonlinear Schr"{o}dinger equations. We will work with concrete objects such as the quantum harmonic oscillator and the discrete Laplacian. The purpose of this course is to bring to attention some of the newer ideas and tools arising in the subject, which as of now are mainly scattered in the literature.
Math 797G: Zeta Functions of Algebraic Varieties
Siman Wong (2007)
Description: In the 1940s Weil discovered that the congruence properties of systems of polynomials with integer coefficients are closely related to the topology and geometry of the {it complex} solution sets of these polynomials. In the 1960s Tate put forth an influential conjecture relating the {it rational solutions} of such polynomials to their congruence properties modulo all integers. These works are formulated in terms of the {it zeta functions } associated to these polynomials. In the 1970s Langland conjectured that these zeta functions are in turn related to the representation theory of linear groups Since then this circle of ideas has been the guiding theme behind much of modern research in number theory, algebraic geometric and representation theory. The goal of this course is to survey these intertwining ideas, with particular emphasis on analyzing concrete examples.
Math 797I: Geophysical Fluid Dynamics
Bruce Turkington (2008)
Description: We will derive the partial differential equations that describe the motions of layers of fluid on a rotating sphere. These equations govern the dominant patterns of weather in the atmosphere and circulation in the oceans. Over the time scales of days they are the basis of predictive meteorology, and over much long time scales and large space scales they are used to model the evolution of global climate. Our presentation will begin from the basic principles of physical science, and we will give full derivations of the governing equations from those principles. Then we will study various regimes of motion that have particular significance for the dynamics of the atmosphere or oceans. In the process we will develop some of the standard tools of advanced applied mathematics, such as perturbation expansions and stability analysis, and we will encounter model building in a relatively sophisticated context. If time permits, we will discuss simple climate models and relate them to the contemporary debate on Earth's climate change. This course will be suitable for mathematics graduate students (who may not have any background in fluid dynamics) and for graduate students from the physical sciences or engineering who are familiar with the standard techniques of applied mathematical analysis.
Math 797U: Representations of semisimple Lie algebras
Jim Humphreys (2003)
Description: The representation theory of semisimple Lie algebras (over the real or complex numbers) has become part of the standard core of modern mathematics, with connections to physics and to many areas of pure and applied mathematics. The classical Cartan-Weyl theory is well-developed but several closely parallel theories (Kac-Moody algebras, -adic representations, quantum groups, modular Lie algebras) are less completely understood. To approach the newer theories through the old one, it is best to work in the module category introduced around 1970 by Joseph Bernstein, I. M. Gelfand, S. I. Gelfand (the BGG category). Starting with a quick overview of simple Lie algebras (based on standard examples), we will derive the classical theorems in an efficient way by studying composition factor multiplicities in Verma modules. Weyl's character formula and Kostant's equivalent formula are easy special cases. But the complete story about multiplicities involves the 1979 Kazhdan - Lusztig Conjecture, which was proved in 1980 using deep results from algebraic geometry (intersection cohomology, perverse sheaves) and has led to analogous open questions in other parts of Lie theory. Further algebraic details about the BGG category will be discussed: projective modules, Jantzen's translation functors and filtrations, Loewy series. This leads toward more recent work on Koszul duality due to Beilinson - Ginzburg - Soergel. Students will be asked to write up some exercises and to make presentations in class.
Math 797W: Algebraic Geometry
Ivan Mirkovic (2004)
Description: This course is an introduction to the vocabulary and methods of algebraic geometry, geared towards the use of algebraic geometry in various areas of mathematics: number theory, representation theory, combinatorics, mathematical physics. The basic vocabulary will evolve from systems of polynomial equations to algebraic varieties and schemes. We will also get introduced to the topology of algebraic varieties: cohomology and algebraic cycles. Our first example will be algebraic curves. This is the best understood part of algebraic geometry since it deals with one-dimensional objects. The highlights: Riemann Roch theorem and the relation to number theory (``geometric class field theory''). The second example will be the flag varieties, i.e., homogeneous compact algebraic varieties. We will consider maps of curves into flag varieties, a topic of current interest related to mathematical physics (``quantum cohomology'') and representation theory.
Math 797X: Riemann Surfaces and Integrable Systems
Franz Pedit (2005)
Description: This course will explore the relevant role Riemann surfaces play in integrable system theory. We will start with a simple geometric problem of polygon evolution in the plane and attach a finite genus Riemann surface to any closed polygon. The evolution is then governed by the Jacobi variety of this Riemann surface and theta functions explicitly parameterize the flow. This is a prototypical situation for the kind of integrable systems related to Riemann surfaces: the phase space consists of Riemann surfaces (of some type) together with their Jacobians as energy shells. If time permits, we will also discuss how problems of smooth geometry (Harmonic maps, self-duality equations, conformal surfaces, KdV equation) fit into this picture. The students are expected to have taken a Riemann surfaces class and preferably at least one semester of the manifold course. Other than that the course is self-contained with emphasis on the basic ideas rather then the most general techniques.
Math 861: Complex Manifolds & Physics
(2003)
Description: This course runs in parallel with the GANG seminar, and involves students reporting on current research topics, as well as lectures by both outside speakers and the faculty member in charge. This spring we plan to work on; geometric analysis of knots and links, with applications in topology, fluid dynamics and microbiology; geometric Brownian motion, with applications in potential theory, Reimann surfaces and minimal surfaces.
Math 897A: Symplectic Topology and Pseudo-Holomorphic Curves
Weimin Chen (2007)
Description: The purpose of this course would be to introduce the students to the basic concepts and results in symplectic geometry/topology, as well as some fundamental materials in Gromov''s theory of pseudo-holomorphic curves. The hope is that after completion of this course, the student would have a necessary background in order to further pursue their research interests in the related ares, such as symplectic or contact topology/geometry, Gromov-Witten invariants, Floer homology, etc. The materials covered in this course may also serve as a useful background in symplectic geometry for those who are interested in algebraic geometry, differential geometry, low-dimensional topology, or mathematical physics. The first half of the course would be on symplectic geometry, which includes: basic definitions and examples, the linear theory (symplectic spaces, Lagrangian subspaces, symplectic vector bundles, compatible almost complex structures), the local structural theorems (Darboux''s theorem, Moser''s stability, symplectic neighborhood theorem), some basic constructions (symplectic reduction, blowing up and down, symplectic connected sum). The second half would be on pseudo-holomorphic curves, which includes: basic definitions and local properties of pseudo-holomorphic curves, Fredholm theory and transversality of moduli space, Gromov''s compactness theorem, pseudo-holomorphic curves in dimension 4, some applications in symplectic topology.
Stat 605: Probability Theory
Description: The subject matter of probability theory is the mathematical analysis of random events, which are empirical phenomena having some statistical regularity but not deterministic regularity. The theory combines aesthetic beauty, deep results, and the ability to model and to predict the behavior of a wide range of physical systems as well as systems arising in technological applications. In order to properly handle applications involving continuous state spaces, a measure-theoretic treatment of probability is required. The purpose of this course is to present such a treatment, which is based on Kolmogorov's axiomatic approach. Topics to be covered include the following: Random variables, expectations, independence, laws of large numbers, central limit theorems, and large deviations, the concepts of conditional probability and conditional expectation, basic properties of certain classes of random processes such as martingales and random walks.
Stat 607: Mathematical Statistics I
Description: The first part of a two-semester graduate level sequence in probability and statistics, this course develops probability theory at an intermediate level (i.e., non measure-theoretic - STAT 605 is a course in measure-theoretic probability) and introduces the basic concepts of statistics. Topics include: general probability concepts; discrete probability; random variables (including special discrete and continuous distributions) and random vectors; independence; laws of large numbers; central limit theorem; statistical models and sampling distributions; and an introduction to point estimation, confidence intervals, and hypothesis testing. Statistical inference will be developed more fully in ST 608. This course is suitable for students in a wide variety of disciplines and will give strong preparation for further courses in statistics, econometrics, and stochastic processes, time series, decision theory, operations research, etc.
Stat 608: Mathematical Statistics II
Description: This is the second part of a two semester sequence on probability and mathematical statistics. ST607 covered probability, basic statistical modelling, and an introduction to the basic methods of statistical inference with application to mainly one sample problem. In ST608 we pick up some additional probability topics as needed and examine further issues in methods of inference including more on likelihood based methods, optimal methods of inference, more large sample methods, Bayesian inference and decision theoretic approaches. The theory is utilized in addressing problems in nonparametric methods, two and multi-sample problems, and categorial, regression and survival models. As with ST607 this is primarily a theory course emphasizing fundamental concepts and techniques.
Stat 640: Sampling Theory and Methods
Description: Introduction to the basic theory and practice of sampling from finite populations. Designs covered include simple random sampling, stratified sampling, systematic sampling, cluster sampling, sampling with unequal probabilities, multi-stage and double sampling. Coverage also includes ratio and regression estimators, the jackknife and other techniques for variance estimation, determination of samples sizes and optimal allocations, nonresponse, regression in complex surveys. The course includes an applied component, with applications and data analysis using SAS, and possibly other packages.
Stat 691P: Statistics Master's Project
Description: This course is designed for students to complete the master's project requirement in statistics, with guidance from faculty. The course will begin with determining student topics. Each student will complete an individual project and possibly a group project or projects. Regular class meetings will involve student presentations on progress of projects, with input from instructors. The project can take many forms; an expository report on a particular area, an examination of methods through simulations, or a detailed statistical analysis of real data. Each project will require a computational component. Final written reports are required as well as an oral presentation.
Stat 697B: Bayesian Data Analysis
John Staudenmayer (2008)
Description: This course will introduce students to Bayesian data analysis - including modeling and computation. We will begin with a description of the components of a Bayesian model and analysis (including the likelihood, prior, posterior, conjugacy, non-informativeness, credible intervals, etc) and illustrate these objects in simple models. After that, we will develop Bayesian approaches to more complicated models including linear regression, mixture models, and generalized linear models. The course will introduce Monte Carlo Markov Chain methods, and students will have the opportunity to learn to use the BUGS and R open source statistical packages for computation. Grading will be based on written and computational problem sets and a final project.
Stat 697F: Topics in Regression
John Buonaccorsi (2007)
Description: This course examines a variety of additional topics in regression beyond those traditionally found in a first regression course. Within the multiple linear regression framework, topics will include models with changing variances and the use of generalized/weighted least squares; models with correlated errors (data over time, repeated measures/random coefficients, hierarchical and mixed models, time-series/cross sections data); measurement error/errors-in-variables in the regressors. A collection of nonlinear type problems will then be considered. These will include traditional intrinsically nonlinear regression with constant variance; nonlinear regression models arising from the perspective of generalized linear models including binary regression (eg. logistic, probit and dose/response-quantal bioassay models) and Poisson regression. Additional topics include bootstrapping, non-parametric regression and regression with survival data. This is a mainly applied course with a focus on a careful description of the models and methods, how they are applied, and on computational aspects. This is not a theory course proving why everything works, although we do make liberal use of matrix/vector notation in expressing models and methods. There is plenty of data analysis using examples from a variety of disciplines. Computing will be primarily in SAS using (but not limited to) procs REG, NLIN, LOGISTIC, PROBIT, MIXED and TSCSREG. While the focus is of course on the specific models in the outline, the course also provides some fundamental tools for doing statistcs, whether for regression or other types of problems. These tools (most alluded to above) include i) the use of matrix/vector notation for expressing models and results ii) matrix based computational methods iii) the handling of nonlinear functions via the delta method, iv) the use of boostrapping as a fundamental tool for statistical inference, v) Wald-type procedures for approximate inferences vi) least squares and iteratively reweighted least squares techniques, vi) likelihood based methods of inference and associated notions of information.
Stat 697G: Survival Analysis
H. K. Hsieh (2005)
Description: Survival analysis, also known as reliability analysis, deals with incomplete (censored) data, where an individual's life length is known to occur only in a certain period of time, or individuals are included in the study only if the event has occurred by a given date. We will discuss estimation and hypothesis tesing issues on the survival and hazard functions for both parametric and non-parametric models: quantiles, proportional hazard regression models, multivariate survival analysis, system reliability, and their applications in biology, medicine and engineering. Computer package and basic computer programming will be introduced. Students' course grades will be based on class participation, homeworks, mid-term exam, and final take-home exam.
Stat 697K: Bioinformatics
Erin Conlon (2006, 2007)
Description: this course is an introduction to the biological, computational and statistical foundations necessary for bioinformatics-related research. We specifically focus on DNA and protein sequence analysis, and microarrays. This course is cross-listed with Mathematics and Statistics, Microbiology and Computer Science. while there are no formal prerequisites, some level of familiarity with molecular biology, statistics, and/or computer programming is recommended.
Stat 697K (See Computer Science 660/691K): Bioinformatics
Erin Conlon (2008)
Description: Credit will be given for either Statistics, Computer Science, or Microbiology (see individual departments). Undergraduate are welcome with permission of instructor. this course is an introduction to the biological, computational and statistical foundations necessary for bioinformatics-related research. We specifically focus on DNA and protein sequence analysis, and microarrays. While there are no formal prerequisites, some level of familiarity with molecular biology, statistics, and/or computer programming is recommended. There will be at least one assignment involving some basic-level computer programming.
Stat 697M: Measurement Error: Methods and Applications
John Buonaccorsi (2007)
Description: In many problems some of the variables in terms of which our model of interest is defined are unobservable; that is there is measurement error (or in the case of a categorical variable, misclassification) present. There is a long history of consideration of such "errors-in-variables" problems for linear regression in both the statistical and econometrics literature. Since the late 80''s there has been explosion of work (much of it motivated by applications in epidemiology but with expanded use in many other disciplines including ecology) which has considerably expanded the treatment of these problems by consideration of larger classes of models for both the underlying true values and the nature of the measurement error. This course examines the impacts of measurement error and methods to correct for measurement error in a variety of settings. The primary focus will be on 1. Estimating proportions for single or multiple samples or in contingency tables in the presence of misclassification. 2. Fitting linear and non-linear regression models with measurement error in the predictor(s) and/or response (this includes some design type models with error in the response only as well as misclassification of categorical predictors). In both settings corrections for measurement error are examined under various ways in which additional information/data is used to estimate the measurement error parameters, including the use of replicates, and internal or external validation data. The use of instrumental variables is also considered for regression problems. Connections to certain aspects of missing data/imputation, double sampling and latent variable models will be made. We will also survey some extensions to other areas including measurement error in mixed models and time series, with the requisite background for these topics provided as needed. Computing will be done using SAS and/or R as well as STATA, which has some of the measurement error corrections procedures built in.
Stat 697R: Regression Modeling
John Staudenmayer (2006, 2007), John Buonaccorsi (2005)
Description: Topics covered include simple and multiple linear regression; correlation; inverse prediction; the use of dummy variables; residuals and diagnostics; model building and variable selection; weighted least squares; the development of regression models and methods in matrix form; an introduction to nonlinear regression. Focus is on a careful understanding and presentation of regression models and methods, interpretation of results and data analysis/statistical computation using SAS (no prior experience assumed). A matrix formulation of the linear regression model is developed during the course to facilitate the presentation of models and results. The necessary basic matrix tools will be given in the course. (Note: While ST697R covers many of the same topics as ST505 it will differ in a number of ways. The pace of the course will be a bit quicker to allow coverage of additionial topics. More importantly, there will be more in the way of applications and computing, including more emphasis on choosing models and interpreting analyses.)
Stat 705: Linear Models I
Description: ST705 is the first of two courses which treat the theory of linear models. The first part of ST705 provides some general background for handling linear models covering i) a review and introduction of some new topics in linear algebra, ii) a discussion of random vectors, multivariate distributions, the multivariate normal, linear and quadratic forms including an introduction to non-central distributions. We then develop the basic theory for estimation, confidence intervals and hypothesis testing, for the general linear model, with a focus on regression models of full rank. Additional topics include model selection, correlation, calibration, residual, analysis and diagnostics, prediction, computational aspects. The applied part of the course is not directed towards extensive data analysis (available in ST505/697R) but in understanding and choosing models and computational implementation of the methods.
Stat 706: Linear Models II
Description: The course follows ST705 and depends on the general theory developed in the earlier part of that course. It differs in that the focus is now on "analysis of variance/design of experiments" models. Coverage includes one-way and multi-way factoral designs; inferences in linear models of less than full rank; reparameterizations; randomized block designs; incomplete designs including latin squares; random effects, nested and mixed models; randomization. Possible additional topics include analysis of covariance, fractional factorials and response surface methodology. The applied part of the course will focus on integrating the theory with computational aspects, including aspects of proc GLM and MIXED (including a careful understanding of how these procedures function) and the use of IML in SAS.
Stat 725: Estimation Theory and Hypothesis Testing/Advanced Theory of Statistics
John Buonaccorsi (2008), John Staudenmayer (2005)
Description: Point and interval estimation, small sample and asymptotic properties of maximum likelihood estimators, generalized estimating equations; bootstrap estimates and confidence intervals; hypothesis testing, optimality properties; basic decision theory, Bayes rules, minimaxity. Applications to nonlinear regression, generalized linear models.
Stat 797M: State Space Markov Models
Michael Lavine (2008)
Description: Data collected at different times or at different locations often have this property: observations that are nearby, in either time or space, are more similar to each other than to distant observations. This course introduces two classes and models. Dynamic Linear Models and Markov Random fields, to model and account for such similarities. After introducing the basic models we will study Bayesian analyses of the models, use them on real data, and introduce generalizations to make them more widely applicable. Students will be expected to use packages and write programs in the statistical software R.
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