Application Deadlines	Current Research Projects Click on the image for more information . Louis F. Rossi
Program policy statement
Catalog information
Major research groups
Travel support
Typical program of studies
Graduate exams
Current graduate students
Graduate courses
Ph.D. Dissertation Abstracts
Graduate Student Travel Request Form
Faculty Supervised Study Numbers

Application Deadlines

In order to receive full consideration for financial aid, prospective students must complete their applications by February 1st for admission in Fall semester and December 1st for admission in Spring semester. Applications received after those dates may still be considered, but financial aid may no longer be available. Please see the online application information. The GRE subject test is not required to apply to our program.

Program Policy Statement

Old program description: The old Blue Book for students admitted in or before Fall of 2003. Note: students admitted in or before Fall of 2003 can participate in the program under the old guidelines or the new guidelines.

Research Areas

Some projects for the summer of 2008 intended for students after their first year are available. The first professor is the contact in each case. If you are interested in working with a professor who is not listed there, please contact him or her directly.

Graduate Student Travel

Each graduate student who passes both the preliminary and candidacy exams will be awarded $1,000 of travel money for the remainder of their studies at Delaware. This money can be spent on a combination of approved travel at the choice of the student.

The student can apply to use this money to pay the costs of attending scientific meetings or other educational/academic events. Forms for applying for travel approval are available from Debbie See. The form must be completed and approved before travel commences. In no cases will students be reimbursed for travel without prior approval of this form.

In no case will any funds beyond the $1,000 total be paid. We recommend that you reserve some of your travel money to attend a national meeting for recruitment purposes late in your time at Delaware.

Students who wish to return to their undergraduate school to help recruit new graduate students may petition the department for extra support.

Students may also petition the Graduate Committee to receive partial funding to attend special training programs/workshops or clinics. Such requests must be clearly documented and will only be granted in unusual cases.

Typical Program of Study

This document outlines a typical path of courses through the applied mathematics and pure mathematics PhD programs. It assumes that the student starts in the fall of academic year one with an undergraduate degree in mathematics.

The actual sequence of courses taken by any given student is likely to differ from this sequence based on individual needs, interests and when the student enters the program.
 

Applied Mathematics

Mathematics

A student interested in Combinatorics may choose

A student interested in Analysis may choose

Graduate Exams

Preliminary Examinations

Those students who wish to continue toward a Ph.D. degree must take two written examinations called the Preliminary Examinations. One written Preliminary Examination covers material from Math600 and Math602 (Advanced Calculus) and another examination covers material from Math672 (Linear Algebra) or from Math 611 (Numerical Linear Algebra). These examinations will be given twice each year, once before the start of the Fall Semester, and once before the start of the Spring Semester. For students entering with a Bachelor's degree, it is required that the Preliminary Examinations be passed by the beginning of the fourth semester of study. Providing it is before the beginning of the fourth semester, a student may take these examinations several times. Only the part not passed needs to be repeated. Students who failed to pass both subject areas of the Preliminary Examinations by the beginning of the fourth semester will be asked to leave the graduate program.

Advanced Calculus

Topology of Euclidean spaces. Bolzano-Weierstrass and Heine-Borel Theorems. Continuity and uniform continuity. Differentiability, Mean value theorems, Taylor's theorem. Sequences and series of real numbers and power series. Pointwise and uniform convergence of sequences and series of functions. Riemann-Stieltjes Integral.

Differentiability of functions of several variables. Inverse function theorem. Implicit function theorem. Multiple integrals. Line and surface integrals. Green's Theorem, Stokes' Theorem, Divergence Theorem.

References:

• Walter Rudin, Principles of Mathematical Analysis (Third edition).
• Tom M. Apostol, Mathematical Analysis (First edition).
• R. Creighton Buck,. Advanced Calculus.

Linear Algebra

Subspaces, bases, dimension, Linear transformations and matrix representations, Linear functionals, adjoints, and dual spaces, Scalar products and orthogonality, Symmetric, hermitian, and unitary operators, Eigenvectors and eigenvalues (spectral theorem), Jordan canonical form

References:

• Sh. Axler, Linear Algebra Done Right, Springer-Verlag
• M. L. Curtis, Abstract Linear Algebra,(skip Chapter 2(B) and Chapter 5), (Springer-Verlag).
• K. Huffman and R. Kunze, Linear Algebra, (skip Chapters 6.5, 7.2, 7.6, 9.4, 9.6, 10.6), (Prentice-Hall).
• S. Lang, Linear Algebra, Third Edition, (Springer-Verlag).

Note: Material skipped in one of these references might be found in other references, but presented in a more elementary form.

Numerical Linear Algebra

Direct and iterative methods for the solution of linear systems, LU factorization, row pivoting, stable QR factorization, solution of linear least squares problems by normal equations and QR, stability and conditioning issues, power and inverse iterations, QR iteration, singular value decomposition, simple iterations for sparse matrices, conjugate gradients and other Krylov subspace iterations.

Suggested References:

• L.N. Trefethen and D. Bau, III, Numerical Linear AlgebraSIAM (see e.g., I, II, III.)
• J.W. Demmel, Applied Numerical Linear Algebra, SIAM (Sections 2.1, 2.2, 2.3, 2.4, 3.1-3.3, 4.4, 6.5, 6.6)
• A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag (Sections 3.1, 3.3, 4.1, 4.2, 4.3, 5.2, 5.3, 5.4, 5.5, 5.8)

Candidacy Exams

This written examination is administered in February. A student entering with a Bachelor's degree must pass the Candidacy Exam by the beginning of his/her 6th semester of study (by the 4th semester of study for those entering with a Master's degree). A second and final attempt is permitted in the following August. Dismissal will be recommended for a student who does not pass the Candidacy Exam on the second try.

In this examination a student must choose 2 topics from Algebra, Analysis, Applied Mathematics, Discrete Mathematics and Probability. The exams are based on MATH 650 and MATH 845 (Algebra), two chosen from MATH 805, MATH 806 and MATH 807 (Analysis), MATH 616, MATH 617 and MATH 810 (Applied Mathematics), MATH 688 and MATH 689 (Discrete Mathematics) and MATH630, MATH631 and MATH850 (Probability). Another subject area may be substituted for one of the above by petition to the graduate committee based on two graduate level courses and supported by a faculty member.

Elementary Group Theory: Theorems of Lagrange, Cayley, Cauchy, Sylow, Hölder; direct, semi-direct products.

Permutation Groups and Group Actions: k-transitivity, primitivity, regular actions, class equation.

Enumeration: Orbit-Stabilizer Theorem, Burnside's Lemma.

Group Quotient Structures and Homomorphism Theorems.

Series, Nilpotency and Solvability.

Structure of Cyclic Groups and Finitely-Generated Abelian Groups: elementary divisors, invariant factors.

Structure of Classical Groups: symmetric group, alternating group, group of semilinear transformations, general linear group, special linear group, projective semilinear and linear groups.

Automorphism Groups of Groups and Fields.

Elementary Ring Theory: ideals, quotient rings, polynomial rings, fields of quotients, modules.

PID's, UFD's, and Euclidean Domains: primes and irreducible elements, factorization and irreducibility of polynomials.

Field Extensions: algebraic, transcendental, separable, and simple extensions; minimal polynomials; splitting fields; primitive elements; uniqueness, construction, and structure of finite fields./

Fundamental Theorem of Galois Theory: statements and applications (details of proof will not be required).

References

• Algebra, A Graduate Course by I. Martin Isaacs (Brooks/Cole): Chapters 1-7, 12, 16, 17, 20A, 20B, 21.
• Abstract Algebra by D. S. Dummit and R. M. Foote (Prentice Hall, second edition): Chapters 1-5, 6.1, 6.2, 7-9, 13.1 -- 13.6, 14.1 -- 14.3.

Syllabus for the Analysis (M805, M806, M807)
Candidacy Exam Approved by the graduate committee on 5/16/99

Note: Students must choose to be examined in two of the following areas when registering to take the exam:

• Real analysis (M805)
• Functional Analaysis (M806)
• Complex Variables (M807)

Real Analysis Component (M805)

• Properties of Lebesgue outer measure and Lebesgue measurable sets in .
• Properties of Lebesgue measurable functions. Concepts of convergence almost everywhere and convergence in measure.
• Properties of Lebesgue integrals and various convergence theorems. Relation between Lebesgue integral and Riemann-Stieltjes integrals.
• Repeated integration. Fubini's and Tonelli's Theorems.
• Differentiation. Vitali's covering lemma and its applications. Absolutely continuous and singular functions.
• Spaces.

References:

• Measure and Integral by R. Wheeden and A. Zygmund
• Real Analysis by H. Royden

Further References:

• Theory of Functions of Real Variables by I. Natanson
• Theory of Functions by E.C. Titchmarsh

Functional Analysis Component (M806)

• Normed linear spaces; Banach spaces and Hilbert spaces
• Linear operators
• Linear functionals; Hahn Banach theorem
• Open mapping theorem; closed graph theorem
• Uniform boundedness principle
• Compact operators Hilbert-Schmidt theorem and Fredholm alternative
• Weak topology

References

Kreyszig, "Introduction to Functional Analysis"

Complex Variables Component (M807)

• Limits, continuity and uniform convergence;
• Differentiation, analytic functions and elementary properties of conformality;
• Mobius transformation;
• Complex integrals, Cauchy integral theorem;
• Cauchy integral formula and its implications;
• Power series, uniform convergence;
• Laurent series;
• singular points;
• The residue theorem and its implications.

References:

• L. Alfors, "Complex Analysis"
• J. Conway, "Functions of a Complex Variable".

Syllabus for the Applied Mathematics (M616, M617, M810) Candidacy Exam
Approved by the graduate committee on 5/16/99

• Fourier Series: Completeness, Parseval equation, convergence theorems, Sturm-Liouville problem, separation of variables.
• The Fourier Transform: Fourier transforms of square integrable functions, inversion theorems, applications to partial differential equations.
• Nonlinear Ordinary Differential Equations: Autonomous systems, critical points and stability, Liapunov's Method.
• Classification of Second Order Partial Differential Equations, Reduction to Canonical Form.
• Hyperbolic Equations: The wave equation, initial value problems, Huygen's principle, initial-boundary value problems in two and three space variables, uniqueness theorems, simple nonlinear problems.
• Parabolic Equations: The heat equation, initial value problems, initial-boundary value problems, uniqueness theorems, maximum principle.
• Elliptic Equations: Laplace's equation, potential theory, Green's functions, maximum principles, scattering theory.
• Asymptotic Expansion of Integrals: Laplace's Method, the method of steepest descent, the method of stationary phase.
• Asymptotic Solutions of Ordinary Differential Equations: WKB method, turning point problems.
• Perturbation Theory for Ordinary and Partial Differential Equations: Multiscale method, matched asymptotic expansions.

References

• A First Course in Partial Differential Equations by H. F. Weinberger: Chapters 1-7, 10.
• Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou and D. W. Thoe: Chapters 5-10.
• A Second Course in Elementary Differential Equations by Paul Waltman: Chapter 2.
• Asymptotic Analysis by J. D. Murray: Chapters 1-4, 6, 7.

Syllabus for the Discrete Mathematics (M688, M689) Candidacy Exam
Approved by the graduate committee in Fall, 2004.

• Combinatorial Enumeration: elementary counting, generating functions, Sieve method.
• Designs, set systems and combinatorial geometries. Linear spaces, De Bruijn-Erdos inequality. Balanced incomplete block designs. Fisher's inequality, Bruck-Ryser-Chowla theorem. Applications to Steiner systems. Latin squares. Hadamard matrices. Finite affine planes and finite projective planes. Desargues' and Pappus' Theorems.
• Graphs and diagraphs. Degree sequences, paths and cycles, Euler tours and Hamiltonian graphs, vertex and edge colorings, planarity, chromatic polynomials. Trees. Connectivity. Matchings and coverings in bipartite graphs: Theorems of König and Hall. Cliques and independent sets, Turan's theorem. Ramsey theory. Networks, flows, cuts.
• Linear algebra methods in combinatorics.
• Probabilistic method in combinatorics.
• Codes and linear codes: Bounds on parameters. Shannon's theorem. Weight enumerators and MacWilliams' theorem. Classes of codes: Hamming, Golay codes.

References

• The book Combinatorics by P.J. Cameron, Cambridge University Press (1994) can serve as a general reference, but it does not cover all the topics in the above syllabus. Topics not covered in Cameron's book may be found in one or more of the following references. Some topics may be found in several of these references, so you may pick and choose your favorite presentation. You need not be familiar with all the references. Most importantly, about 85% of the material will be covered in a typical offering of M688/689.
• generatingfunctionology by H. S. Wilf, Second Edition, Academic Press (1994): Chapters 1, 2, 4.1, 4.5-4.7, 4.10.
• Enumerative Combinatorics by R.P. Stanley, Wadworth & Brooks (1986): Chapters 1, 2.1-2.3.
• Modern Graph Theory by B. Bollobas, Springer (1998): Chapters 1.1-1.4, 3.1-3.4, 4.1-4.3, 5.1, 5.2, 6.1 (Theorems 1-4), 6.3, 6.4 (Theorems 20, 21, 27), 7.1.
• Graph Theory with Applications by J.A.Bondy and U.S.R. Murty, North-Holland (1976): Chapters 1.1-1.7, 2.-2.4, 3.1, 3.2, 4.1, 4.2, 5.1-5.3, 6.1, 6.2, 7.1-7.3, 8.1. 8.2, 8.4, 8.5, 9.1-9.3, 9.5 (statements only), 10.1-10.3, 11.1-11.4.
• Linear Algebra Methods in Combinatorics by L. Babai and P. Frankl, Department of Computer Science, University of Chicago (preprint, version 1992): Chapters 1.1, 1.2, 1.5, 4.3, 4.4, 5.4-5.7, 9.3.
• The Probabilistic Method by N. Alon and J.H. Spencer, John Willey & Sons, Inc., 2nd edition (2000): Chapters 1, 2.1-2.5, p.38, 4, 14.1.
• Introduction to the Theory of Error-Correcting Codes by V. Pless, John Willey & Sons (1982): Chapters 1, 2.2-2.4, 4, 8.1, 8.2.
• A course in combinatorics by Van Lint and Wilson,Cambridge Univ. Press, 2nd edition (2000): Chapters: 1-5, 10, 13-23, 26-29, 33.

Syllabus for the Probability (M630, M631, M850) Candidacy Exam
Approved by the department on October 18, 2005

• Probability space and measure; sigma algebras; Caratheodory’s extension theorem; Kolmogorov’s extension theorem; independence.
• Random variables; expected value; joint distributions; moment generating functions; characteristic functions; Chebyshev’s inequality; large deviations.
• Modes of convergence (weak, in probability, L^p, almost sure) and properties.
• Weak laws of large numbers for sequences and triangular arrays; L² weak law; applications to analysis and combinatorics.
• Borel-Cantelli lemmas; strong law of large numbers; convergence of random series.
• Weak convergence; central limit theorem for sequences and triangular arrays; Poisson convergence; applications.
• Stopping times; Wald’s equation; conditional probability/expectation; Martingales; Radon-Nikodym theorem; optional stopping theorems; applications.
• Stochastic processes and their properties; discrete and continuous time Markov chains; birth-death process and Poisson process; renewal theory.
• Brownian motion and related Gaussian/diffusion processes; Donsker’s theorem.
• Stochastic simulations

References

• Introduction to Probability Models by Sheldon Ross (Eighth edition): Chapters 1-7, 10-11.
• Probability and random processes, by Geoffrey R Grimmett and David R. Stirzaker (Third edition): Chapter 1-10, 12-13.
• Probability and random processes, problems and solutions, by Geoffrey R Grimmett and David R. Stirzaker.
• Probability: Theory and Examples, by Richard Durrett (Third edition): Chapters 1-4, 7.1

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Last Modified: March 7, 2008
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