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Department of Mathematics

Course Descriptions |
Departmental Degree Requirements

**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, Green’s, Stokes’, divergence theorems, conservation of energy, and potential energy functions. Emphasizes physical interpretation, Sturm-Liouville problems and Fourier series, special functions, orthogonal polynomials, and Green’s 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: Picard’s method and basic existence and uniqueness theorems; linear systems; the phase plane and Sturm’s theorems; the Poincaré-Bendixon theorem; and Lyapunov’s 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, Weierstrass’ theorem, inverse function theorem and implicit function theorem, integration of differential forms, and Stokes’ 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; Milnor’s 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 master’s research, taken before a thesis director has been selected.

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

For master’s 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