Graduate School of Arts and Sciences - Mathematics and Statistics

Department of Mathematics and Statistics

The Graduate Program
MA in Mathematics, Including Statistics
MA in Mathematical Finance
MA in Actuarial Science
PhD in Mathematics, Including Statistics and Probability
Courses
Directed Study and Research
Related Courses in Other Departments

The following list reflects the 2006/2007 faculty.

Chairman Ralph B. D’Agostino

Associate Chairman Ashis Gangopadhyay

Director, Graduate Studies Program  Paul Blanchard

Director, Undergraduate Mathematical Instruction
 Glen Richard Hall

Director, Statistics Program  
Eric Kolaczyk

Director, Statistics and Consulting Unit  
Ralph B. D’Agostino

Executive Director, Biostatistics Program  
Ralph B. D’Agostino

Director, Mathematical Finance Program  
Andrew Lyasoff

Associate Director, Mathematical Finance Program
Paolo Guasoni

Faculty

Paul Blanchard Director, Graduate Studies; Associate Professor of Mathematics, College of Arts and Sciences. BA, Brown University; PhD, Yale University

Gail A. Carpenter Director of Graduate Studies, Department of Cognitive and Neural Systems; Professor of Mathematics and Cognitive and Neural Systems, College of Arts and Sciences. BA, University of Colorado; MA, PhD, University of Wisconsin, Madison

Ralph B. D’Agostino Chairman, Department of Mathematics and Statistics; Director of Statistics and Consulting Unit; Executive Director, Biostatistics Program; Director, Statistics and Research Unit, Center for Psychiatric Rehabilitation; Professor of Mathematics and Statistics, College of Arts and Sciences; Professor of Biostatistics and Epidemiology, School of Public Health. BA, MA, Boston University; PhD, Harvard University

Robert L. Devaney Professor of Mathematics, College of Arts and Sciences. BA, College of the Holy Cross; PhD, University of California, Berkeley

Uri Tzvi Eden Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BS, California Institute of Technology; SM, PhD, Harvard University

David Fried Professor of Mathematics, College of Arts and Sciences. BA, MS, University of Chicago; PhD, University of California, Berkeley

Isaac Fried Professor of Mathematics, College of Arts and Sciences. Bsc, Technion, Israel Institute of Technology; MScAe, Ecole Nationale Supérieure de l’Aeronautique (France); PhD, Massachusetts Institute of Technology

Ashis Gangopadhyay Associate Chairman, Department of Mathematics and Statistics; Associate Professor of Mathematics and Statistics, College of Arts and Sciences. BStat, MStat, Indian Statistical University (India); PhD, University of California, Davis

Stephen Grossberg Chairman, Department of Cognitive and Neural Systems; Director, Center for Adaptive Systems; Wang Professor of Cognitive and Neural Systems, Professor of Mathematics and Psychology, College of Arts and Sciences; Professor of Biomedical Engineering, College of Engineering. BA, Dartmouth College; MS, Stanford University; PhD, Rockefeller University

Paolo Guasoni Associate Director, Mathematical Finance Program; Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. Laurea in Mathematics, University of Pisa (Italy); MA, Columbia University; PhD, Scuola Normale Superiore (Italy)

Glen Richard Hall Director, Undergraduate Mathematical Instruction; Associate Professor of Mathematics, College of Arts and Sciences. BA, Carleton College; PhD, University of Minnesota

Thomas Hawkins Professor of Mathematics, College of Arts and Sciences. BA, Houghton College; MS, University of Rochester; PhD, University of Wisconsin

Akihiro Kanamori Professor of Mathematics, College of Arts and Sciences. BS, California Institute of Technology; PhD, University of Cambridge (England)

Tasso Kaper Professor of Mathematics, College of Arts and Sciences. BSc, University of Chicago; PhD, California Institute of Technology

Konstantinos Kardaras Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BSc, MSc, University of Athens (Greece); PhD, Columbia University

Takashi Kimura Associate Professor of Mathematics, College of Arts and Sciences. BA, California Institute of Technology; PhD, State University of New York at Stony Brook

Eric Kolaczyk Director, Statistics Program; Associate Professor of Mathematics and Statistics, College of Arts and Sciences. BS, University of Chicago; MS, PhD, Stanford University

Mark Kon Professor of Mathematics, College of Arts and Sciences. BA, Cornell University; PhD, Massachusetts Institute of Technology

Nancy Kopell William Goodwin Aurelio Professor of Mathematics and Sciences, Co-Director, Center for Biodynamics, College of Arts and Sciences. BA, Cornell University; MA, PhD, University of California, Berkeley

Dirk Kreimer Director, Center for Mathematical Physics, Professor of Mathematics and Physics, College of Arts and Sciences; PhD, University of Mainz (Germany)

Andrew Lyasoff Director, Mathematical Finance Program, Associate Professor of Mathematics, College of Arts and Sciences. BS, MS, PhD, Sofia University St. Kliment Ohridski (Bulgaria)

Diane M. Meuser Associate Professor of Mathematics, College of Arts and Sciences. BA, State University of New York, Albany; MA, PhD, Johns Hopkins University

Robert Pollack Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BS, Washington University; MA, PhD, Harvard University

Emma Previato Professor of Mathematics, College of Arts and Sciences. Laurea in Mathematics, University of Padua (Italy); PhD, Harvard University

Surajit Ray Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BSc, Presidency College (India); MStat, Indian Statistical Institute (India); PhD, Pennsylvania State University

David Rohrlich Professor of Mathematics, College of Arts and Sciences. BA, Haverford College; PhD, Yale University

Steven Rosenberg Professor of Mathematics, College of Arts and Sciences. BA, Hampshire College; MA, University of Massachusetts; PhD, University of California, Berkeley

Glenn Stevens Professor of Mathematics, College of Arts and Sciences. BA, University of California, Santa Barbara; PhD, Harvard University

Lisa Sullivan Assistant Dean for Undergraduate Education in Public Health, Associate Chair, Biostatistics, Associate Professor of Biostatistics, School of Public Health; Associate Professor of Mathematics and Statistics, College of Arts and Sciences. BA, University of New Hampshire; MA, PhD, Boston University

Maciej Szczesny Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BSc, University of Toronto (Canada); PhD, University of California, Berkeley

Murad Taqqu Professor of Mathematics, College of Arts and Sciences. BA, University of Lausanne (Switzerland); MA, PhD, Columbia University

Clarence Eugene Wayne Professor of Mathematics, College of Arts and Sciences. BA, University of Virginia; MA, PhD, Harvard University

Daniel C. Weiner Associate Professor of Mathematics and Statistics, College of Arts and Sciences. BS, Rensselaer Polytechnic Institute; MA, PhD, University of Wisconsin, Madison

Affiliated Faculty

Steven Homer Professor of Computer Science, College of Arts and Sciences. AB, University of California, Berkeley; PhD, Massachusetts Institute of Technology

Affiliated Faculty (Statistics and Consulting Unit)

Arlene S. Ash Research Professor of Public Health (Epidemiology and Biostatistics). AB, Harvard College; MS, Washington University; PhD, University of Illinois, Chicago Circle

Alexa Beiser Professor of Biostatistics, School of Public Health. MA, University of California, San Diego; PhD, Boston University

Theodore Colton Chairman Emeritus, Department of Epidemiology, School of Public Health; Professor of Epidemiology and Biostatistics, School of Public Health. MS, University of North Carolina; ScD, Johns Hopkins University

Merrill Elias Research Professor of Epidemiology, Department of Mathematics and Statistics, Statistics and Consulting Unit, College of Arts and Sciences. BA, Allegheny College; MS, PhD, Purdue University; MPH, Boston University

Philimon Gona Research Assistant Professor of Statistics, Department of Mathematics and Statistics, College of Arts and Sciences. MPH, Boston University; MA, PhD, Boston University

Chao-Yu Guo Research Assistant Professor of Statistics, Department of Mathematics and Statistics, College of Arts and Sciences. BS, MS, National Chen-Chi University (Taiwan); PhD, Boston University

Timothy C. Heeren Professor of Biostatistics, School of Public Health. BA, Carleton College; PhD, Boston University

Martin Larson Research Professor of Statistics, Department of Mathematics and Statistics, College of Arts and Sciences. AB, SM, SD, Harvard University

Joseph Massaro Assistant Professor of Biostatistics, School of Public Health. MA, PhD, Boston University

Michael Pencina Research Assistant Professor of Statistics, College of Arts and Sciences. MA, Warsaw University (Poland); PhD, Boston University

Janice M. Weinberg Associate Professor of Biostatistics, School of Public Health. MS, University of North Carolina; ScD, Harvard University

Emeriti

Robin E. Esch Professor Emeritus of Mathematics, College of Arts and Sciences. AB, AM, PhD, Harvard University

Barry Granoff Associate Professor Emeritus of Mathematics, College of Arts and Sciences. BS, Fairleigh Dickinson University; MS, PhD, New York University

Robert Willcutt Associate Professor Emeritus of Mathematics, College of Arts and Sciences. BS, Iowa State; MS, PhD, Indiana University

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The Graduate Program

The Department of Mathematics and Statistics offers graduate coursework and degree programs across a broad range of subject areas within the mathematical sciences. A master’s-level curriculum rich in both theory and application is available to graduate students in mathematics, the natural sciences, engineering, and mathematics education, as well as to advanced undergraduates in mathematics. At the doctoral level instruction is frequently in seminar and tutorial formats, with individual faculty members meeting with small groups of students.

The curriculum in mathematics provides the intellectual core on which all programs are based. Current faculty research interests include arithmetic and analytic number theory, algebraic geometry, differential geometry and topology, dynamical systems, mathematical biology, mathematical physics, mathematical logic, ordinary and partial  differential equations, probability theory, and the history of mathematics. In addition to these areas, coursework is provided in  real and complex analysis, functional analysis, Lie groups and algebras, and algebraic topology.

The program in dynamical systems and applied analysis provides a framework in which students may bring tools of modern analysis, geometry, and topology to bear on a broad range of problems. Areas of current research interest include dynamical systems, celestial mechanics, complex dynamics, ordinary and partial differential equations, biomathematics, and neural networks.

The program in probability theory provides mathematical background for understanding the random phenomena that arise in science and engineering. It also provides a link between mathematics and statistics. The degree program in mathematical statistics provides the mathematical foundations of modern statistical methodologies. The program in applied statistics is designed to produce practitioners broadly trained in the application of statistics. Research interests of the faculty include stochastic processes, estimation theory, hypothesis testing, mathematical finance, biostatistics, robust statistics, multivariate analysis, life-testing and reliability theory, data analysis, time series, regression, and the design of experiments.

Admission

Prospective applicants are urged to visit the Department of Mathematics and Statistics for information which explains the requirements, practices, and offerings of the department in greater detail.

Ordinarily, applications should be received by the Graduate School Office no later than January 15. Late applications will be considered if there is some special reason. Late applicants should correspond directly with the department.

Applicants are required to submit results of the Graduate Record Examinations (GRE) (including the Subject Test in Mathematics). The examination should be taken in October or earlier, so that the scores are received by January. Notices of admission with fellowship or assistantship awards will be made on or about April 1. Applicants requesting financial assistance should be sure to have all application materials submitted by January 15. After this date the possibilities of financial assistance are greatly reduced. Students are accepted for January admission, and financial aid may be available at that time, although opportunities for financial aid are greatly reduced.

Further information is available from the Department of Mathematics and Statistics, 111 Cummington Street, Boston, MA 02215; 617-353-2560; e-mail: dgs@math.bu.edu.

MA in Mathematics, Including Statistics

Admissions Tests and Prerequisites The candidates must have met the requirements for a bachelor’s degree with the equivalent of a concentration in mathematics. GRE results (including the Mathematics Subject Test) are required. For more information, see the preceding material.

Course Requirements Eight semester courses (32 credits) approved for graduate study are required; these should constitute a reasonably coherent program relevant to the objectives of the student. Ordinarily, all courses are in mathematics, or probability and statistics. However, students may petition the departmental graduate committee to use courses from other departments to partially satisfy this requirement.

Comprehensive Examination This examination, which is offered at the end of each year, serves as a master’s degree examination. The student may choose to take the examination either in mathematics or in statistics. Students judged to be inadequately prepared for the first examination will be allowed to take the examination again. More detailed information will be provided upon admission.

MA, PhD in Mathematical Finance

The Program in Mathematical Finance is an interdisciplinary graduate program offered by the Graduate School of Arts and Sciences. Please refer to the Program in Mathematical Finance.

MA in Actuarial Science

The program is offered by the Metropolitan College. Please refer to the Boston University Metropolitan College Bulletin or to the Department of Actuarial Science.

PhD in Mathematics, Including Statistics and Probability

Admission Tests and Prerequisites Strong candidates with a bachelor’s degree and students with a master’s degree in mathematics, statistics, or the equivalent may be admitted directly into the PhD program. GRE results (including the Mathematics Subject Test) are required.

Course Requirements Eight semester courses (32 credits) beyond those taken to fulfill the master’s degree are required. These shall provide a reasonable breadth of background in mathematics, or probability and statistics, and mastery of the basic tools relevant to the research area selected.

Language Requirement The candidate must demonstrate proficiency in one language other than their native tongue. Foreign students may use English to fulfill the language requirement.

Qualifying Examination Students must take this examination within two years of the start of their 700-level study. More detailed information will be provided upon admission to the PhD program.

Residency Requirement, Dissertation Prospectus, and Final Oral Examination

Please see General Requirements for the PhD on this site.

Dissertation An original thesis is required. A proposal outlining a tentative program of doctoral research approved by a faculty member must be submitted in the first year of PhD candidacy.

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Courses

Statistics courses are listed separately. For computer science courses, see the “Department of Computer Science” section in this bulletin. Course offerings in mathematics are selected from the following list.

CAS MF 502 Fundamentals of Finance

Prereq:CAS MA 123, MA 124, MA 225 or MA 230, and MA 242 or MA 442. Financial systems, financial statements, time value of money, interest rates, return sales, spot rates, forward rates, inflation rates, cost of capital rates, tax rates, bonds, cash-flow models, risk management. This course is part of the Mathematical Finance Program but is open to all students. Kardaras. 4 cr, Summer 2

CAS MA 505 History of Mathematics

Prereq: one year of college-level calculus. Patterns of mathematical thought from antiquity to the seventeenth century. Emphasis throughout on the background and origins of the mathematical revolution of the seventeenth century, in which Descartes, Newton, and Leibniz played key roles. Offered 2008/2009

CAS MA 511 Introduction to Analysis I

Prereq: CAS MA 225 or MA 230. Fundamental concepts of mathematical reasoning. Properties of the real number system, elementary point-set theory, and metric spaces. Limits, sequences, series, convergence, uniform convergence, and continuity. Differentiability for functions of a single variable, and Riemann-Stieltjes integration. Weiner. 4 cr, 1st sem.

CAS MA 512 Introduction to Analysis II

Prereq: CAS MA 511. Background of CAS MA 511 used to develop further topics of calculus. Exponential and logarithmic functions, Taylor series, power series, real analytic functions. Differential and integral calculus for functions of several variables. Line and surface integrals, divergence theorem, Stokes’ theorem, inverse and implicit function theorems, and change of variables. Fourier analysis. Weiner. 4 cr, 2nd sem.

CAS MA 528 Introduction to Modern Geometry

Prereq: CAS MA 225 or MA 230 or consent of instructor. Foundations of Euclidean geometry. Transformation and symmetries in the plane; inverse and projective planes; coordinates; conics and quadrics; the Golden Section; intermediary and Dedekind’s axiom; and models for non-Euclidean geometries. Previato. 4 cr, 2nd sem.

CAS MA 531 Mathematical Logic

The syntax and semantics of sentential and quantificational logic, culminating in the Godel Completeness Theorem. The Godel Incompleteness Theorem and its ramifications for computability and philosophy. Students may receive credit for only one of the following: CAS MA 531, PH 461, or PH 661. Kanamori. 4 cr, 1st sem.

CAS MA 532 Foundations of Mathematics

Prereq: CAS MA 531. Axiomatic set theory as a foundation for mathematics and as a field of mathematics: Axiom of Choice, the Continuum Hypothesis, and consistency results. Students may receive credit for only one of the following courses: CAS MA 532, PH 462, or PH 662. Kanamori. 4 cr, 2nd sem.

CAS MA 539 Methods of Scientific Computing

Prereq: CAS MA 225 or MA 230, MA 226 or MA 231, MA 242 or MA 442, and CAS CS 330, or consent of instructor. Topics include computational linear algebra, fast Fourier transform, wavelets, data compression, numerical integration, simulated annealing, random number generation, Monte Carlo methods. Meets with CAS CS 539. Offered 2008/2009

CAS MA 541 Modern Algebra I

Basic properties of groups, Sylow theorems, basic properties of rings and ideals, Euclidean rings, and polynomial rings. Stevens. 4 cr, 1st sem.

CAS MA 542 Modern Algebra II

Prereq: CAS MA 541. Vector spaces and modules, Galois theory, linear transformations and matrices, canonical forms, and bilinear and quadratic forms. Pollack. 4 cr, 2nd sem.

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CAS MA 547 Topics in Number Theory

Prereq: CAS MA 242 or MA 442; Coreq: CAS MA 548 and consent of instructor. An exploration of rational arithmetic and its generalizations. Foundations of arithmetic, Euclid’s algorithm; the fundamental theorem of arithmetic; arithmetic modulo m; continued fractions; Diophantine approximation; Pell’s equation; sums of squares; the arithmetic of polynomials over a field; quadratic reciprocity; arithmetic in quadratic number field; lattice point-free regions; Minkowski’s theorem on convex bodies. 4 cr, Summer 2

CAS MA 548 Problem Solving in Number Theory

Coreq: CAS MA 547. Mathematical heuristics, including good use of language and symbolism, techniques of exploration and discovery, through intensive work on a large assortment of unusually challenging problems in number theory. Students practice the art of mathematical discovery—numerical exploration, formulation and critique of conjectures, and techniques of proof and generalization. 4 cr, Summer 2

CAS MA 549 Geometry and Symmetry

Prereq: consent of instructor. Problem-oriented seminar in modern geometry focusing on invariants of transformation groups. Specific topics may include: Euclidean and plane geometry; conics; tilings; finite, projective, spherical, and/or hyperbolic geometry; applications to number theory; Platonic Solids. Rosenberg. 4 cr, Summer 2

CAS MA 555 Numerical Analysis I

Prereq: CAS MA 225 or MA 230. Numerical solutions of equations, iterative methods, analysis of sequences. Theory of interpolation and functional approximation, divided differences. Numerical differentiation and integration. Polynomial theory. Ordinary differential equations. I. Fried. 4 cr, 2nd sem.

CAS MA 556 Numerical Analysis II

Prereq: CAS MA 242 or MA 442 and MA 555, or consent of instructor. Numerical linear algebra; norms, elimination methods, error analysis, conditioning, eigenvalues, iterative methods, least squares and nonlinear functional minimization. Partial differential equations, boundary-value and initial-value problems. Finite-element methods. Legendre and Chebyshev polynomials. Treatment in greater depth of selected topics from CAS MA 555. I. Fried. 4 cr, 1st sem.

CAS MA 557 Mathematical Structures in Physics I

Prereq: CAS MA 226 or MA 231, MA 242 or MA 442, or one of CAS PY 313, PY 354. Relativistic wave equations, quantum equations of motion, Feynman graphs, combinatorics of perturbative expansions and Hopf algebras, renormalization and elimination of divergences, locality of fields, scaling transformations and renormalization group, basic applications to particle physics, and condensed matter theory. Kreimer. 4 cr, 1st sem.

CAS MA 561 Methods of Applied Mathematics I

Prereq: CAS MA 226 or MA 231. Derivation and analysis of the classical equations of mathematical physics; heat equation, wave equation, and potential equation. Initial boundary value problems, method of separation of variables, eigenvalue problems, and eigenfunction expansions. Fourier analysis. Existence and uniqueness of solution. Wayne. 4 cr, 1st sem.

CAS MA 562 Methods of Applied Mathematics II

Prereq: CAS MA 561. Calculus of variations, first-order nonlinear partial differential equations, Hamilton-Jacobi theory, Rayleigh-Ritz procedure, and perturbation methods. Kon. 4 cr, 2nd sem.

CAS MA 563 Introduction to Differential Geometry

Prereq: CAS MA 226 or MA 231 and MA 411 or consent of instructor. Study of local properties of curves and surfaces in the three-dimensional Euclidean space; curvature, torsion, Frenet equations, tangent and normal planes; first and second fundamental forms; developable surfaces, principal, mean, and Gaussian curvature; vector fields, covariant differentiation, geodesics, and surfaces of constant curvature. Kimura. 4 cr, 1st sem.

CAS MA 564 Introduction to Topology

Prereq: CAS MA 411 or consent of instructor. An introduction to topological spaces and continuity. Compactness and connectedness. Metrizable topological spaces. Product topology and Tychonoff’s theorem. The fundamental group and van Kampen’s theorem. Covering spaces and the universal cover. Szczesny. 4 cr, 2nd sem.

CAS MA 565 Mathematical Models in the Life Sciences

Prereq: CAS MA 226 or 231. An introduction to mathematical modeling, using applications in the biological sciences. Mathematics includes linear difference and differential equations, and an introduction to nonlinear phenomena and qualitative methods. An elementary knowledge of differential equations and linear algebra is assumed. Kopell. 4 cr, 2nd sem.

CAS MA 566 Geometric Methods in Mechanics

Prereq: CAS MA 226 or MA 231, MA 242 or MA 442, MA 411 or consent of instructor. Modern geometric theories applied to motion of physical objects. Differential forms. Symplectic manifolds. Lie Groups and their Lie Algebras. Hamiltonian and Lagrangian systems; Liouville’s theorem, Poincare’s return theorem, Noether’s theorem. Additional topics according to instructor. Kimura. 4 cr, 1st sem.

CAS MA 569 Optimization Methods of Operations Research

Prereq: CAS MA 225 or MA 230 and MA 242 or MA 442. Optimization of linear functions: linear programming, simplex method, transportation problems, assignment problems. Networks: shortest path, minimum spanning tree, maximum flow. Integer programming and dynamic programming. Nonlinear optimization: Kuhn-Tucker conditions and Lagrange’s method. Lyasoff. 4 cr, 1st sem.

CAS MA 570 Stochastic Methods of Operations Research

Prereq: CAS MA 225 or MA 230 and MA 242 or MA 442. Markov chains, Poisson processes, transition probabilities, steady state distributions, queuing theory, inventory theory, time series, simulation, simulated annealing. TBA. 4 cr, 2nd sem.

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CAS MF 572 Introduction to Mathematical Finance

Prereq: CAS MA 226 or MA 231 and MA 381 or MA 581 or MA 590. An introduction to mathematical finance emphasizing the rigors of the analytical methods used in finance. Mathematics of Modern Portfolio Theory; Capital Asset Pricing Model (CAPM); Arbitrage Pricing Theory (APT); options, futures, and swaps; and risk management are discussed in depth and at a rigorous and analytical level. Valuation and hedging models are derived based on concepts from Itô Calculus. Guasoni. 4 cr, 1st sem.

CAS MA 573 Qualitative Theory of Ordinary Differential Equations

Prereq: CAS MA 226 or MA 231, MA 242 or MA 442 and MA 411. Eigenvalues, eigenvectors, and Jordan normal forms. Linear systems of differential equations. Phase portrait, Hamiltonian systems, and stability theory. Applications to systems arising in mechanics, economics, ecology, and electrical circuit theory. Van Baalen. 4 cr, 1st sem.

CAS MA 574 Applied Nonlinear Dynamics

Prereq: CAS MA 573. Attractors and invariant measures for nonlinear dynamical systems. Measures of chaos such as Lyapunov exponents. Time series analysis. Multiple time scales and singular perturbation theory. Synchronization in coupled oscillators. Strong emphasis on applications to realistic biological and mechanical systems. Kaper. 4 cr, 2nd sem.

CAS MA 577 Mathematics of Financial Derivatives

Prereq: CAS MA 581 or consent of instructor. Develops the probabilistic tools used in finance and presents the methodologies that are used in the pricing of financial derivatives. No previous knowledge of finance is required. Taqqu. 4 cr, 2nd sem.

CAS MA 590 Introduction to Probability Theory

Prereq: CAS MA 225 or CAS MA 230. Combinatorics; conditional probability; independence; discrete and continuous random variables; sigma algebras; joint, marginal, and conditional distributions. Conditional and unconditional expectations and variance; derived distributions; characteristic functions; convergence of random variables; limit theorems; unbiased estimates of mean and variance; and Cochran’s Theorem. (Cannot be taken for credit in addition to MA 381 or MA 581.) This course is part of the Mathematical Finance Program but is open to all students with stamped approval. Guasoni. 4 cr, Summer 2

CAS MF 593 Statistical Analysis of Financial Data

Prereq: CAS MA 381 or CAS MA 581 or CAS MA 590. Statistical/probabilistic techniques commonly used in financial engineering and risk management. Topics include: estimation of diffusion processes, elements of linear (ARMA) and nonlineary ((G)ARCH) time series-analysis, multivariate data anlysis, extreme value theory and Monte Carlo methods. Guasoni. 4 cr, 1st sem.

CAS MF 594 Stochastic Optimal Control and Investment

Prereq: CAS MA 225 or MA 230 and MA 242 or MA 442. Dynamic hedging, option valuation, Brownian motion and stochastic dynamic programming, Bellmann’s equation, contingent claims analysis, optimal stopping rules, dynamic equilibrium, sequential investment. Lyasoff. 4 cr, 2nd sem.

GRS MA 647 Research Methods in Mathematics I

Prereq: CAS MA 547 and CAS MA 548 or consent of instructor. Methods of mathematical research via prolonged study of one selected mathematical topic. Topics are usually chosen from number theory or combinatorics. Written and oral research presentations. D. Fried. 2 cr, Summer 2

GRS MA 648 Research Methods in Mathematics II

Prereq: GRS MA 647 or consent of instructor. Methods of mathematical research via prolonged, directed study of one selected mathematical topic, distinct from that chosen for GRS MA 647. Topics are usually chosen from geometry, number theory, or combinatorics, and may involve open problems. Written and oral research presentation. Staff. 2 cr, 1st sem.

CAS MA 671 Chaotic Dynamical Systems

Prereq: CAS MA 225 or MA 230 or equivalent. This course is not open to CAS students; it is intended for graduate students in disciplines outside of mathematics. Iterations of functions of one or several variables. Periodicity, stability, chaos, fractals, bifurcations. Julia sets and the Madelbrot set. Students are required to perform several experiments on personal computers. Devaney. 4 cr, 1st sem.

GRS MA 699 Teaching College Mathematics I

The goals, contents, and methods of instruction in mathematics. General teaching-learning issues. Required of all teaching fellows. Staff. 2 cr, 1st & 2nd sem.

GRS MA 711 Real Analysis

Prereq: CAS MA 512 or equivalent mathematical experience. Measure theory and integration on measure spaces, specialization to integration on locally compact spaces, and the Haar integral. Lp spaces, duality, and representation theorems. Introduction to Banach and Hilbert spaces, open mapping theorem, spectral theorem for Hermitian operators, and compact and Fredholm operators. Hawkins. 4 cr, 1st sem.

GRS MA 713 Functions of a Complex Variable

Prereq: advanced calculus or equivalent mathematical experience. The theory of analytic functions. Integral theorems, contour integration, conformal mapping, and analytic continuation. Blanchard. 4 cr, 2nd sem.

GRS MA 717 Functional Analysis

Prereq or coreq: GRS MA 711 or equivalent. Theory of Banach and Hilbert spaces, and Hahn-Banach and separation theorems. Dual spaces. Banach contraction mapping theorem. Reflexivity and Krein-Milman theorem. Operator theory. Brouwer-Schauder fixed-point theorems. Applications to probability, dynamical systems, and applied mathematics. Kon. 4 cr, 1st sem.

GRS MA 721 Differential Topology I

Prereq: CAS MA 511, MA 512, or equivalent. Differential manifolds, tangent bundles, transversality, winding numbers, and vector bundles. Previato. 4 cr, 1st sem.

GRS MA 722 Differential Topology II

Prereq: GRS MA 721. Intersection theory, Lefschetz fixed point theory, integration on manifolds, vector fields and flows, and Frobenius’ theorem. Previato. 4 cr, 2nd sem.

GRS MA 725 Differential Geometry I

Prereq: GRS MA 721 or consent of instructor. Geometry of surfaces in Euclidean space; geodesics and curvature of Riemannian manifolds; topological restrictions on curvature. Offered 2008/2009

GRS MA 726 Differential Geometry II

Prereq: GRS MA 725. Topics include connections on vector bundles, moving frames, Hodge theory, spectral geometry, and characteristic classes. Offered 2008/2009

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GRS MA 727 Algebraic Topology I

Prereq: CAS MA 564 or equivalent. Covers singular and simplical homology theory. Cohomology and cup products. Duality on manifolds. Lefschetz and fixed-point formula. D. Fried. 4 cr, 1st sem.

GRS MA 728 Algebraic Topology II

Prereq: GRS MA 727. Topics include homotopy theory, theory of characteristic classes and covering spaces, and cobordism theory. D. Fried. 4 cr, 2nd sem.

GRS MA 731 Lie Groups and Lie Algebras

Prereq: GRS MA 721 and MA 741. Classical Lie groups, associated Lie algebras, exponential map, closed subgroups and homogeneous spaces, classification of simple Lie algebras, and elementary representation theory of Lie algebras. Selection of applications to analysis, geometry, or algebra. Offered 2008/2009

GRS MA 741 Algebra I

Basic properties of groups, rings, fields, and modules. Specific topics include the Jordan-Holder and Sylow theorems, local rings, theory of localization, modules over PIDs, and Galois theory. Rohrlich. 4 cr, 1st sem.

GRS MA 742 Algebra II

Prereq: GRS MA 741 or consent of instructor. Advanced topics in algebra. Linear and multilinear algebra, commutative algebra, and an introduction to category theory and homological algebra. Further topics may include representation of groups, completions, real fields, and elementary algebraic number theory and algebraic geometry. Rohrlich. 4 cr, 2nd sem.

GRS MA 743 Algebraic Number Theory I

Prereq: GRS MA 741 or consent of instructor. Algebraic integers, completions, ramification and the discriminant, cyclotomic and quadratic fields, ideal class groups, Dirichlet’s unit theorem, ideles, and adeles. Further topics are chosen from analytic number theory, class field theory, and the theory of Diophantine equations. Offered 2008/2009

GRS MA 744 Algebraic Number Theory II

Prereq: GRS MA 743. Advanced topics in number theory. Topics chosen from: zeta functions of number fields of algebraic varieties; arithmetic of elliptic curves; modular forms and modular curves; class field theory; and Iwasawa theory. Offered 2008/2009

GRS MA 745 Algebraic Geometry I

Prereq: GRS MA 741 or consent of instructor. Affine and projective varieties, morphisms and rational maps, nonsingular varieties, Bezout’s theorem, and an introduction to sheaves and schemes. Further topics are chosen from the advanced theory of schemes, algebraic curves, Riemann-Roch theorem, algebraic surfaces, and sheaf cohomology. Szczesny. 4 cr, 1st sem.

GRS MA 746 Algebraic Geometry II

Prereq: GRS MA 745. Continuation of topics in algebraic geometry begun in GRS MA 745, including sheaves, schemes, sheaf cohomology, and further study of algebraic curves and surfaces. TBA. 4 cr, 2nd sem.

GRS MA 770 Mathematical and Statistical Methods of Bioinformatics

Prereq: advanced undergraduate math/statistics major or GRS student status, CAS MA 225 and CAS MA 242 or consent of instructor. Mathematical and statistical bases of bioinformatic methods and their applications. Hidden Markov models, kernel methods, mathematics of machine learning approaches, probabilistic sequence alignment, Markov chain Monte Carlo and Gibbs sampling, mathematics of phylogenetic trees, and statistical methods in microarray analysis. Kon. 4 cr, 2nd sem.

GRS MA 771 Introduction to Dynamical Systems

Diffeomorphisms and flows; periodic points, nonwandering points, and recurrent points; hyperbolicity, topological conjugacy, and structural stability; stable manifold theorem; symbolic dynamics; Axiom A and chaotic systems. TBA. 4 cr, 2nd sem.

GRS MA 772 Mathematical Finance Theory

Prereq: GRS MA 711, GRS MA 717, GRS MF 795 or consent of instructor. Martingale and local martingale measures. Notions of Arbitrage. Fundamental Theorem of Asset Pricing in the semimartingale setting. Superhedging and optional decomposition. Snell envelopes and American options. Predictable representation and complete markets. Convex duality and utility maximization. Risk measures. Kardaras. 4 cr, 2nd sem.

GRS MA 775 Ordinary Differential Equations

Stable and center manifolds theorem, linearization of vector fields, variational equations, Floquet theory and Poincaré maps for periodic orbits, bifurcation of rest points, averaging theory, topics from singular perturbations, Hamiltonian systems, nonlinear oscillations, normal forms, and applications. Kaper. 4 cr, 1st sem.

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GRS MA 776 Partial Differential Equations

Prereq or coreq: GRS MA 711 or consent of instructor. Hyperbolic, elliptic, and parabolic equations. Characteristics and seperation of variables. Eigenvalue problems, Fourier techniques, Sobolev spaces, and potential theory. Introduction to pseudodifferential operators. Offered 2008/2009

GRS MA 779 Probability Theory I

Prereq: CAS MA 511 or consent of instructor. Introduction to probability with measure theoretic foundations. Fundamentals of measure theory. Probability space. Measurable functions and random variables. Expectation and conditional expectation. Zero-one laws and Borel-Cantelli lemmas. Characteristic functions. Modes of convergence. Uniform integrability. Skorokhod representation theorem. Basic limit theorems. Taqqu. 4 cr, 1st sem.

GRS MA 780 Probability Theory II

Prereq: GRS MA 779 or consent of instructor. Probability topics important in applications and research. Laws of large numbers. Three series theorem. Central limit theorems for independent and non-identically distributed random variables. Speed of convergence. Large deviations. Laws of the iterated logarithm. Stable and infinitely divisible distributions. Discrete time martingales and applications. Taqqu. 4 cr, 2nd sem.

GRS MA 822 Topics in Geometry and Topology

Advanced seminar in topics in geometry, topology, and mathematical physics of current research interest. Rosenberg. 4 cr, 2nd sem.

GRS MA 831 Topics in Mathematical Physics

Prereq: CAS MA 557 or consent of instructor. Covers advanced topics in mathematical physics. Focuses first on quantum gauge symmetry in the context of quantum field theory, then discusses aspects of self-similarity in non-perturbative field theory and natural sciences in general. Kreimer. 4 cr, 1st sem.

GRS MA 841, 842 Seminar: Algebra

Not offered 2007/2008

GRS MA 843 Topics in Number Theory I

Prereq: consent of instructor. Topics such as modular forms, special values of L-series, p-adic L-functions, and p-adic variation are studied in this advanced number theory course. Offered 2008/2009

GRS MA 844 Topics in Number Theory II

Prereq: consent of instructor. Topics such as Selmer groups, Iwasawa theory, Euler systems, and main conjectures are studied in this advanced number theory course. Offered 2008/2009

GRS MA 861, 862 Seminar: Applied Mathematics

Not offered 2007/2008

GRS MA 871, 872 Seminar: Dynamical Systems

Devaney. Variable cr, 1st sem.

GRS MA 876 Seminar: Partial Differential Equations

Wayne. Variable cr, 2nd sem.

Directed Study and Research

GRS MA 905, 906 Directed Study: History of Mathematics

Variable cr.

GRS MA 911, 912 Directed Study: Real Analysis

Variable cr.

GRS MA 913, 914 Directed Study: Complex Analysis

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GRS MA 917, 918 Directed Study: Functional Analysis

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GRS MA 921, 922 Directed Study: Differential Topology

Variable cr.

GRS MA 925, 926 Directed Study: Differential Geometry

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GRS MA 927, 928 Directed Study: Algebraic Topology

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GRS MA 931, 932 Directed Study: Logic and Theoretical Computer Science

Variable cr.

GRS MA 941, 942 Directed Study: Algebra

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GRS MA 943, 944 Directed Study: Number Theory

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GRS MA 945, 946 Directed Study: Algebraic Geometry

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GRS MA 951, 952 Directed Study: Numerical Analysis

Variable cr.

GRS MA 955, 956 Directed Study: Theory of Computation

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GRS MA 963, 964 Directed Study: Applied Analysis

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GRS MA 967, 968 Directed Study: Mathematical Physics

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GRS MA 969, 970 Directed Study: Operations Research

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GRS MA 971, 972 Directed Study: Ordinary Differential Equations

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GRS MA 975, 976 Directed Study: Dynamical Systems

Variable cr.

GRS MA 977, 978 Directed Study: Partial Differential Equations

Variable cr.

GRS MA 979, 980 Directed Study: Probability

Variable cr.

Statistics Courses

CAS MA 568 Statistical Analysis of Point Process Data

Prereq: CAS MA 213 and 214, or consent of instructor. Introduces the theory of point processes and develops practical problem solving skills to construct models, assess goodness-of-fit, and perform estimation from point process data. Applications to neural data, earthquake analysis, financial modeling, and queuing theory. Eden. 4 cr, 1st sem.

CAS MA 570 Stochastic Methods of Operations Research

Prereq: CAS MA 225 or 230 and MA 242 or 442. Poisson processes, Markov chains, queuing theory. Matrix differential equations, differential-difference equations, probability- and moment-generating functions, single- and multiple-channel queues, steady-state and transient distributions. TBA. 4 cr, 2nd sem.

CAS MF 572 Introduction to Mathematical Finance

A rigorous mathematical introduction to developments in the field of finance. Mathematics of modern portfolio theory, capital asset pricing model, and arbitrage pricing theory. Derivation of pricing models for options, futures, and swaps based on concepts from Itô Calculus. Guasoni. 4 cr, 1st sem.

CAS MA 575 Linear Models

Prereq: CAS MA 214, 242 and 581, or consent of instructor. Post-introductory course in linear models, with focus on both principles and practice. Simple and multiple linear regression, weighted and generalized least squares, polynomials and factors, transformations, regression diagnostics, variable selection, and a selection from topics on extensions of linear models. Kolaczyk. 4 cr, 1st sem.

CAS MA 576 Generalized Linear Models

Prereq: CAS MA 575. Covers topics in linear models beyond MA 575: generalized linear models, analysis of binary and polytomous data, log-linear models, multivariate response models, non-linear models, graphical models and relevant model selection techniques. Additional topics in modern regression as time allows. Ray. 4 cr, 2nd sem.

CAS MA 577 Mathematics of Financial Derivatives

CAS MA 578 Bayesian Statistics

Prereq: CAS MA 581 and MA 582. The principles and methods of Bayesian statistics. Subjective probability, Bayes rule, posterior distributions, predictive distributions. Computationally based inference using Monte Carlo integration, and Markov chain simulation. Hierarchical models, mixture models, model checking, and methods for Bayesian model selection. TBA. 4 cr, 2nd sem.

CAS MA 581 Probability

Prereq: CAS MA 225 or MA 230 or consent of instructor. Basic probability, conditional probability, independence. Discrete and continuous random variables, mean and variance, functions of random variables, moment generating function. Jointly distributed random variables, conditional distributions, independent random variables. Methods of transformations, law of large numbers, central limit theorem. Cannot be taken for credit in addition to CAS MA 381. Weiner. 4 cr, 1st sem.

CAS MA 582 Mathematical Statistics

Prereq: CAS MA 381 or MA 581. Point estimation including unbiasedness, efficiency, consistency, sufficiency, minimum variance unbiased estimator, Rao-Blackwell theorem, and Rao-Cramer inequality. Also includes maximum likelihood and method of moment estimations; interval estimation; tests of hypothesis, uniformly most powerful tests, uniformly most powerful unbiased tests, likelihood ratio test, and chi-square test. Ginovyan. 4 cr, 2nd sem.

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CAS MA 583 Introduction to Stochastic Processes

Prereq: CAS MA 381 or MA 581 or consent of instructor. Basic concepts and techniques of stochastic processes as they are most often used to construct models for a variety of problems of practical interest. Topics include Markov chains, Poisson process, birth and death processes, queuing theory, renewal processes, and reliability. Eden. 4 cr, 2nd sem.

CAS MA 584 Multivariate Statistical Analysis

Prereq: CAS MA 242, CAS MA 213 and CAS MA 581 or consent of instructor. Presents statistical concepts and methods and their application for the exploration, regression, testing, visualization, and clustering of multivariate data. Both classical and modern techniques developed, including methods for analysis of high dimensional and non-euclidean data. Ray. 4 cr, 2nd sem.

CAS MA 585 Time Series Analysis

Prereq: CAS MA 581 or consent of instructor. Autocorrelation and partial autocorrelation functions; stationary and nonstationary processes; ARIMA and Seasonal ARIMA model identification, estimation, diagnostics, and forecasting. Modeling financial data via ARCH and GARCH models. Volatility estimation; Additional topics, including long-range dependence and state-space models. Gangopadhyay. 4 cr, 2nd sem.

CAS MA 586 The Design of Experiments

Prereq: CAS MA 582 or consent of instructor. Randomized blocks; Latin and Graeco-Latin squares, factorial arrangements with confounding and fractional replication; split-plot, crossover, and response surface designs. Treatment of missing data, group sizes, relative efficiency, and relationship between design and analysis. Offered 2008/2009

CAS MA 587 Sampling Design: Theory and Methods

Prereq: CAS MA 582 or consent of instructor. Stratified, cluster, systematic, multistage, double, and inverse sampling; optimum sample size, relative efficiency, sampling with unequal probabilities, types of estimators (ratio and regression) and their properties. Measurement error nonresponse and randomized response models. Ginovyan. 4 cr, 1st sem.

CAS MA 588 Nonparametric Statistics

Prereq: CAS MA 582 or consent of instructor. The theory and logic in the development of nonparametric techniques including order statistics, tests based on runs, goodness of fit, rank-order (for location and scale), measures of association, analysis of variance, asymptotic relative efficiency. Offered 2008/2009

CAS MA 614 Statistical Methods

For graduate students in education and the social sciences. Not open to CAS students. Students may receive credit for no more than one of the following courses: CAS MA 116, MA 214, and MA 614. Second course in statistics, embodying basic statistical methods used in educational and social science research. Reviews all basic concepts covered in a first statistics course and presents in detail more advanced topics such as analysis of variance, covariance, experimental design, correlation, regression, and selected nonparametric techniques. A problem-solving course; students carry out analysis of data taken from educational and other social science sources. Heeren. 4 cr, 1st sem.

CAS MA 684 Applied Multiple Regression and Multivariable Methods

Prereq: one year of statistics. Application of multivariable data analytic techniques. Multiple linear regression and correlation, confounding and interaction, variable selection, categorical predictors and outcomes, logistic regression, factor analysis, MANOVA, discriminant analysis, regression with longitudinal data, repeated measures ANOVA. Heeren. 4 cr, 2nd sem.

CAS MA 685 Advanced Topics in Applied Statistical Analysis

Prereq: CAS MA 684 or consent of instructor. Continues topics of MA 684 at a more advanced level. Canonical correlation, multivariate analysis of variance, and multivariate regressions. Categorical dependent variables techniques; discriminant analysis, logistic regression, and log-linear analysis. Factor analysis; principal-axes, rotations, and factor scores. Cluster analysis. Power analysis. Extensive use of statistical software. D’Agostino. 4 cr, 1st sem.

GRS MA 750 Advanced Statistical Methods I

Prereq: CAS MA 575 and CAS MA 581, or consent of instructor. First in a two-semester PhD sequence on post-classical statistical methods and their applications. Selection from topics in non- and semi-parametric modeling and inference, such as smoothing, splines, generalized additive models, projection pursuit, and classification and regression trees. Gangopadhyay. 4 cr, 1st sem.

GRS MA 751 Advanced Statistical Methods II

Prereq: CAS MA 575 and CAS MA 581, or consent of instructor. Second in a two-semester PhD sequence on post-classical statistical methods and their applications. Selection from topics in statistical learning, such as regularized basis methods, kernel methods, boosting, neural networks, support vector machines, and graphical models. Kolaczyk. 4 cr, 2nd sem.

GRS MA 770 Mathematical and Statistical Methods of Bioinformatics

GRS MA 772 Mathematical Finance Theory

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GRS MA 779 Probability Theory I

GRS MA 780 Probability Theory II

GRS MA 781 Estimation Theory

Prereq: CAS MA 581, MA 582, or consent of instructor. Review of probability, populations, samples, sampling distributions, and delta theorems. Parametic point estimation. Rao-Cramer inequality, sufficient statistics, Rao-Blackwell theorem, maximum likelihood estimation, least squares estimation, and general linear model of full rank. Confidence intervals. Bayesian analysis and decision theory. Gangopadhyay. 4 cr, 1st sem.

GRS MA 782 Hypothesis Testing

Prereq: GRS MA 781 or consent of instructor. Parametric hypothesis testing, uniformly and locally the most powerful tests, similar tests, invariant tests, likelihood ratio tests, linear model testing, asymptotic theory of likelihood ratio, and chi-squared test. Logit and log-lin analysis of contingency tables. Gangopadhyay. 4 cr, 2nd sem.

GRS MA 791 Recent Advances in Probability and Statistics I

Prereq: consent of instructor. Participants discuss ongoing research, as well as important results that have recently appeared in the literature. Taqqu. 4 cr, 1st sem.

GRS MA 792 Recent Advances in Probability and Statistics II

Prereq: consent of instructor. Participants discuss onging research, as well as important results that have recently appeared in the literature. Taqqu. 4 cr, 2nd sem.

GRS MF 795 Stochastic Methods of Mathematical Finance

Prereq: CAS MA 225 or MA 230, MA 511, and MA 581. The essentials in stochastic calculus; Weiner’s process, martingales, stochastic integrals, Itô’s lemma, Girsanov’s transformation formula, diffusion processes. Most of the theory is motivated with examples from finance and engineering. The course provides all prerequisites needed for an in-depth study of mathematical finance. Lyasoff. 4 cr, 1st sem.

GRS MF 796 Computational Methods of Mathematical Finance

Prereq: GRS MF 795. The essentials of the mathematics theory of option pricing: basic option models, Black-Scholes analysis, American options, exotic and general path-dependent options, Asian options, options with transaction costs. Kardaras. 4 cr, 2nd sem.

GRS MA 881, 882 Seminar: Statistics

Ray. Variable cr, 1st sem.

GRS MA 883, 884 Seminar: Probability and Statistics

Not offered 2007/2008

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Directed Study and Research

GRS MA 991, 992 Directed Study: Statistical Inference and Probability

Variable cr.

Related Courses in Other Departments

CAS CS 511 Object-Oriented Software Principles

Prereq: CAS CS 320 or CS 411 or consent of instructor. Specification, programming, and analysis of large-scale, reliable, and reusable Java software using object-oriented design principles. Topics may include object-oriented programming, object models, memory models, inheritance, exceptions, namespaces, data abstraction, design against failure, design patterns, reasoning about objects. 4 cr, 2nd sem.

CAS CN 550 Neural and Computational Models of Recognition, Memory, and Attention

Prereq: CAS CN 510 or consent of instructor. Develops neural-network models of how internal representations of sensory events and cognitive hypotheses are learned and remembered as well as models of how such representations enable recognition and recall these events. Various neural and statistical pattern-recognition models are analyzed. Special attention is given to stable self-organization of pattern-recognition and recall codes by Adaptive Resonance Theory (ART) models. Mathematical techniques and definitions to support fluent access to the neural network and pattern-recognition literature are developed throughout the course. Experimental data and theoretical predictions from cognitive psychology, neuropsychology, and neurophysiology of normal and abnormal individuals are also analyzed. Coursework emphasizes skill development, including writing, computational analysis, teamwork, and verbal communication. Carpenter. 4 cr, 2nd sem.

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31 October 2007
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