5: Graduate School of Arts and Sciences

General Information | Programs and Degrees Offered | Admission Information
Financial Assistance | Graduate Academic Regulations
Requirements for Specific Graduate Degrees | Departments and Programs | Faculty

Non-Departmental | Anthropology | Art | Asian and Middle Eastern | Asian Studies | Astronomy
Biochemistry | Biology | Biological and Physical Sciences | Biophysics | Cell and Molecular Biology
Cell Biology | Chemistry | Classics | Commerce | Drama | Economics | English | Environmental Sciences
French | German | Government and Foreign Affairs | Health Evaluation Sciences | History | Linguistics
Mathematics | Microbiology | Molecular Physiology and Biological Physics | Music | Neuroscience
Pharmacology | Philosophy | Physics | Psychology | Religious Studies | Russian and East European Studies
Slavic | Sociology | Spanish, Italian, and Portuguese | Statistics | Surgery

Department of Mathematics
Course Descriptions | Departmental Degree Requirements

Course Descriptions

MATH 501 - (3) (Y)
The History of the Calculus

Prerequisite: MATH 221 and MATH 351 or permission of instructor
Study of the evolution of the various mathematical ideas leading up to the development of the calculus in the seventeenth century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials.

MATH 503 - (3) (Y)
The History of Mathematics

Prerequisite: MATH 221 and MATH 351 or permission of instructor
Study of the development of mathematics from classical antiquity through the end of the nineteenth century, focusing on the critical periods in the evolution of such areas as geometry, number theory, algebra, probability, and set theory. Emphasizes primary source materials.

MATH 509 - (3) (Y)
Mathematical Probability

Prerequisites: Three semesters of calculus, and graduate standing or departmental approval. Students who have received credit for MATH 311 may not take MATH 509 for credit
Study of the development and analysis of probability models through the basic concepts of sample spaces, random variables, probability distributions, expectations, and conditional probability. Additional topics include distributions of transformed variables, moment generating functions, and the central limit theorem.

MATH 510 - (3) (Y)
Mathematical Statistics

Prerequisites: MATH 509 or equivalent, and graduate standing or departmental approval
Study of methods of estimation, general concepts of hypothesis testing, linear models and estimation by least squares, categorical data, and nonparametric statistics.

MATH 511 - (3) (Y)
Stochastic Processes

Prerequisite: MATH 311 or permission of instructor
Topics in probability selected from: Random walks, Markov processes, Brownian motion, Poisson processes, branching processes, stationary time series, linear filtering and prediction, queuing processes, and renewal theory.

MATH 514 - (3) (Y)
Mathematics of Derivative Securities

Prerequisite: MATH 221 or MATH 122 or its equivalent, and a knowledge of probability and statistics. MATH 311 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

MATH 521 - (3) (Y)
Advanced Calculus and Applied Mathematics

Prerequisites: MATH 221, MATH 225
Topics include vector analysis, Greenís, Stokesí, divergence theorems, conservation of energy, and potential energy functions. Emphasizes physical interpretation, Sturm-Liouville problems and Fourier series, special functions, orthogonal polynomials, and Greenís functions.

MATH 522 - (3) (Y)
Partial Differential Equations and Applied Mathematics

Prerequisites: MATH 521 (MATH 351 recommended)
Introduction to partial differential equations, Fourier transforms. Topics include separation of variables, boundary value problems, classification of partial differential equations in two variables, Laplace and Poisson equations, and heat and wave equations.

MATH 525 - (3) (IR)
Advanced Ordinary Differential Equations

Prerequisites: MATH 221, MATH 225, MATH 351 or permission of instructor
Study of the qualitative geometrical theory of ordinary differential equations. Topics include all or most of the following: Picardís method and basic existence and uniqueness theorems; linear systems; the phase plane and Sturmís theorems; the Poincarť-Bendixon theorem; and Lyapunovís method and stability. Other topics presented as time permits.

MATH 526 - (3) (IR)
Partial Differential Equations

Prerequisite: MATH 221, MATH 225 and MATH 351 or permission of instructor
A theoretical introduction from a classical viewpoint. Topics include harmonic and subharmonic functions; wave and heat equations; Cauchy-Kowalewski and Holmgren theorems; characteristics; and the Hamilton-Jacobi theory.

MATH 530 - (3) (IR)
Computer Methods in Numerical Analysis

Prerequisites: MATH 351, MATH 430, and computer proficiency
A study of the underlying mathematical principles, and the use of sophisticated software for numerical problems such as spline interpolation, ordinary differential equations, nonlinear equations, optimization, and singular-value decomposition of a matrix.

MATH 531, 532 - (3) (Y)
Introduction to Real Analysis I, II

Prerequisites: MATH 221, MATH 351
Topics include the basic topology of Euclidean spaces, continuity and differentiation of functions on Euclidean spaces, Riemann-Stieltjes integration, convergence of sequences and series of functions. Equicontinuous families of functions, Weierstrassí theorem, inverse function theorem and implicit function theorem, integration of differential forms, and Stokesí Theorem.

MATH 551, 552 - (3) (Y)
Introduction to Abstract Algebra I,II

Prerequisite: MATH 351 or permission of instructor
Introduction to algebraic systems, including groups, rings, fields, vector spaces and their general properties including subsystems, quotient systems, homomorphisms. Study of basic examples such as permutation groups, polynomial rings, groups and rings of matrices. Additional topics may include applications to linear algebra and number theory.

MATH 570 - (3) (Y)
Introduction to Geometry

Prerequisite: MATH 221, MATH 351 or permission of instructor
Study of topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic, and non-Euclidean geometry.

MATH 572 - (3) (Y)
Introduction to Differential Geometry

Prerequisite: MATH 221, MATH 351 or permission of instructor
Study of topics selected are from the theory of curves and surfaces in Euclidean space and the theory of manifolds.

MATH 577 - (3) (Y)
General Topology

Prerequisite: MATH 221; corequisite: MATH 551 or the equivalent
Topics include topological spaces and continuous functions; product and quotient topologies; compactness and connectedness; separation and metrization; and the fundamental group and covering spaces.

MATH 583 - (3) (SI)
Seminar

Prerequisite: Permission of instructor
Presentation of selected topics in mathematics.

MATH 596 - (3) (S)
Supervised Study in Mathematics

Prerequisite: Permission of instructor and graduate standing
In exceptional circumstances, a student may undertake a rigorous program of supervised study designed to expose the student to a particular area of mathematics. Regular homework assignments and scheduled examinations are required.

MATH 731 - (4) (Y)
Real Analysis and Linear Spaces I

Prerequisite: MATH 531 or equivalent
Introduction to measure and integration theory.

MATH 732 - (3) (Y)
Real Analysis and Linear Spaces II

Prerequisites: MATH 731, MATH 734 or equivalent
Study of additional topics in measure theory. Banach and Hilbert spaces, and Fourier analysis.

MATH 734 - (4) (SI)
Complex Analysis I

Study of the fundamental theorems of analytic function theory.

MATH 735 - (3) (Y)
Complex Analysis II

Prerequisite: MATH 734 or equivalent
Study of the Riemann mapping theorem, meromorphic and entire functions, topics in analytic function theory.

MATH 736 - (3) (Y)
Mathematical Theory of Probability

Prerequisite: MATH 731 or equivalent
Rigorous introduction to probability, using techniques of measure theory. Includes limit theorems, martingales, and stochastic processes.

MATH 741 - (3) (Y)
Functional Analysis I

Prerequisites: MATH 734 and MATH 731 or equivalent
Study of the basic principles of linear analysis, including spectral theory of compact and self adjoint operators.

MATH 742 - (3) (SI)
Functional Analysis II

Prerequisite: MATH 741 or equivalent
Study of the spectral theory of unbounded operators, semigroups, and distribution theory.

MATH 751, 752, (4) (Y)
Algebra I, II

Prerequisites: MATH 551, 552 or equivalent
Study of groups, rings, fields, modules, tensor products, and multilinear functions.

MATH 753 - (3) (Y)
Algebra III

Prerequisites: MATH 751, 752 or equivalent
Study of the Wedderburn theory, commutative algebra, topics in advanced algebra.

MATH 760 - (3) (SI)
Homological Algebra

Study of modules, algebras; Ext and Tor; cohomology of groups and algebras; differential graded modules, algebras, coalgebras; spectral sequences; and homological dimension.

MATH 780 - (3) (Y)
Differential Topology

Prerequisites: MATH 531, MATH 577 or the equivalent
Study of the basic theory of smooth manifolds and functions; tangent bundles and vector fields; embeddings, immersions, and transversality.

MATH 781 - (3) (Y)
Algebraic Topology: Homology Theory

Prerequisite: MATH 577
Topics include singular homology and cohomology; simplicial and CW-theory; cohomology ring; cap products and duality.

MATH 782 - (3) (Y)
Algebraic Topology: Homotopy Theory

Prerequisite: MATH 781
Topics include fibrations and cofibrations; homotopy groups; cohomology operations; Eilenberg-MacLane spaces; obstruction theory and spectral sequences.

MATH 783 - (3) (Y)
Algebraic Topology: Fiber Bundles

Prerequisite: MATH 780
Topics include coordinate bundles; principal bundles and classifying spaces; vector bundles and characteristic classes; elementary K-theory.

MATH 825 - (3) (SI)
Differential Equations

Study of topics in the theory of ordinary and partial differential equations.

MATH 830 - (3) (SI)
Topics in Function Theory

Study of topics in real and complex function theory.

MATH 831, 832 - (3) (Y)
Operator Theory I, II

Study of topics in the theory of operators on a Hilbert space and related areas of function theory.

MATH 836, 837 - (3) (Y)
Topics in Probability Theory and Stochastics Processes

Study of topics in probability; stochastic processes and ergodic theory.

MATH 840 - (3) (SI)
Harmonic Analysis

Study of Banach and C* algebras, topological vector spaces, locally compact groups, Fourier analysis. Topics selected by instructor.

MATH 845 - (3) (Y)
Topics in Mathematical Physics

Application of functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.

MATH 851 - (3) (SI)
Group Theory

Study of the basic structure theory of groups, especially finite groups.

MATH 852 - (3) (SI)
Representation Theory

Study of the foundations of representation and character theory of finite groups.

MATH 853 - (3) (SI)
Algebraic Combinatorics

Geometries, generating functions, partitions, and error-correcting codes and graphs are studied by using algebraic methods involving group theory, number theory, linear algebra and others.

MATH 855 - (3) (SI)
Theory of Algebras

Study of the basic structure theory of associative or nonassociative algebras.

MATH 860 - (3) (SI)
Commutative Algebra

Study of the foundations of commutative algebra, algebraic number theory, or algebraic geometry.

MATH 862 - (3) (SI)
Algebraic Geometry

Study of the foundations of algebraic geometry.

MATH 865 -(3) (SI)
Algebraic K-Theory

Topics include projective class groups and Whitehead groups; Milnorís K2 and symbols; higher K-theory and finite fields.

MATH 870 - (3) (Y)
Lie Groups

Study of basic results concerning Lie groups, Lie algebras, and the correspondence between them.

MATH 871 - (3) (Y)
Lie Algebras

Study of basic structure theory of Lie algebras.

MATH 872 - (3) (SI)
Differential Geometry

Study of differential geometry in the large; connections; Riemannian geometry; Gauss-Bonnet formula; differential forms, and other topics.

MATH 875 - (3) (SI)
Topology of Manifolds

Study of manifolds from the topological, piecewise-linear, or smooth point of view; topics selected from imbeddings, smoothing theory, Morse theory, index theory, and s-cobordism.

MATH 880 - (3) (SI)
Generalized Cohomology Theory

Topics include the axiomatic generalized cohomology theory; representability and spectra; spectra and ring spectra; orientability of bundles in generalized cohomology theory; Adams spectral sequence, and stable homotopy.

MATH 883 - (3) (SI)
Cobordism and K-Theory

Study of classical cobordism theories; Pontryagin-Thom construction; bordism and cobordism of spaces; K-theory and Bott periodicity; formal groups, and cobordism.

MATH 885 - (3) (Y)
Topics in Algebraic Topology

Study of selected advanced topics in algebraic topology.

MATH 888 - (3) (SI)
Transformation Groups

Study of groups of transformations operating on a space; properties of fixed point sets, orbit spaces; and local and global invariants.

MATH 896 - (3-12) (Y)
Thesis

MATH 897 - (3-12) (Y)
Non-Topical Research, Preparation for Research

For masterís research, taken before a thesis director has been selected.

MATH 898 - (3-12) (Y)
Non-Topical Research

For masterís thesis, taken under the supervision of a thesis director.

MATH 931 (3) (Y)
Operator Theory Seminar

MATH 936 - (3) (Y)
Probability Seminar

MATH 941 - (3) (Y)
Analysis Seminar

MATH 945 - (3) (Y)
Mathematical Physics Seminar

MATH 950 - (3) (Y)
Algebra Seminar

MATH 980 - (3) (Y)
Topology Seminar

MATH 996 - (3-9) (Y)
Independent Research

MATH 997 - (3-12) (Y)
Non-Topical Research, Preparation for Doctoral Research

For doctoral research, taken before a dissertation director has been selected.

MATH 999 - (3-12) (Y)
Non-Topical Research

Fordoctoral dissertation, taken under the supervision of a dissertation director.

The Mathematics Colloquium is held weekly, the sessions being devoted to research activities of students and faculty members, and to reports by visiting mathematicians on current work of interest.


Continue to: Departmental Degree Requirements
Return to: Chapter 5 Index