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College of Science and Technology 306 Barton Hall, Degree Programs: isc.temple.edu/grad/Programs
/stgrid.htm Departments: Chemistry Computer
& Information Geology Mathematics Physics
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Doctor of Philosophy Degree Requirements a) Students must complete 18 graduatecourses, chosen in consultation with an adviser, and including a year each of real analysis, complex analysis, and algebra. b) Students must spend at least one full yearin residence, and pass both a written comprehensive qualifying exam (CQE) and an oral preliminary exam, prior to being admitted to candidacy. The written comprehensive exam must be taken after no more than two years of graduate studies at Temple, can only be taken twice, and must be passed by the third year at Temple. The preliminary exam must be taken before the end of the fourth year and can be repeated only once. c) Students must demonstrate a reading knowledge of two of the following languages: Chinese, French, German, Japanese, Russian. d) Completion of the above requirements and acceptance of a dissertation proposal, qualifies the student as a doctoral candidate. The student must then work on the dissertation and eventually defend it successfully in a public lecture. While working on the dissertation, the student must register for Math 999, Dissertation Research. The student must complete the total of 6 credits of Dissertation Research. This can be done in one semester or in as long as 3 years of graduate studies. e) All requirements for the Ph.D. must be fulfilled within seven years of graduate study at Temple. Further details about requirements for the M.A. and Ph.D. degrees are available in the department.
Course Descriptions -Mathematics 414. Introduction to Numerical Analysis. (3 s.h.) Roots of nonlinear equations, errors, their source and propagation, linear systems, approximation and interpolation of functions, numerical integration. Offered every two years in the spring. 417-418. Concepts of Analysis. (3 s.h.) Advanced calculus in one and several real variables. Topics include topology of metric spaces, continuity, sequences and series of numbers and functions, convergence, including uniform convergence. Ascoli and Stone-Weierstrass theorems. Integration and Fourier series. Inverse and implicit function theorems, differential forms, Stokes theorem. Offered every year. 462. Ordinary Differential Equations. (3 s.h.) Existence and uniqueness theorems, continuous and smooth dependence on parameters, linear differential equations, asymptotic behavior of solutions, isolated singularities, nonlinear equations, Sturm-Liouville problems, numerical solution of ODEs. Offered every spring. 477. Linear Algebra. (3 s.h.) Vector spaces and subspaces over the real and complex numbers; linear independence and bases; linear mappings; dual and quotient spaces; fields and general vector spaces; polynomials, ideals and factorization of polynomials; determinant; Jordan canonical form. Fundamentals of multilinear algebra. Offered every fall. 501-502. Seminar. (3 s.h.)Challenging problems from many different areas of mathematics are posed and discussed. Offered sporadically.
503-504. Number Theory. (3 s.h.) An introduction to the ideas and techniques of number theory, elementary, analytic, and algebraic. Dirichlet's theorem on primes in an arithmetic progression, the prime number theorem, algebraic number fields. Offered every year. 513-514. Numerical Linear Algebra. (3 s.h.) The syllabus includes iterative methods, classical methods, nonnegative matrices. Semi-iterative methods. Multigrid methods. Conjugate gradient methods. Preconditioning. Domain decomposition. Direct Methods. Sparse Matrix techniques. Graph theory. Eigenvalue Problems. Offered every two years. 515-516. Methods of Applied Mathematics. (3 or 6 s.h.) Various applied mathematical methods and topics discussed: Laplace and Fourier transforms, calculus of variations, integral equations, boundary value problems. Offered sporadically. 519-520. Numerical Differential Equations. (3 or 6 s.h.) Various methods for numerical integration of ordinary and partial differential equations are discussed. Offered every two years. 535-536. Differential Geometry and Topology. (3 or 6 s.h.) Prerequisites: Math. 247, 248, 365. Elementary theory of smooth manifolds. Singular cohomology and DeRham's theorem. Fundamental group and covering spaces. Hodge theory. Offered every two years. 540. Probability Theory. (3 s.h.) Prerequisite: Math 233 or permission of instructor. With a rigorous approach the course covers the axioms, random variables, expectation and variance. Limit theorems are developed through characteristic functions. Offered every year. 548. Stochastic Processes. (3 s.h.) Prerequisite: Math. 540. Random sequences and functions; linear theory; limit theorems; Markov processes; branching processes; queuing processes. Offered every two years in the spring. 557-558. Real Analysis. (3 or 6 s.h.) Prerequisite: Math. 417 or equivalent. The syllabus coincides with the syllabus for the Ph.D. Examination in Real Analysis. Offered every year. 559-560. Functions of a Complex Variable. (3 or 6 s.h.) Prerequisite: Math. 347 or equivalent. Analytic functions. Conformal mapping. Analytic continuation. Topics in univalent functions, elliptic functions, Riemann surfaces, analytic number theory. Nevanlinna theory, several complex variables. Offered every year. 561-562. Partial Differential Equations. (3 s.h.) The classical theory of partial differential equations. Elliptic, parabolic, and hyperbolic operations. Offered every two years. 563. Topics in Applied Mathematics. (3 s.h.) Variable content course. Offered sporadically. 565-566. Topology. (3 s.h.) Prerequisite: Math. 417. Point set topology through the Urysohn Metrization Theorem; fundamental group and covering spaces. Differential forms; the DeRham groups. Offered every year. 573-574. Topics in Applied Mathematics. (3 or 6 s.h.) Prerequisite: permission of instructor. Variable topics, such as control theory and transform theory, will be treated. Offered sporadically. 575-576. Abstract Algebra. (3 or 6 s.h.) Prerequisite: Math. 205 or equivalent. Groups, rings, modules, fields; Galois theory; linear algebra. Offered every year. 591-594. Independent Study. (1, 2 or 3 s.h.) Note: Under normal circumstances it is assumed that a student has taken basic courses on the 500 level before entering any of the 600-level courses. Except when noted to the contrary, 600-level courses are offered sporadically. 601. Theory of Groups. (3 s.h.)
603-604. Topics in Algebra. (3 or 6 s.h.) Variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, algebraic geometry. 605-606. Lie Groups. (3 or 6 s.h.) The theory of Lie groups, from the analytic, algebraic, and applied point of view. 615-616. Selected Topics in Complex Variable Theory. (3 or 6 s.h.)
617-618. Topics in Number Theory. (3 or 6 s.h.) Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions. Offered every two years. 621-622. Functional Analysis. (3 or 6 s.h.) Prerequisites: Math. 557, 558, and Math. 565 or permission of instructor. Offered sporadically. 624. Topics in Analysis. (3 s.h.) Variable content course. Offered sporadically. 625-626. Topics in Functional Analysis.
629-630. Seminar in Real Analysis. (3 or 6 s.h.)
631-632. Seminar in Complex Analysis. (3 or 6 s.h.)
635-636. Topics in Differential Equations. (3 or 6 s.h.) Variable content course. Offered sporadically. 637-638. Modular Functions. (3 s.h.) Modular forms, Einsenstein and Poincaré series, Theta series, Dirichlet series, Hecke operators. Offered every two years. 643-644. Algebraic Topology. (3 or 6 s.h.) Advanced topics in algebraic topology. Offered sporadically. 651. Advanced Probability Theory. (3 s.h.) Variable content course. 657-658. Harmonic Analysis. (3 s.h.) A year long course to explore the real-variable techniques developed in Harmonic Analysis to study smoothness properties of functions and the behavior of certain spaces under the action of some operators. These techniques are also essential in many applications to PDE's and several complex variables. Offered every two years. 659-660. Several Complex Variables. (3 s.h.) Holomorphic functions of several complex variables, domains of holomorphy, pseudoconvexity, analytic varieties, CR manifolds. Offered sporadically. 673-674. Topics in Differential Geometry. (3 or 6 s.h.) Advanced topics in differential geometry. Offered sporadically. 681-682. Topics in Numerical Analysis. (3 or 6 s.h.)
683-684. Combinatorial Mathematics. (3 or 6 s.h.) Variable content course. Offered sporadically. 697-698. Seminar in Probability. (3 or 6 s.h.)
799. Preliminary Examination Preparation. (1 to 6 s.h.)
899. Pre-Dissertation Research. (1 to 6 s.h.)
950-959. Dissertation. (1, 2, 3 or 6 s.h.)
999. Dissertation Research.
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