# Doctoral Program - Additional Information

Doctoral Program - Additional Information

Graduate Committee

The current members of the department's Graduate Committee are Professors Bauer, Dentcheva, and Zabarankin. Professor Bauer chairs the committee. Graduate students (in particular doctoral students who do not have a thesis advisor yet) are invited to consult them with any questions or concerns.

Stevens Society of Mathematicians

The Stevens Society of Mathematicians (SSM) is an organization devoted to disseminating "post-classroom" mathematics to the Stevens community and the greater scientific and mathematical community. The group is an active student chapter of SIAM, the Society for Industrial and Applied Mathematics.

The General Exam

The General Exam is a written exam that must be passed before starting work on the dissertation. Students should take the general exam as soon as they are ready. Except in unusual circumstances you should have taken the exam by the time you have completed thirty graduate credits toward your Ph.D. degree or within one year after first enrolling in the graduate program at Stevens, whichever occurs later. The general exam is offered twice a year, usually during the third week of January and the third week of September. Any student who wishes to take the exam during either the winter or the fall session must communicate his or her intention at least a month in advance to the Department's Administrative Assistant. The exact date and room will be announced three weeks before the examination. One failure of the General Exam is allowed. A second failure, however, will result in the student being dropped from the Ph.D. Program. At this point, he/she can still obtain a Master's degree, upon completion of the required course work.

The exam consists of two parts taken during the same day. The first part is from 9:00am to 12:00pm and will consist of about six problems similar to the type of questions that appear on final exams of the required courses. The second part of the exam is from 2:00pm to 4:00pm and consists of about four questions covering the same subject areas as the first part of the examination. However, the solutions to the problems may involve a little more "ingenuity" and the ability to see connections between the different subject areas.

The test will cover three subjects: Analysis, Complex Variable and Algebra. The courses needed to prepare for the general examination are: MA 547 & 548 (Advanced Calculus I & II), MA 605 & 606 (Algebra I & II) , MA 635 & 636 (Real Analysis I & II), and MA 681 (Complex Analysis), or their equivalent. Preparation for this examination include reviewing topics in advanced calculus, linear algebra, algebra (without Galois theory), real and complex analysis. The purpose of the exam is to ensure that the student is well-versed on fundamental subjects in mathematics before moving on to research work.

For each of the three subjects, the following are topics that might appear on the general examination. Textbook suggestions are also given. At the end some sample exam questions are attached.

Analysis

Elements of set theory, Zorn's lemma, well-ordering, the real number system, elements of topology, metric spaces, Hausdorff spaces, compactness, limits, continuity, uniform continuity, the Bolzano-Weierstrass theorem, derivatives and differentials, mean value theorems, Taylor expansions, inverse mapping theorem, implicit function theorem, sequences and series of functions, the Ascoli-Arzelö theorem, the Riemann integral, the Lebesgue measure, measurable and integrable functions, convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma, criterion for Riemann integrability, Tonelli's and Fubini's theorems, theorems on local existence, uniqueness and continuity of solutions of ODEs.

Suggested texts:

W. Rudin, *Principles of Mathematical Analysis*, McGraw-Hill.

E. Hille, *Analysis, vols. 1 & 2*, Blaisdell.

T. M. Apostol, *Mathematical Analysis; A Modern Approach to Advanced Calculus*, Addison-Wesley.

S. Lang, *Analysis I, Analysis II*, Addison-Wesley.

H.L. Royden,* Real Analysis*, Prentice Hall.

Complex Variables

Derivative, Cauchy-Riemann equations, definition of analyticity, harmonic functions, branches of complex functions, the Cauchy-Goursat theorem, the fundamental theorems of integration, integral representation of analytic functions, Morera's theorem, Liouville's theorem, the principle of the argument, Rouche's theorem, Laurent and Taylor series, singularities, zeros, and poles, residue theorem, calculation of integrals. Poisson's integral formula, analytic continuation, Picard's theorem.

Suggested texts:

J. B. Conway, *Functions of One Complex Variable, Functions of One Complex Variable II*, Springer-Verlag.

Lars V. Ahlfors, *Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable*, McGraw-Hill.

Walter Rudin, *Real and Complex Analysis*, McGraw-Hill College.

E. Hille, *Analytic Function Theory*, Ginn.

Algebra

Group theory, basic theory of rings and their ideals, polynomial rings, algebras, vector spaces, matrices, Gaussian elimination, linear algebra, determinants and diagonalizability, eigenvalues and eigenvectors, fields, finite fields, algebraic extensions.

Suggested texts:

S. Lang, *Algebra*, Addison-Wesley.

T. W. Hungerford; *Algebra*, Springer-Verlag.

P. B. Bhattacharya, S. K. Jain, and S. R. Nagpual; *Basic Abstract Algebra*, Cambridge U. P.

N. Jacobson, *Basic Algebra, vol. 1*, W. H. Freeman.

I.N. Herstein ,* Topics in Algebra,* 2nd ed., John Wiley and Sons Inc.

David Dummit and Richard M. Foote *Abstract Algebra,* John Wiley and Sons Inc.