## 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)