Doctor
of Philosophy
Requirements An advisory
committee consisting of five members of the graduate faculty
will be appointed for each graduate student who declares his
or her intention to become a candidate for the degree. The candidate
must complete a minimum of 48 course hours of graduate work,
exclusive of the dissertation. This must include the sequences
Modern Algebra (MATH 525, 526); Theory of Functions of Real Variables
(MATH 653, 654); and Theory of Functions of Complex Variables
(MATH 655, 656). Of the 48 course hours, 36 must be in courses
open only to graduate students. Reading knowledge of one foreign
language is required; French, Russian, or German is recommended.
This requirement may be satisfied by the completion of six hours
of an undergraduate language at the sophomore level or by making
an appropriate score on the Graduate School Foreign Language
Test of the Educational Testing
Service.
Written comprehensive exams will be administered covering the required sequences
and one other approved sequence. Successful completion of these comprehensive
exams is required. In addition, the candidate must satisfy the advisory committee
as to the extent of the candidate's research ability and activity, as well
as the suitability and excellence of course work presented. Prospective students
are advised
to communicate and consult freely.
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. MODERN ALGEBRA I. General properties
of groups. (3).
526. MODERN 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. Highschool 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).
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).
653. THEORY OF FUNCTIONS OF REAL VARIABLES
I. Lebesgue measure and integration; differentiation;
bounded variation and absolute continuity
of functions. (3)
654. THEORY OF FUNCTIONS OF REAL VARIABLES
II. General measure theory. (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)
675. 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).
676. ADVANCED MATHEMATICAL STATISTICS II.
Multivariate distributions and regression
systems; multiple and partial correlation;
sampling theory; statistical hypotheses;
power and efficiency of tests. (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)
682. 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)
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).
775. 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).
797. DISSERTATION. (1-18)
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