Graduate Programs

The City

Certificate in Advanced Computational Science Requirements

• Departmental Regulations
  • Syllabus for the First Qualifying Examination
  • Syllabus for the Second Qualifying Examination

January 1992 . Updated April 1997, March 2002, Feb. 2004, Sept. 2005, Aug. 2007

The main steps in obtaining a Ph.D. are passing the qualifying examinations, writing a thesis, and passing a final oral examination on this thesis. The departmental regulations concerning each of these are given below. The regulations are interpreted by the Graduate Studies Committee which, on written petition from a student, may permit deviations from the rules, provided there are exceptional circumstances.

In addition to the departmental regulations, there are university requirements which must also be satisfied.

A student is considered to have a deficiency if in his first semester as a graduate student at SUNYAB he officially enrolls in and completes Math 519 (introductory algebra) or Math 531 (introductory real variables).

The student should base his decision on whether to take these courses on advice from the Director of Graduate Studies and on evaluation of his knowledge in algebra and analysis by the relevant area committees.

At the time of admission to the Graduate School at SUNYAB, the Director of Graduate Studies may decide that certain students have advanced standing of one or two semesters of graduate work, depending on the Graduate School requirements. This will be done after examining the graduate records of the students and taking account of his previous courses, the institutions where he studied, his proficiency in English (TOEFL), etc. It will be clear from what follows that such students will have to fulfill various requirements more quickly than normally admitted students.

The sum of the semesters of graduate work as defined by (i) and (ii) below yields the total semesters of graduate work which will simply be called "semesters of graduate work".

(i) A student admitted with graduate coursework may credited with one or two semesters of graduate work, according to Graduate School requirements.

(ii) For every semester at SUNYAB that a student is registered for fewer than nine credit hours, the credit hours are to be totaled and divided by nine. The result, rounded down to the next integer, will also be counted as semesters of graduate work. In no event will a student be said to have completed more than two semesters of academic work in one calendar year.

This is a three-hour written examination based on a syllabus covering introductory real variables at the level of Math 331-332, introductory abstract algebra at about the level of Math 419-420, and linear algebra. The examination will be given twice a year, during the first week of each semester.

The purpose of the first examination is to assist the Director and the student in deciding soon after the student's entry into Graduate School, whether or not he should be in the Ph.D. program.

Normally, to remain in the Ph.D. program, a student is required to pass this examination within his first two semesters of graduate work. A student who entered with a deficiency is not required to pass this examination until the first opportunity after he has completed two semesters of graduate work.

This consists of two three-hour area examinations, selected by each student from the following four choices: ALGEBRA, ANALYSIS, GEOMETRY/TOPOLOGY, and DIFFERENTIAL EQUATIONS. It is the purpose of the second qualifying examination to insure that each student has a rudimentary command of at least two "core" areas of mathematics.

To remain in the Ph.D. program a student is required to obtain a grade of A or B for one of the area examinations no later than the beginning of his fourth semester of graduate work and an average of at least B for both of the area exams no later than the beginning of his fifth semester. The Students may repeat the examinations, within the time limit, without penalty and are encouraged to take at least one of the examinations as early as possible.

During the semester in which he completes the Second Qualifying Examination, each student will select a major professor, who is a member of the graduate faculty, in consultation with the Director of Graduate Studies. The latter and the major professor will then choose the student's doctoral committee, consisting of at least three members of the faculty with the major professor as chairman.

The student's doctoral committee will set the requirements for admission to candidacy. These are subject to the approval of the Director of Graduate Studies and may include, but are not restricted to, any of the following: an oral examination on "research level" material, a project, a series of lectures on "research level" mathematics, or a written qualifying examination in another department. These requirements must be satisfied by the end of the sixth semester of graduate work.

None.

Before the final oral exam, each student should pass, with a grade of A, B , or S, two one-semester graduate course in subjects other than those of his or her second qualifying exam. These courses are to be approved by the Director of Graduate Studies.

Each Ph.D. student must complete 72-credit hours from: (a) selected 500 level Mathematics courses; (b) 600-800 level Mathematics courses, with the exception of thesis guidance, seminar courses, and other courses of this nature; (c) courses designated by his/her major professor.

The final departmental steps in attaining the degree of Doctor of Philosophy are:

1. completion of a thesis satisfactory to the major professor and the student's doctoral committee;
2. approval by the Graduate School that the student proceed to examination on his/her thesis at a final oral examination;
3. submission of the thesis to each member of the doctoral committee at least three weeks prior to the Final Oral Examination;
4. passing that examination.

1. Analysis

   The real number system, Least Upper Bound Axiom, limits, lim sup and lim inf, sequences and series of real numbers.

   Topology of Rⁿ including compactness, connectedness, completeness, Cauchy-Schwartz inequality, distance and norm. Functions of one and several variables, continuity, uniform continuity. Differentiability in one and several variables, derivatives and applications, partial derivatives and Jacobian matrix, chain rule, change of variables. Implicit and inverse function theorems, multivariate differential calculus. Sequences and series of functions, pointwise convergence, uniform convergence, power series, Taylor's Theorem with remainder.

   Riemann Sums, Riemann integral in one and several variables, improper integrals, change of variables and Jacobian determinant. Iterated and multiple integrals, interchange of order of integration. Arc length and surface area, line integrals, surface integrals. Gauss, Green's, Stokes', and Divergence theorems.

   References:

   1. R. G. Bartle, The Elements of Real Analysis
   2. M. Spivak Calculus
   3. W. Rudin, Principles of Mathematics
   4. J. Munkres, Analysis on Manifolds

   
   

2. Algebra (including linear algebra)

   Groups, decomposition of a group into cosets with respect to a subgroup, group homomorphisms, cyclic groups, order and index, normal subgroups, kernel, range, isomorphism, automorphisms. Fundamental theorems for homomorphisms, Sylow Theorems, Fundamental Theorem of Abelian Groups. Rings, modules, and homomorphisms between them. Ideals in a commutative ring (with 1). The residue class ring R/I with respect to an ideal I.

   Minimal and maximal ideals, prime ideals. Quotient field of an integral domain. The construction of polynomial rings over a commutative ring. Unique factorization domains, principal Ideal domains, Euclidean algorithm. Bilinear and multilinear forms, the determinant. Lemma of Gauss, Eisenstein criterion. Irreducibility of polynomials over finite fields. Algebraic field extensions, characteristic, roots of unity.

   Vector spaces, matrices, linear independence, spanning sets, bases, dimension, inner product spaces, orthogonality, linear transformations (vector space homomorphisms), kernels (null spaces), range, rank, eigenvalues (characteristic values), eigenvectors (characteristic vectors), minimal polynomial, diagonalization, Jordan canonical form, nilpotence.

   References:

   1. Herstein, Topics in Algebra
   2. Fraleigh, A First Course in Abstract Algebra
   3. G. Strang, Linear Algebra and Its Applications
   4. J. Gallian, Contemporary Abstract Algebra

1. Semi-continuous functions. Measures, σ-algebras, measurable sets and functions, Borel sets, measure spaces. Lebesgue measure and integration, Lusin's theorem, Egoroff's theorem, Vitali-Caratheodory theorem.
    
2. L^p spaces, bounded linear functionals on L^p. Elementary Hilbert space theory, subspaces, representation theorems, orthonormal systems. Elementary Banach space theory including Baire's theorem, uniform boundedness principle, open mapping theorem, Hahn-Banach theorem.
    
3. Radon-Nikodym Theorem. Product measures, Fubini's Theorem. Functions of bounded variation and absolutely continuous functions.

B. Complex Analysis

1. Complex numbers, analytic functions, Cauchy Riemann equations, relation between harmonic and analytic functions.
    
2. Complex integration, Cauchy integral theorem and formulas, Morera's theorem, Liouville's Theorem, maximum modulus principle.
    
3. Power series and Laurent series representations of analytic functions. Zeros, Isolated singularities. Identity theorem.
    
4. Residue theorem, evaluation of definite integrals, argument principle.
    
5. Entire functions. Casorati-Weierstrass theorem.
    
6. Conformal mapping, linear fractional transformations.

1. Groups. symmetry groups, homomorphism theorems, Sylow theorems, group actions on sets.
2. Rings. various examples (e.g., rings of continuous or analytic functions), unique factorization domains, Gauss' lemma, Eisenstein criterion, Noetherian rings, Artinian rings, Semi-simple rings, Wedderburn-Artin theorems, group rings, Maschke's theorem.
3. Modules. tensor products, exterior powers, projective and injective modules, Nakayama's lemma, modules over principal ideal domains, modules over semi-simple rings, group representations Jordan and rational canonical forms, Cayley-Hamilton theorem, determinants.
4. Fields. field extensions, finite multiplicative subgroups of a field, structure of finite fields, irreducibility of the cyclotomic polynomials, Galois theory, algebraic closure, transcendental extensions.
5. Category theory. representable functors, adjoint functors, universal properties, Yoneda's lemma.

References:

1. E. Artin, Galois Theory
2. S. Lang, Algebra
3. N. Jacobson, Basic Algebra I and II

A. Algebraic Topology

1. Homotopy, fundamental group, covering spaces, Van Kampen's theorem
2. Simplicial and cell complexes, singular homology and cohomology groups
3. The exact homology sequence, the excision theorem, Mayer-Vietoris sequence, Jordan-Brouwer separation theorem
4. Statements and applications of the Künneth theorem and the Universal Coefficient theorem
5. Orientation of manifolds, cup product, Poincaré-Lefschetz duality, Lefschetz Fixed Point theorem

Suggested References:

1. Greenburg and Harper, Algebraic Topology: A First Course
2. Munkres, Elements of Algebraic Topology

B. Differential Geometry

1. Theory of curves and surfaces in R³ , Gauss-Bonnet theorem in dimension 2
2. Manifolds, implicit and inverse function theorems
3. Tangent bundles, vector fields and Lie derivatives, Frobenius theorem
4. Differential forms, Stokes' theorem, de Rham cohomology

Suggested References:

1. O'Neill, Elementary Differential Geometry
2. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I

A. Ordinary Differential Equations

1. Existence and uniqueness of solutions to initial value problems for single equations and systems
2. Solution of linear first order systems, especially constant coefficient systems
3. Qualitative analysis for nonlinear systems, phase portraits, classification of equilibrium states, Poincaré-Bendixson theorem, Lyapunov functions, Lienard and van der Pol equations
4. Floquet theory and the stability of periodic solutions, stable manifold theorem, invariant manifolds
5. Sturm-Liouville and two-point boundary value problems

Suggested References:

1. Hale, Ordinary Differential Equations
2. Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra
3. Perko, Differential Equations and Dynamical Systems

B. Partial Differential Equations

1. Linear and non-linear equations of first order, characteristics, Hamilton-Jacobi equations, equations of geometrical optics
2. Classification of PDE
3. Fundamental solutions of elliptic and parabolic equations, especially the Laplace, Helmholtz and heat equations
4. Dirichlet and Neumann problems for Laplace, Helmholtz and heat equations, maximum principle and uniqueness theorems for elliptic and parabolic equations
5. Solution of the initial value problem for the wave equation, conservation of energy and uniqueness theorems for the wave equation, Huyghen's principle
6. Fredholm Alternative and eigenfunction expansion with applications to elliptic, parabolic and hyperbolic equations

Suggested References:

1. Evans Partial Differential Equations
2. John, Partial Differential Equations
3. Smoller, Shock Waves and Reaction-Diffusion Equations, chaps. 1-9

Graduate Programs

The City

Certificate in Advanced Computational Science Requirements