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: The course will focus on rings and modules, including a review of linear algebra, multilinear algebra, basic commutative algebra, and introduction to homological algebra. We will move fast, and a lot of homework should be expected. The non-commutative aspects (Galois theory, representation theory of finite groups, etc.) will be covered in Math 612.
Math 612: Algebra II
Description: This is a second half of the standard graduate algebra sequence. We will concentrate on Galois theory and on basic 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, elliptic functions, 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: This course is a continuation of Math 623, and will cover the following topics: differentiation of functions of bounded variation and the fundamental theorem of calculus for Lebegue-integrable functions, theory of L^p spaces, introduction to functional analysis and operator theory in Banach and Hilbert space, distributional derivatives and Sobolev space, Fourier series representation of L^2 functions and applications to elementary partial differential equations.
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 646: Ergodic Theory
Luc Rey-Bellet (2005)
Description:
Math 651: Numerical Analysis I
Description: The analysis and application of the fundamental numerical methods used to solve a common body of problems in science. The following topics will be included: Methods for solving large linear systems (direct and iterative methods), interpolation and least squares, nonlinear equations and systems, numerical integration and the solving of initial value problems for ordinary differential equations. Grading will be based on homework, exams 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.
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 697EG: Evolutionary Game Theory
Luc Rey-Bellet (2009)
Description: This course will be an introduction to game theory in its modern evolutionary version. We will first discuss the basic static theory of noncooperative game theory (utility theory, Nash equilibria, minimax theorem). We will classify and analyze the space of games and discuss in details a number of simple games such as Prisonner's dilemna, Chicken's game, Rock-Paper-Scissors and so on. We will introduce and analyze various time-evolutions based on the use of game-theoretic ideas in biology, economics , and physics. We will use methods from probability as well as from ODE's to analyze the behavior of the dynamics.
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, 2009)
Description: This course is an introduction to Lie groups, Lie algebras, and their representation theory. Topics will include the following: basics of Lie groups and Lie algebras, structure theory and classification of complex semisimple Lie algebras, and representation 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 697LA: Introduction to Lie Algebras
Eric Sommers (2010)
Description: An introduction to the classification of semisimple Lie algebras and their representation theory. Topics covered include root systems and the Weyl group; survey of the classical Lie algebras; structure theory and classification; and highest weight theory.
Math 697U: Stochastic Processes
Luc Rey-Bellet (2007, 2006, 2010)
Description: This class is an introduction to stochastic processes and Monte-Carlo methods. One of the goal of this class is to develop the mathematical tools to analyze the short-time and long-time behavior of systems subject to random forces. In particular we consider Markov chains on countable state space and discuss the concepts of ineducibility, aperiodicity, recurrence as well as law of large numbers (ergodicidy) and its refinements (central-limit theorom and large deviations). We illustrate the theory with various examples such as random walks, birth and death processes, queueing and branching processes, stochastic Ising models and so on. We will also discuss Monte-Carlo methods which are efficient stochastic algorithms designed to simulate a given probability distribution. If time permits we will discuss Brownian motion and martingales.
Math 697Y/597Y: Nonlinear Dynamical Lattices
Panos Kevrekidis (2009)
Description: The aim of this course will be to give an overview of the mathematical background, physical applications and numerical computations associated with nonlinear dynamical lattices, especially of the Hamiltonian type. We will start from finite dimensional Hamiltonian systems, discuss their symmetries and Lagrangian/Hamiltonian structure, and then extend considerations to infinite dimensional systems of differential-difference equations and subsequently to their PDE analogs. Prototypical case examples of the former type will be the famous FPU model and the discrete nonlinear Schrodinger equation (around which much of the analysis of the course will be developed), while among the latter type, the Korteweg-de Vries equation and the continuum nonlinear Schrodinger equation will be examined. We will analyze the symmetries, conservation laws, solitary wave solutions, linearization spectral properties and dynamics of such equations and attempt to connect them with physical applications from nonlinear optics, fluid mechanics and atomic physics, as well as develop computational tools (such as fixed point, linearization or time-integration algorithms) about how to address them. Grading will be based on homework (50%) and semester-long, research-type projects based on recent literature on the subject of nonlinear dynamical lattices (50%).
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. In the first half of the semester we will discuss differential forms, integration of manifolds and deRham cohomology. The second half will be an introduction to Riemannian geometry.
After discussing the classical theory of surfaces we will study basic notions such as connections, curvature, and geodesics in arbitrary Riemannian manifolds. Additional topics, if time permits.
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 uses the modern methods of analysis to study linear partial differential equations. Emphasis is placed on the theory of existence, uniqueness and stability of solutions to boundary-value problems and initial-value problems. The theory is developed in the context of the prototype PDEs of mathematical physics--the Laplace/Poisson equation, the Heat equation, and the Wave equation. Elements of the theory of distributions are presented along with some functional analysis. Elliptic problems are studies in Hilbert-Sobolev spaces using the variational formulation, and the associated spectral theory is developed. 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. Real 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 791N: Algebraic Number Theory
(2009)
Description: Many classical problems in number theory concern rational solutions of polynomial equations with rational coefficients. But to solve these equations we need to introduce auxiliary algebraic numbers (I.e. roots of polynomials with Q-coefficients). In modern terminology this means working with finite field extensions of the rationals. 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 Dedeking domains, class numbers and class groups, units in rings of integers, valuations and local fields, and zeta-and L-functions.
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 797AT: Algebraic Topology
Tom Braden (2010)
Description: An introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, vector spaces, rings, and homomorphism between them). Emphasis will be placed on being able to compute these invariants, not just on their definitions and associated theorems.
Topics will include: Homotopy, fundamental group and covering spaces, simplical and cell complexes, singular and simplicial homology, long exact sequences and excision, cohomology, Künneth formulas, Poincaré duality.
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 797V: Complex Alg. Surfaces
Eyal Markman (2009)
Description: We develop the theory of (complex) algebraic surfaces, with the goal of
understanding Enriques' classification of surfaces.
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 797Z: Theory of Psuedo-holomorphic Curves
Weimin Chen (2009)
Description: This is a sequel of the topics course taught in Spring 07. I intend to cover the following materials:
1. Basic theory of elliptic PDEs on R^n
2. Fredholm theory of nonlinear analysis on manifolds
3. Local properties of J-holomorphic curves, moduli, and Gromov compactness.
4. Gromov-Witten invariants and Floer homology
5. Applications
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: a) random variables, expectation, independence, laws of large numbers, weak convergence, central limit theorems, and large deviations; b) the concepts of conditional probability and conditional expectation; c) basic properties of certain classes of random processes such as Markov chains 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 likelihood, sampling distributions, point estimation, confidence intervals, and hypothesis testing. Statistical inference will be developed more fully in Stat 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. Stat 607 covered basic probability and basic statistical modelling and inference. Stat 608 covers more advanced probability and statistics topics. As with Stat 607 this is primarily a theory course emphasizing fundamental concepts and techniques, including computing in the R statistical environment
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 691A: Sem-Cross Disciplinary Research
Michael Lavine (2009)
Description: Students will work in teams to collaborate with researchers in other disciplines. Each research project will have a team of two students, one faculty statistician, and one researcher from another discipline. Students will be assigned to teams according to their skills and interests. Each team will work together for one semester and be responsible for its own schedule, work plan, and final report. In addition, the whole class will meet weekly for teams to update each other on their progress and problems. Students will learn about several areas of application and the statistical methods employed by each team. Students in the course will probably learn new statistical methods, a discipline where statistics is applied, how to work collaboratively, how to use R, and how to present oral and written reports.
Stat 691P: Statistics Master's Project
Description: Students will work in teams to collaborate with researchers in other disciplines. Each research project will have a team of two students, one faculty statistician, and one researcher from another discipline. Students will be assigned to teams according to their skills and interests. Each team will work together for one semester and will be responsible for its own schedule, work plan, and final report. In addition, the whole class will meet weekly for teams to update each other on their progress and problems. That way, all students will learn about several areas of application and about the statistical methods employed by each team.
Students in the course will probably learn new statistical methods, a discipline where statistics is applied, how to work collaboratively, how to use R, and how to present oral and written reports. Each team will make an oral presentation to the department's faculty at the end of the semester.
Stat 697B: Bayesian Statistics
John Staudenmayer (2009, 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, 2009)
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, regression methods for data with correlated errors (data over time, repeated measures/random coefficients, hierarchical and mixed models, time-series/cross sectional data) and measurement error in the predictors. Various nonlinear regression problems are also considered, including traditional intrinsically nonlinear regression with constant variance, generalized linear models, including binary regression (eg., logistic, probit and dose/response-quantal bioassay models) and Poisson regression, and nonlinear regressio with correlated/clustered data. Additional topics include bootstrapping and non-parametric regression.
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 describing models and methods. There is abundant data analysis with examples from a variety of disciplines. Computing will be primarily in SAS using (but not limited to) procs REG, AUTOREG, NLIN, LOGISTIC, PROBIT, MIXED, TSCSREG, GENMOD and LOESS. Student's have the option of using R for computing if they prefer.
While the focus is of course on the specific regression models in the outline, the course provides and emphasizes a number of fundamental statistical tools;
including i) the use of matrix/vector notation for expressing models and results and matrix based computational methods ii) the handling of nonlinear functions via the delta method, iii) issues in the use of bootstrapping as a tool for
statistical inference, iv) Wald-type procedures for approximate inferences v) 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 691K/648): 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 697L: Categ. Data Anal
Anna Liu (2010)
Description: Distribution and inference for binomial and multinomial
variables with contingency tables, generalized linear models, logistic
regression for binary responses, logit models for multiple response
categories, loglinear models, inference for matched-pairs and correlated
clustered data. Textbook chapters 1-12 are to be covered.
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: Theoretical development of the following topics: 1) Matrix notation, generalized inverse. 2) The multivariate normal distribution, distributions of linear and quadratic forms of multivariate normal vectors. 3) Estimability of parameters in a linear model, least squares estimates and their statistical properties; Gauss-Markov theorem; simultaneous confidence intervals. 4) Testing linear hypotheses, the full reduced model method. 5) Model selections, non-linear models.
Computation: SAS or its equivalent.
Grading: Class Participation * (20%); Home-Works (20%); 1st Test (15%); 2nd Test (15%); Final Exam (30%). * Any one who misses three or more classes will be disqualified to receive an "A" in this course.
Other Major References:
Generalized Linear, and Mixed Models, 2nd Ed., C. E: McCulloch, S. R. Searle, John M Neuhaus, Wiley 2008.
Theory and Application of the Linear Model, by Franklin A. Graybill, Duxbury, 1976.
Linear Models, by S. R. Searle, Wiley 1971.
The Analysis of Variance, by H. Scheffe, Wiley, 1959.
Linear Regression Analysis, by Seber and Lee, Wiley, 2003.
Stat 706: Linear Models II
Description: 1. Review of major theorems proved in S705. 2) Two-factor fixed effects models: Complete designs and incomplete designs; main effects and interactions; testing for interactions, and interpretations of mail effects if interactions are significant; computation of expected mean squares; power of the F-test and estimation of sample sizes. 3) One and two factor with random effects: Estimation of variance components; best linear unbiased predictors (BLUP); justification for some sums of squares to have chi-square distributions; construction of F-tests; BIB and nested designs. 4) Some Large sample results: Asymptotic LR test, information matrix, Wald tests, score tests. 5) Linear mixed models: Model formulation; estimation techniques - ML, REML, EM, Bayesian, etc;; growth curves analysis. 6) Generalized linear models: Discrete response models (e.g., binary regression); link function, estimation and asymtotic confidence intervals and tests of hypotheses. 7) Multivariate techniques: Principal components, structure models, discrimination.
Stat 725: Estimation Theory and Hypothesis Testing/Advanced Theory of Statistics
John Buonaccorsi (2008), (2009), 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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