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.

Prerequisite: MATH 310 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.

Prerequisite: MATH 231 or MATH 122 and a knowledge of probability and statistics. MATH 310 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

Prerequisites: MATH 231, 325; 351 recommended.
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.

Prerequisites: MATH 521
Introduces complex variables and partial differential equations. Topics include analytic functions, complex integration, power series, residues, conformal mapping; separation of variables, boundary value problems, Laplace's equation, wave equation, and heat equation.

Prerequisites: MATH 231, 225, 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.

Prerequisite: MATH 231, 225 and 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.

Prerequisites: MATH 351, 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.

Prerequisites: MATH 231, 351
Includes the basic topology of Euclidean spaces, continuity and differentiation of functions of a single variable, Riemann-Stieltjes integration, and convergence of sequences and series.

Prerequisites: MATH 531
Differential and integral calculus in Euclidean spaces, implicit and inverse function theorems, differential forms, and Stokes' Theorem.

Prerequisite: MATH 351 or permission of instructor
This course includes a systematic review of the material usually considered in MATH 351 such as matrices, determinants, systems of linear equations, vector spaces, and linear operators. However, these concepts will be developed over general fields and more theoretical aspects will be emphasized. The centerpiece of the course is the theory of canonical forms, including the Jordan canonical form and the rational canonical form. Another important topic is general bilinear forms on vector spaces. Time permitting, some applications of linear algebra in differential equations, probability, etc., are considered.

Prerequisite: MATH 351 or permission of instructor
Focuses on structural properties of basic algebraic systems such as groups, rings and fields. A special emphasis is made on polynomials in one and several variables, including irreducible polynomials, unique factorization and symmetric polynomials. Time permitting, such topics as group representations or algebras over a field may be included.

Prerequisite: MATH 132 or equivalent and graduate standing
Surveys groups, rings, and fields, and presents applications to other areas of mathematics, such as geometry and number theory. Explores the rational, real, and complex number systems, and the algebra of polynomials.

Prerequisite: MATH 231, 351 or permission of instructor
Study of topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic, and non-Euclidean geometry.

Prerequisite: MATH 231, 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.

Prerequisite: MATH 231; corequisite: MATH 551 or the equivalent
Topological spaces and continuous functions, connectedness, compactness, countability and separation axioms, and function spaces. Time permitting, more advanced examples of topological spaces, such as projectives spaces, as well as an introduction to the fundamental group will be covered.

Prerequisite: Permission of instructor
Presentation of selected topics 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.

Prerequisite: MATH 531 and linear algebra, or the equivalent
Topics include well-posedness and stability of dynamical flows, attractors, invariant manifolds and their properties, dissipative and Hamiltonian systems.

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

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

Study of the fundamental theorems of analytic function theory.

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

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

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.

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

Prerequisite: MATH 531
An introduction to classical mechanics, with topics in statistical and quantum mechanics, as time permits.

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

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

Prerequisites:
Further topics in algebra.

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

Prerequisites: MATH 577
The fundamental group and covering spaces. Simplicial and singular homology. Euler characteristic and degree. Classical applications including fixed point theorems.

Prerequisite: MATH 780
The universal coefficient theorem, the Künneth formula. The cohomology ring: cup and cap products. Manifolds, orientations, the fundamental class, Poincaré duality.

Prerequisite: MATH 781 or permission of instructor
Smooth manifolds and functions, tangent bundles and vector fields. Embeddings, immersions, transversality. Regular values, critical points, degree of maps. Vector bundles, characteristic classes. Differential forms and de Rham cohomology.

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

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

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

Study of topics in real and complex function theory.

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

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

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

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

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

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

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.

Prerequisite: MATH 752
General methods of analyzing groups viewed as discrete subgroups of real algebraic subgroups. Additional topics include the congruence subgroup problem.

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

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

Study of the foundations of algebraic geometry.

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

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

Study of basic structure theory of Lie algebras.

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

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.

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.

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

Study of selected advanced topics in algebraic topology.

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


For master's research, taken before a thesis director has been selected.

For master's thesis, taken under the supervision of a thesis director.









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

For doctoral dissertation, taken under the supervision of a dissertation director.

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