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Mathematics

Graduate Courses

501. GENERAL TOPOLOGY I. . Metric spaces, Baire’s theorem, topological spaces, continuity, separation axioms, connectedness, compactness, quotient and product topologies. Prerequisite: Math 305 with minimum grade of C. (3)

502. GENERAL TOPOLOGY II. Algebraic invariants in topology. Prerequisite: Math 501 with minimum grade of C. (3)

513. THEORY OF NUMBERS I. Divisibility; properties of prime numbers; congruences and modular arithmetic; quadratic reciprocity; representation of integers as sums of squares. Prerequisite: Math 305. (3)

514. THEORY OF NUMBERS II. Arithmetic functions and their distribution; distribution of prime numbers; Dirichlet characters and primes in arithmetic progression; partitions. Prerequisite: Math 513, Math 555 or approval of the instructor. (3)

519. MATRICES. Basic matrix theory, eigenvalues, eigenvectors, normal and Hermitian matrices, similarity, Sylvester’s Law of Inertia, normal forms, functions of matrices. (3).

520. LINEAR ALGEBRA. An introduction to vector spaces and linear transformations; eigenvalues and the spectral theorem. (3).

525. INTRODUCTION TO ABSTRACT ALGEBRA I. General properties of groups. (3).

526. INTRODUCTION TO ABSTRACT ALGEBRA II. General properties of rings and fields. Prerequisite: MATH 525. (3).

533. TOPICS IN EUCLIDEAN GEOMETRY. A study of incidence geometry; distance and congruence; separation; angular measure; congruences between triangles; inequalities; parallel postulate; similarities between triangles; circle area. Prerequisite: MATH 305 or graduate standing. (3).

537. NON-EUCLIDEAN GEOMETRY. Brief review of the foundation of Euclidean plane geometry with special emphasis given the Fifth Postulate; hyperbolic plane geometry; elliptic plane geometry. (3).

540. HISTORY OF MATHEMATICS. Development of mathematics, especially algebra, geometry, and analysis; lives and works of Euclid, Pythagoras, Cardan, Descartes, Newton, Euler, and Gauss. Prerequisite: Math 305 or consent of instructor. (3).

545. SELECTED TOPICS IN MATHEMATICS FOR SECONDARY SCHOOL TEACHERS. High school subjects from an advanced point of view; their relation to the more advanced subjects. (3).

555. ADVANCED CALCULUS I. Suprema and infima on the real line, limits, liminf and limsup of a sequence of reals, convergent sequences, Cauchy sequences, series, absolute and conditional convergence of series. Prerequisite requirements for this course may also be satisfied by consent of instructor. Prerequisite: Math 305 with minimum grade of C. (3)

556. ADVANCED CALCULUS II. Limit of a function, metric spaces, limits in metric spaces, complete metric spaces, uniform continuity, pointwise and uniform convergence of sequences and series of functions, power series. Prerequisite: Math 555 with minimum grade of C. (3)

567. INTRODUCTION TO FUNCTIONAL ANALYSIS I. Hilbert spaces, Banach spaces , Hahn-Banach Theorem, Banach Steinhaus Theorem, Open Mapping Theorem, weak topologies, Banach-Alaoglu Theorem, Classical Banach spaces. Prerequisite: Math 556 with minimum grade of C. Prerequisite requirements for this course may also be satisfied by consent of instructor. (3)

568. INTRODUCTION TO FUNCTIONAL ANALYSIS II.Topics in Banach space theory. Prerequisite: Math 567 with minimum grade of C. (3)

569. THEORY OF INTEGRALS. Continuity, quasi-continuity, measure, variation, Stieltjes integrals, Lebesgue integrals. (3).

571. FINITE DIFFERENCES. Principles of differencing, summation, and the standard interpolation formulas and procedures. (3).

572. INTRODUCTION TO PROBABILITY AND STATISTICS. Emphasis on standard statistical methods and the application of probability to statistical problems. Prerequisite: MATH 264. (3).

573. APPLIED PROBABILITY. Emphasis on understanding the theory of probability and knowing how to apply it. Proofs are given only when they are simple and illuminating. Among topics covered are joint, marginal, and conditional distributions, conditional and unconditional moments, independence, the weak law of large numbers, Tchebycheff’s inequality, Central Limit Theorem. Prerequisite: MATH 264. (3).

574. PROBABILITY. Topics introduced in MATH 573 will be covered at a more sophisticated mathematical level. Additional topics will include the Borel-Cantelli Lemma, the Strong Law of Large Numbers, characteristic functions (Fourier transforms). Prerequisite: MATH 573. (3).

575. MATHEMATICAL STATISTICS I. Mathematical treatment of statistical and moment characteristics; probability models; random variables; distribution theory; correlation; central limit theory; multi-parameter models. Prerequisite: Math 262 with minimum grade of C. (3)

576. Mathematical treatment of statistical inference; maximum likelihood estimation and maximum likelihood ratio test; minimum variance unbiased estimators; most powerful tests; asymptotic normality and efficiency; Baysian statistics. Prerequisite: Math 575 with minimum grade of C. (3)

577. APPLIED STOCHASTIC PROCESSES. Emphasis on the application of the theory of stochastic processes to problems in engineering, physics, and economics. Discrete and continuous time Markov processes, Brownian Motion, Ergodic theory for Stationary processes. Prerequisite: MATH 573 or consent of instructor. (3).

578. STOCHASTIC PROCESSES. Topics will include General Diffusions, Martingales, and Stochastic Differential Equations. (3).

590. TECHNIQUES IN TEACHING COLLEGE MATHEMATICS. Directed studies of methods in the presentation of college mathematics topics, teaching and testing techniques. Z grade. This course is required of all teaching assistants, each semester, and may not be used for credit toward a degree. Prerequisite: departmental consent. (1-3).

597. SPECIAL PROBLEMS. (1-3).

598, 599. SPECIAL PROBLEMS. (Same as MATH 597).

625. MODERN ALGEBRA I. Advanced group theory. (3)

626. MODERN ALGEBRA II. Advanced theory of rings and fields, including Galois theory and modules. (3)

631. FOUNDATIONS OF GEOMETRY. Development of Euclidean geometry in two and three dimensions using the axiomatic method; introduction to high dimensional Euclidean geometry and to non-Euclidean geometrics. (3).

639. PROJECTIVE GEOMETRY. Fundamental propositions of projective geometry from synthetic and analytic point of view; principle of duality; poles and polars; cross ratios; theorems of Desargues, Pascal, Brianchon; involutions. (3).

647. TOPICS IN MODERN MATHEMATICS. Survey of the more recent developments in pure and applied mathematics. Prerequisite: consent of instructor. (3).

649. CONTINUED FRACTIONS. Arithmetic theory; analytic theory; applications to Lyapunov theory. Prerequisite: consent of instructor. (3).

655. THEORY OF FUNCTIONS OF COMPLEX VARIABLES I. Complex numbers, analytic functions, complex integration, Cauchy’s theorem and integral formula, Liouvilles’s theorem, maximum modulus principle, Schwarz’s lemma, sequences and series of analytic functions, isolated singularities, the residue theorem. (3)

656. THEORY OF FUNCTIONS OF COMPLEX VARIABLES II. Conformal mappings, analytic continuation, harmonic functions, infinite products. (3)

661. NUMERICAL ANALYSIS I. Numerical linear algebra, error analysis, computation of eigenvalues, and eigenvectors, finite differences. (3)

662. NUMERICAL ANALYSIS II. Techniques for ordinary and partial differential equations, stability and convergence analysis. (3)

663. SPECIAL FUNCTIONS. Advanced study of gamma functions; hypergeometric functions; generating function; theory and application of cylinder functions and spherical harmonics.
(3).

667. FUNCTIONAL ANALYSIS II. Topological vector spaces (tvs), complete tvs, product and quotient tvs, separation theorems for convex sets, locally convex spaces, Krein-Milman theorem, linear operators, dual pairs and Mackey-Arens theorem, Alaoglu-Bourbaki thorem, bornological and barreled spaces.

668. FUNCTIONAL ANALYSIS II. Topics in applied functional analysis.

669. PARTIAL DIFFERENTIAL EQUATIONS I. Classical theories of wave and heat equations. Prerequisite: MATH 353 or MATH 555. (3).

670. PARTIAL DIFFERENTIAL EQUATIONS II. Hilbert space methods for boundary value problems. Prerequisite: MATH 669. (3).

673. ADVANCED PROBABILITY I. Topics in probability are treated at an advanced level. Measure theoretic foundations, infinitely divisible laws, stable laws. Corequisite: Math 654. (3)

674. ADVANCED PROBABILITY II. Multidimensional central limit theorem, strong laws, law of the iterated logarithm. Prerequisite: Math 673 with minimum grade of C. (3)

677. ADVANCED STOCHASTIC PROCESSES I. Topics in the theory of stochastic processes, separability, martingales, stochastic integrals, the Wiener process. Prerequisite: Math 674 with minimum grade of C. (3)

678 ADVANCED STOCHASTIC PROCESSES II. Gaussian processes, random walk, Ornstein-Uhlenbeck process, semi-group theory for diffusions. Prerequisite: Math 677 with minimum grade of C. (3)

679. STATISTICAL BIOINFORMATICS. Introduction to bioinformatics—an interdisciplinary study that combines techniques and knowledge in mathematical, statistical, computational, and life sciences in order to understand the biological significance of genetic sequence data. Prerequisite: Math 575 with minimum grade of C. (3)(3).

681. GRAPH THEORY I. Primarily topics in Matroid Theory including duality, minors, connectivity, graphic matroids, representable matroids, and matroid structure. Connections between the class of matroids with the classes of graphs and projective geometries are also studied. (3)

697. THESIS. (1-12).

700. SEMINAR IN TOPOLOGY. Prerequisite: consent of instructor. (May be repeated for
credit). (3).

710. SEMINAR IN ALGEBRA. Prerequisite: consent of instructor. (May be repeated for credit).
(3).

750. SEMINAR IN ANALYSIS. Prerequisite: consent of instructor. (May be repeated for
credit). (3).

753. THEORY OF FUNCTIONS OF REAL VARIABLES I. Lebesgue measure and integration; differentiation; bounded variation and absolute continuity of functions. (3)

754. THEORY OF FUNCTIONS OF REAL VARIABLES II. General measure theory. (3)

775. ADVANCED MATHEMATICAL STATISTICS I. Univariate distribution functions and their characteristics; moment generating functions and semi-invariants; Pearson’s system; Gram-Charlier series; inversion theorems. (3).

776. ADVANCED MATHEMATICAL STATISTICS II. Multivariate distributions and regression systems; multiple and partial correlation; sampling theory; statistical hypotheses; power and efficiency of tests. (3).

777. SEMINAR IN STATISTICS. (May be repeated for credit up to a maximum of 9 hours).
(3).

780. SEMINAR IN GRAPH THEORY. Prerequisite: consent of instructor. (May be repeated for credit up to a maximum of 9 hours). (3).

782. GRAPH THEORY II. Topics in Graph Theory including trees, connectivity, matchings, paths, cycles, coverings, planarity, graph colorings, networks and directed graphs. Extremal graph structure, applications, and algorithms will also be studied. (3)

797. DISSERTATION. (1-18)