Ph.D. Candidate, Washington
University in St. Louis
Graduate Courses taken or graded (1991-97):
Mathematics 404 Numerical Methods (S 94).
Calculus of finite differences. Interpolation. Numeric
integration. Optimization. Systems of algebraic equations. Systems of ordinary
differential equations.
Mathematics 407 An Introduction to Differential Geometry (grader
S 95).
Properties of space curves and surfaces.
Mathematics 408 Nonparametric Statistics (S 93).
Order statistics, Pitman randomization, rank-order test,
runs test, goodness-of-fit tests.
Mathematics 411 Advanced Calculus I (F 91, grader F 93).
Limits and infinite series; limits of sequences, convergence
of series. Convergence test. Series of functions. Taylor series. Fourier
series; orthogonality, Bessel's inequality, convergence. Applications.
Weierstrass's approximation theorem. Integration; Stieltjes integrals,
mean value theorems, improper integrals, divergent integrals, special functions.
Mathematics 412 Advanced Calculus II (S 92, grader S 94).
Differential calculus of functions of several variables;
partial derivatives, the chain rule, differentials, Taylor's theorem, maxima
and minima, Lagrange multipliers. Integral calculus of functions of several
variables; multiple integrals, iterated integrals, line and surface integrals,
exact differentials, theorems of Gauss, Stokes and Green, change of variables
in integration. The implicit function theorem.
Mathematics 417 Introduction to Topology & Modern Analysis I
(F 91).
Introduction to set theory, transfinite methods, cardinal
and ordinal numbers; metric spaces, Euclidean spaces, spaces of continuous
functions; complete spaces, Baire Category theorem, contraction mapping
theorem; compactness, separability, bases and subbases, Lindelof's theorem;
introduction to topological spaces.
Mathematics 418 Introduction to Topology & Modern Analysis II
(S 92).
Compactness, separation and connectedness in topological
spaces, with emphasis on their significance in metric spaces; product and
quotient spaces; approximation theorems; Tychonoff's theorem, Ascoli's
theorem, Tietze's extension theorem, Stone-Cech compactification.
Mathematics 421 Theory of Functions of a Complex Variable I ( F
92, Ph.D.Q.).
Elementary properties of complex numbers. Limits, continuity
and differentiability of complex valued functions. Line integrals. Cauchy's
theorem and applications. Expansions in power series. Local behavior of
regular functions. Isolated singularities. The point at infinity. Elementary
functions. Introduction to conformal mappings. Calculus of residues and
its applications.
Mathematics 422 Theory of Functions of a Complex Variable II (S
93, Ph.D.Q.).
Normal families. Riemann mapping theorem. Poisson integral.
Jensen's theorem. Analytic continuation. Univalent functions. Integral
and meromorphic functions. Elliptic functions.
Mathematics 429 Linear Algebra (F 91).
Linear equations and matrices: row-reduction, matrix
multiplication, matrix inversion. Vector spaces: bases, coordinates, row
equivalence. Linear transformations: isomorphism, matrix representations,
similarity. Determinants: product formula, Cayley-Hamilton theorem, eigenvectors,
eigenvalues. Inner product spaces: linear functionals and adjoints, unitary
and normal operators, spectral theorem.
Mathematics 430 Abstract Algebra (grader S 93).
Integers: Euclidean algorithm, unique factorization,
congruence. Groups: matrix groups, permutation groups, subgroups, cosets,
Lagrange's theorem, Cayley's theorem, homomorphism, normal subgroups, quotient
groups, Sylow's existence theorem. Rings: matrix rings, polynomial rings,
homomorphism, ideals, maximal and prime ideals, field of quotients, principal
ideal domains. Fields: characteristic, dimension, geometric constructions,
splitting a polynomial, finite fields.
Mathematics 431 Algebra I (F 92).
Groups: Actions of groups, class formula, symmetric groups,
Sylow theorems, Permutation groups, Abelian groups, Jordan-Holder Theorem.
Rings and Modules: Polynomial rings, localisation, Noetherian rings, Hilbert
basis theorem, Principal ideal domains, Dedekind domains, Unique factorisation
domains, Gauss' lemma, Integral extensions, Noether normalisation theorem.
Fields: Extension of fields, finite fields, Galois theory, Kummer theory.
More ring theory: Hilbert Nullstellensatz, Primary decomposition.
An introduction to algebraic structure: groups, rings,
integral domains, division rings, fields. Applications to curcuit design
and analysis, algebraic coding theory, atomic physics.
Mathematics 434 Survival Analysis, (grader F 95).
Textbook: Elisa T. Lee - Statistical Methods for survival
data analysis, Second Edition. John Wiley & Sons 1992.
Mathematics 439 Linear Statistical Models (F 92).
Unified treatment of those statistical methods having
their basis in linear algebra. General linear hypothesis, joint confidence
regions, experimental design models, variance components models.
Mathematics 441 Geometry I ( F 93 Ph.D.Q.).
Differentiability, mean value theorem and chain rule
for mappings of Rn into Rm. Special
properties of infinite order smooth (C infinity) functions and mappings,
tangent vectors at a point of Rn. Inverse function theorem,
implicit function theorem and theorem on rank. Definition and examples
of differentiable manifolds and submanifolds. Tangent space to a manifold,
immersions and embeddings. Vector fields and one-parameter groups, existence
theorem for ordinary differential equations. Lie groups as an example of
a manifold, action of a group on a manifold. Frobenius' theorem. Covectors
and covector fields, bilinear forms on a manifold. Partitions of unity
and applications.
Mathematics 442 Geometry II (S 94, Ph.D.Q.).
Tensor fields and their behavior under mappings. Tensor
product and exterior product. The calculus of differential forms. Stokes'
Theorem for manifolds with boundary. Covariant differentiation on Riemannian
manifolds, parallel transport, geodesics. Examples from classical differential
geometry and Lie groups. Riemannian curvature and symmetric spaces. Further
topics from the geometry of Riemannian manifolds and Lie Groups.
Mathematics 451 Measure Theory and Functional Analysis I (F 93 Ph.D.Q.).
Set functions and the construction of measures. The Hahn-Jordan
decomposition. Lebesgue and Lebesgue-Stieltjes measure. Definition of the
integral and convergence theorems. Product measures and Fubini's theorem.
Convergence in the space measurable functions: pointwise convergence, convergence
in measure, convergence in Lp. Egoroff's theorem. The Riesz-Fischer
theorem Lp spaces and the Riesz representation theorem.
Mathematics 452 Measure Theory and Functional Analysis II (S 94,
Ph.D.Q.).
Topological groups and Haar measure. Topological spaces.
Hahn-Banach theorem. Weak topologies. The closed graph theorem. The Banach-Steinhaus
theorem. Locally convex spaces and the Krein-Milman theorem. Measures on
locally compact spaces.
Mathematics 475 Statistical Computation (F 94).
Practice and theory of statistical computation. 1st half
of course is a thourough introduction to statistical software, emphasizing
SAS. 2nd half covers the computational matrix algebra underlying most statistical
models, computational aspects of maximum likelihood, and the resampling
methods known as the jackknife and bootstrap.
Mathematics 493 Probability (grader F 94).
Theory and application of mathematical probability.
Mathematics 494 Mathematical Statistics (grader S 95).
Parametric and nonparametric significance and hypothesis
testing; order statistics; theory of estimation; theory of runs, sampling
schemes, analysis of variance, sequential analysis.
Engineering and Policy 571 Data Analysis and Experimental Design
(F 94).
Techniques of data analysis and experimental design for
decision making. Topics include: statistical design, regression, analysis
of variance, variance components, response surface methodology and elementary
probability modeling.
Mathematics 500 (reading course) Probability Theory I (F 94 Ph.D.Q.).
textbook: Richard Durrett - Probability: Theory and Examples,
Duxbury Press 1991.
Mathematics 500 (reading course) Probability Theory II (S 95 Ph.D.Q.).
textbook: Richard Durrett - Probability: Theory and Examples,
Duxbury Press 1991.
Mathematics 500 (reading course) Survival Analysis (S 95).
Textbook: Elisa T. Lee - Statistical Methods for survival
data analysis, Second Edition. John Wiley & Sons 1992.
Mathematics 590 (reading course) Biostatistics (S 95).
textbook: Bernard Rosner - Fundamentals of Biostatistics,
Fourth Edition, Duxbury Press 1995. Topics include: prevalence and incidence,
prospective and retrospective studies, point and interval estimation: normal
theory and exact methods, one-sample and multi-sample inference, categorical
data analysis, multiple logistic regression, person-time data estimation
and inference, introduction to survival analysis.
Mathematics 597 Teaching Seminar (S 92).
Minor Oral Examination: Bootstrap methods for statistical estimation
(4/17/96 Ph.D.Q.)
Advisor: Stanley Sawyer.
Other committee member: Edward Spitznagel.
Major Oral Examination: Survival analysis and martingales (11/13/96
11/20/96 Ph.D.Q.)