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Chair: Christopher Hunter;
Associate Chair: Wright;
Director of Applied Mathematics: Navon;
Director of Basic Mathematics: Stiles;
Professors: Bellenot, Bowers, Bryant, Case, Gilmer, Heil, Huckaba, Hunter, Hussaini,
Kopriva, Loper, Mesterton-Gibbons, Mott, Navon, Nichols, Oberlin, Quine, Sumners, Tam,
Wright, Young;
Associate Professors: Aluffi, Blumsack, Erlebacher, Klassen, Magnan, McMichael,
Mio, Nolder;
Assistant Professor: Hironaka;
Visiting Professor: Seppala;
Visiting Associate Professor: Woodruff;
Visiting Assistant Professor: van Hoeij;
Service Professor: McWilliams, Novinger;
Associate in Mathematics: Blackwelder, Boyd, Burgess, Dodaro, Wooland;
Assistants in Mathematics: Jowhar, MacLeod, Rogers;
Professors Emeriti: Heerema, Howard, Kreimer, McArthur;
Courtesy Professors: Chen, Lacher, Levitz, Lin
Graduate study in the Department of Mathematics opens a variety of opportunities with each of its degrees: master of science, master of arts, doctor of philosophy. At the doctoral level a student may specialize in algebra, analysis, topology, symbolic computation, applied mathematics, or may participate in interdisciplinary programs of study and research. The master's programs prepare students for future graduate study or for work in a wide variety of professional venues including industry, teaching, government and finance.
The Florida State University has an active group of algebraists, one of whom is a Robert O. Lawton Distinguished Professor. Their research interests include commutative algebra, ring theory, field theory, Hopf algebras, partially ordered groups, homological algebra, algebraic geometry, and computational algebra. Each semester the advanced graduate students in algebra participate in the algebra seminar with various members of the faculty, studying an advanced topic of common interest.
Analysis has a rich tradition at The Florida State University. Research interests currently include classical and modern complex analysis, concrete and abstract harmonic analysis, and linear space theory from nonlocally convex spaces to the geometric theory of Banach spaces. In addition to the standard core courses in analysis and two weekly seminars, advanced topics courses and directed individual readings offer students opportunities to pursue an individualized course of study.
The topologists share a common interest in the area of geometric topology with publications in knot theory, finite and infinite dimensional manifold theory, 3-manifolds, group actions on manifolds, and simple homotopy theory. One is a Robert O. Lawton Distinguished Professor, and another a Distinguished Research Professor. Graduate courses, advanced seminars, and individual studies are offered on a regular basis, providing students the opportunity to master the basic concepts in geometric and algebraic topology and to pursue selected topics in areas of special interest.
The research focus of the symbolic computation group is in computational complex and real analysis and in computational algebra. Riemann surfaces, algebraic curves, differential equations and Hopf algebras are currently central themes of interest. Symbolic computation is a new addition to the academic curriculum and it is being developed as an independent discipline having close ties to several of the core areas of pure mathematics. Graduate courses, independent studies and a research seminar are being offered each semester.
Mathematicians who direct students and postdocs in mathematical and computational biology have a variety of interests: evolutionary game theory and animal biology; molecular biology and biophysics including analysis of human brain functional data, protein geometry, analysis of NMR data, knotting of DNA, and polymer systems. Topics courses, specialized seminars, and study in other departments add opportunities for students to develop deeper understanding in these interdisciplinary areas. Faculty affiliations include The Florida State University Institute of Molecular Biophysics, the Program in Mathematics and Molecular Biology, and the National High Magnetic Field Laboratory. Their students have opportunities for joint research with the bench scientists from chemistry, biology and physics, internships with companies, and specialized fellowships and training grants.
The applied mathematics program reflects the interests and strengths of its faculty. One is Eminent Scholar Chair in High Performance Computing, another is a McKenzie Professor, and two others are Distinguished Research Professors. The major focus of the program is on the application of mathematics to physical problems, and on the mathematical and computational methods needed to do this effectively. There is a strong emphasis on fluid dynamics as a consequence of cooperative relations with the Departments of Geological Sciences, Mechanical Engineering, Meteorology, and Oceanography, and with the interdisciplinary Geophysical Fluid Dynamics Institute. Students from these departments enroll in applied mathematics classes, and the applied mathematics program encourages its students to take courses, relevant to their chosen area of interest, in other departments. Faculty interests include acoustics and aircraft noise, combustion, the internal dynamics of galaxies, and the fluid dynamics of the Earths core and mantle. There is a heavy emphasis on computational mathematics, and there will be a strong interrelation with the interdisciplinary Program in Computational Science and Engineering that is being developed and is under consideration by the Board of Regents. Several faculty specialize in the development and use of computational methods in a variety of applications, and in high performance computing. Students are introduced to research problems in two series of seminars, one in scientific computing and one in applied mathematics. There is an introductory seminar for beginning graduate students which gives them an overview of the program.
Many members of the faculty attract external grant support from various funding agencies. A considerable number of graduate students are supported as research assistants, and there are special fellowship opportunities in some areas.
Students of both pure and applied mathematics are supported by a wide range of computing resources. The university provides Internet access for all of its students, as well as access to a number of leading databases ranging from AMS Math Reviews to the Encyclopedia Britannica. These resources are available via the Internet. The University operates a number of computers and computer labs which are available to all of its students. The Department of Mathematics operates its own network of computers for both graduates and undergraduates. The graduate computer lab includes many high-end graphics workstations and a large multi-processor server. The department supports a wide range of mathematical software from symbolic packages like Maple to visualization tools like IDL and public domain favorites like gnuplot. The Supercomputer Computations Research Institute (SCRI) has additional computer resources which are often available for mathematical projects.
For additional information, see the departments web site at: http://www.math.fsu.edu Mathematics
Please review all college-wide degree requirements summarized in the "College of Arts and Sciences" section of this Graduate Bulletin.
A student who proposes to do graduate work in the department is required to take the aptitude test of the Graduate Record Examinations (GRE) and make a minimum combined score of 1100 with a minimum score of 700 on the quantitative aptitude part.
Graduate students in mathematics are strongly encouraged to perform some teaching as part of their professional development; teaching experience is seen by many prospective employers as a positive indication. The department is normally able to support a large number of its graduate students as teaching assistants, while many are supported as research assistants or by various fellowships.
A program for the master's degree is planned in conference with an adviser appointed by the chair of the department. It may be either a course-type program, consisting of thirty-two (32) or more semester hours of graduate courses, or a thesis-type program, consisting of thirty (30) or more semester hours of graduate courses including a minimum of six (6) semester hours in MAT 5971r for the writing of an acceptable thesis. Credit in MAT 5971r cannot be counted toward a course-type program.
A program of either type will include at least twenty-two (22) semester hours in courses offered by the department, for students following option a) or b). The courses MAT 5946r and MAT 5911r are not applicable toward the program. No 4000 level course in the department may be counted toward the master's degree. The student will select one of the following options (except that a student who has successfully completed MAT 8964 will be deemed to have qualified for a master's degree, subject to University regulations).
A master's degree student under any option will have a supervisory committee appointed by the chair of the department and consisting of at least three graduate faculty members, of whom at least two (one in option e)must be members of the department. If taking a course-type program, the student will be required to pass a master's comprehensive examination including both written and oral parts, although the students committee may waive the oral part. In option (e) the comprehensive exam is replaced by a project. If taking a thesis-type program, the student will be required to pass a master's thesis defense but not a master's comprehensive examination.
A student expecting to teach in college should have at least the equivalent of a master's degree. For a position in a four-year college, it is recommended that additional course work covering the basic material for the doctoral preliminary examination be included.
A program leading to the doctor of philosophy degree is available under either of two options: a) specialization in algebra, analysis, symbolic computation, or topology; or b) specialization in applied mathematics.
The receipt of a master's degree is one of the requirements for the doctorate. (Subject to University regulations, the successful completion of the doctoral preliminary examination will qualify the student for a master's degree.)
All members of the graduate faculty of the department are invited to participate in the student's preliminary examination, as well as in the examination in defense of dissertation.
Doctoral preliminary examinations in mathematics, option a), are given in two parts. The first part is a series of qualifying exams in algebra, complex analysis, real analysis, topology and symbolic computation. The student must take at least three qualifying exams, of which two should be taken after two years and the third completed by the midpoint of the third year of the students graduate program. The second part is an advanced topics examination, which concentrates on the students chosen area of research and which must be completed in a year and a half beyond the qualifying exams.
The student should complete the courses MAA 5406, 5616, 5617; MAS 5307, 5308, 5311, 5312; MTG 5316, 5326, 5346; and either MAA 5407 or 5506. Many of these courses should be completed before taking the qualifying exam in the appropriate area, as well as some other courses not listed here; a list of courses covered on the various qualifying exams can be obtained from the department.
Under option b), the students program of studies will be planned in conjunction with the major professor and supervisory committee. The doctoral preliminary examination will include both a written part and an oral part. The written part covers fundamental analytical concepts and methods, fundamental numerical concepts and methods, and formulation, modeling, and applications. A student should normally take the written part during the third year of graduate studies. The oral part investigates the students aptitude and preparation in the proposed dissertation area and entails the presentation and defense of a prospectus.
Under option a), the students major professor and supervisory committee will be appointed by the department chair after the student has passed the doctoral preliminary examination. In the case of a student following option b), the chair, in consultation with the director of applied mathematics, will appoint a three-member supervisory committee when the student begins graduate study; the committee will be expanded to a membership of five at the end of the students first year.
A doctoral student under either option is required to demonstrate to the supervisory committee proficiency in a minor area of study; normally this can be accomplished by completing six (6) semester hours in an approved mathematics related minor program of studies with a grade point average (GPA) of at least 3.0. If the minor area is in mathematics itself, these hours must be outside the list of courses required for the doctoral preliminary examination in the students option. At the discretion of the students doctoral supervisory committee, the student may be required to demonstrate competence in research tools appropriate to the students program of studies. Such tools may include a reading knowledge of one or more foreign languages, technological skills or other competencies. From the time of passing the doctoral preliminary examination until receipt of the doctorate, students are required to enroll in the advanced seminar in their area (numbered 6939) in each semester of residence in which the seminar is offered; this seminar enrollment must include at least three semesters.
After students have passed the doctoral preliminary examination, the writing of a dissertation becomes their major concern, although further course work and participation in seminars are usually required. The defense of dissertation must be held within five years after completion of the doctoral preliminary examination; if this time limit is not met, the student may be required to repeat the examination. To be awarded the doctor of philosophy degree, students must have demonstrated the capacity to do original and independent scholarly investigation in their area of specialization and must have satisfied all general requirements of the University.
MAA 5306, 5307. Advanced Calculus I, II (3, 3). Prerequisites: MAC 2313, MAS 3105, MGF 3301. Functions, sequences, limits, continuity, uniform continuity; differentiation; integration; convergence, uniform convergence.
MAA 5406, 5407. Theory of Functions of a Complex Variable I, II (3, 3). Prerequisite: MAA 4227 or 5307. Algebra and geometry of complex numbers; elementary functions and their mappings. Analytic functions; integration in the complex plane; Cauchys integral theorem and related theorems. Representation theorems including the Taylor and Laurent expansions. Calculus of residues. Entire and meromorphic functions.
MAA 5506. Functional Analysis I (3). Prerequisite: MAA 5616. Theory of linear topological spaces, with emphasis on Banach and Hilbert spaces. Hahn-Banach theorem; uniform boundedness theorem; open mapping theorem; weak topologies; Krein-Milman theorem.
MAA 5616, 5617. Measure and Integration I, II (3, 3). Prerequisite: MAA 4227 or 5307. Lebesgue measure and integration; Banach spaces of integrable functions; abstract measure and integration.
MAA 5721. Computer Analysis (3). Prerequisites: MAA 5307 or 4227; MAA 4402. Automatic differentiation, automatic integration, indefinite summation; applications to partial differential equations; advanced topics in complex analysis.
MAD 5305. Graph Theory (3). Prerequisite: Graduate standing. Graphs and digraphs, trees and connectivity, Euler and Hamilton tours, colorings, matchings, planarity and Ramseys theorem, applications. A proof-oriented course that assumes no previous exposure to graph theory but assumes a certain level of mathematical maturity.
MAD 5420. Numerical Optimization (3). Prerequisites: MAS 3105, MAC 2313, Fortran. Unconstrained minimization: one-dimensional, multivariate, including steepest-descent, Newtons method, Quasi-Newton methods, conjugate-gradient methods, and relevant theoretical convergence theorems. Constrained minimization: Kuhn-Tucker theorems, penalty and barrier methods, duality, and augmented Lagrangian methods. Introduction to global minimization.
MAD 5427. Numerical Optimal Control of Partial Differential Equations (3). Prerequisites: MAD 5739; MAS 3105. Euler Lagrange equations, adjoint method algorithm. Optimal control of systems governed by elliptic, parabolic, hyperbolic PDEs. Control of initial and boundary conditions. Adjoint sensitivity analysis. Optimal parameter estimation, Kalman filter for parameter identification. Automatic differentiation techniques.
MAD 5708. Numerical Analysis II (3). Prerequisites: MAD 3703, MAP 2302. Approximation theory, numerical solution of nonlinear systems, boundary value problems, and initial value problems for ordinary differential equations.
MAD 5738, 5739. Numerical Solution of Partial Differential Equations I, II (3, 3). Prerequisites: MAD 5708, MAP 5346. Finite difference methods for parabolic, elliptic, and hyperbolic problems; consistency, convergence, stability.
MAD 5745. Spectral Methods for Partial Differential Equations (3). Prerequisites: MAD 5738, MAP 5431 (recommended). Fourier and orthogonal polynomial spectral methods for the solution of elliptic, parabolic, and hyperbolic equations. Spectral approximation theory. Psuedospectral method and aliasing removal. Applications to fluid flow.
MAD 5757. High Order Finite Difference Methods for Computational Acoustics and Fluid Dynamics (3). Prerequisite: MAD 5738. High order spatial and temporal discretization; artificial selective damping; numerical stability; radiation, inflow and outflow boundary conditions; wall and time-domain impedance boundary conditions; nonlinear acoustic waves; design of computation algorithms for direct numerical simulation.
MAP 5177. Actuarial Models (3). Prerequisites: MAP 4170; STA 4322. Life probabilities; tables; mortality laws and contingent payments; life annuities; premium principles and net premium reserves for continuous, descrete and semi-continuous life insurances; multiple life models; multiple decrement theory (theory of competing risks) and applications to pension plans; pricing and nonforfeiture models.
MAP 5207. Optimization (3). Prerequisites: MAC 2313, MAD 3703, MAS 3105. Linear programming, unconstrained optimization, searching strategies, equality and inequality constrained problems.
MAP 5217. Calculus of Variations (3). Prerequisites: MAP 2302; MAA 5306 or MAP 5207. Fundamental problems, weak and strong extrema, necessary and sufficient conditions, Hamilton-Jacobi theory, dynamic programming, control theory, and Pontryagins maximum principle.
MAP 5336. Qualitative Theory of Ordinary Differential Equations (3). Prerequisites: MAC 2313, MAP 2302, MAS 3105. Existence and uniqueness of solutions of first order equations, linear systems, autonomous systems, phase portraits, and stability of 2-dimensional systems; general stability results for nonautonomous equations; other special topics.
MAP 5345. Elementary Partial Differential Equations I (3). Prerequisites: MAC 2313; MAP 2302 or 3305. Separation of variables; Fourier series; Sturm-Liouville problems; multidimensional initial boundary value problems; nonhomogeneous problems; Bessel functions and Legendre polynomials.
MAP 5346. Elementary Partial Differential Equations II (3). Prerequisite: MAP 4341 or 5345. Solution of first order quasi-linear partial differential equations; classification and reduction to normal form of linear second order equations; Greens function; infinite domain problems; the wave equation; radiation condition; spherical harmonics.
MAP 5395. Finite Element Methods (3). Prerequisites: MAS 3105, MAP 2302, MAP 4341, Fortran. Methods of weighted residuals, finite element analysis of one and two-dimensional problems, isoparametric elements, time dependent problems, algorithms for parabolic and hyperbolic problems, applications, advanced Galerkin techniques.
MAP 5423. Complex Variables, Asymptotic Expansions, and Integral Transforms (3). Prerequisites: MAP 4341 or 5345; MAA 4402 or 5406. Ordinary differential equations in the complex plane; special functions. Asymptotic methods: Laplaces method, steepest descent, stationary phase, WKB. Integral transforms: Fourier, Laplace, Hankel.
MAP 5431. Introduction to Fluid Dynamics (3). Prerequisites: PHY 3048C, MAP 4153; Corequisite: MAP 4341, 5345, or consent of instructor. Physical properties of viscous fluids, hydrostatics, kinematics of slow fields, governing equations. Boussinesq approximation, Buckingham Pi theorem. Dynamics of viscous incompressible fluids: vorticity, boundary layer flow, similarity.
MAP 5441. Perturbation Theory (3). Prerequisite: MAP 4342 or 5346. Regular and singular perturbation problems; methods of averaging, matched asymptotic expansions, multiple scales, strained coordinates, and WKBJ; applications to ordinary and partial differential equations and fluid dynamics.
MAP 5486. Mathematical Bioeconomics (3). Prerequisites: MAC 2313, MAS 3105, MAP 2302, STA 4442. Biological and economic dynamics of renewable resource management. Optimal control theory. Theory of resource regulation. Growth and aging. Multispecies models. Stochastic models.
MAP 5512. Hydrodynamic Stability (3). Prerequisite: MAP 5431. Stability of nearly parallel flows; propagation characteristics of instability waves; effects of rotation, thermal, and density stratification on hydrodynamic stability; computation methods.
MAP 5513. Wave Propagation Theory (3). Prerequisites: MAP 4342 or 5346; 5431; or consent of instructor. Phase and group velocities, dispersion, reflection, characteristics, shock formation, momentum and energy transport, and nonlinear effects. Applications such as acoustics, water waves, internal waves, Rossby waves, and seismic waves. The Korteweg-DeVries equation and solutions.
MAP 5601. Introduction to Financial Mathematics (3). Prerequisites: MAC 2313; MAP 2302; MAS 3105; STA 4442; or equivalents. Course covers partial differential equations, discussion of Brownian motion, Black-Scholes analysis, optimal control, Euler equation.
MAS 5307, 5308. Groups, Rings, and Vector Spaces I, II (3, 3). Prerequisites: MAS 2103 or 3105; MAS 4302. Quotient groups, group mappings; permutation groups, Sylows theorem. Ring homomorphisms, ideals, quotient rings; fields; extension fields. Vector spaces; dual spaces. Algebra of linear transformations; theory of linear transformations.
MAS 5311, 5312. Abstract Algebra I, II (3, 3). Prerequisite: MAS 5308. Groups, group mappings; direct products, linear algebras; rings and ring mappings; extensions of rings and fields; factorization theory; groups with operators; Galois theory; structure of fields; valuations.
MAS 5331r, 5332r. Algebraic Structures I, II (3, 3). Prerequisite: MAS 5312. An intensive study of the structure of one or more of the following algebraic systems: groups, rings, fields. Each course may be repeated to a maximum of six (6) semester hours.
MAS 5731. Computer Algebra (3). Prerequisite: MAS 4303. Factorization of polynomials; decomposition of polynomials; the method of Groebner bases, applications; computing with algebraic numbers.
MAT 5907r. Directed Individual Study (1 - 4). (S/U grade only.) May be repeated to a maximum of twelve (12) semester hours.
MAT 5911r. Supervised Research (1 - 9). (S/U grade only.) Cannot be applied to the master's degree. May be repeated to a maximum of nine (9) semester hours.
MAT 5920r. Colloquium (0). (S/U grade only.) A series of lectures given by faculty and visitors addressing various topics of mathematical interest.
MAT 5932r. Selected Topics in Mathematics (13). May be repeated to a maximum of twelve (12) semester hours.
MAT 5941. Internship in College Teaching (1 - 3). (S/U grade only.)
MAT 5945r. Internship in Financial Mathematics (1 - 3). (S/U grade only.) Prerequisite: Instructor approval. Supervised internship individually assigned to accommodate students professional development in financial mathematics. May be repeated for a maximum of three (3) semester hours.
MAT 5946r. Supervised Teaching (1 - 9). (S/U grade only.) May be repeated to a maximum of nine (9) semester hours.
MAT 5971r. Thesis (36). (S/U grade only.) A minimum of six (6) semester hours credit is required.
MHF 5206. Foundations of Mathematics (3). Zermelo-Fraenkel axioms for set theory. Finite and infinite sets. Ordinal numbers, cardinal numbers. The axiom of choice and some of its equivalents.
MHF 5306. Mathematical Logic I (3). Prerequisite: MAS 3301 or consent of instructor. Propositional and predicate logic, models. Godels completeness theorem and related theorems. Applications to modern algebra. Non-standard analysis.
MHF 5307. Mathematical Logic II (3). Prerequisite: MHF 4302 or 5306. Primitive recursive and recursive functions. Formal number theory. Godels incompleteness theorem. Tarskis theorem. Undecidability of the predicate calculus.
MTG 5316. Elementary Topology I (3). Prerequisite: MAC 2313. Topological spaces, metric spaces, connectedness, compactness, separation properties, topology of the plane, product spaces.
MTG 5317. Elementary Topology II (3). Prerequisite: MTG 4302 or 5316. Function spaces, Hilbert space, quotient spaces, continua, paracompactness and metrizability, nets and filters, the fundamental group.
MTG 5326, 5327. Topology I, II (3, 3). Prerequisite: MTG 4302 or 5316. Fundamental group and covering spaces, simplicial and CW complexes, elementary homotopy theory, elementary homology theory.
MTG 5346, 5347. Algebraic Topology I, II (3, 3). Prerequisite: MTG 5327. Singular homology and cohomology, orientation of manifolds, cup and cap products, Poincare and Lefschetz duality, acyclic models.
MTG 5376r. Topological Structures (3). Prerequisite: MTG 5327. A study of one or more of the following structures: topological, P.L. or smooth manifolds, Riemannian geometry, homotopy theory, obstruction theory, fibre bundles. May be repeated to a maximum of six (6) semester hours.
OCP 5253. Fluid Dynamics: Geophysical Applications (3). Prerequisites: MAP 5431, 5346; or consent of instructor. Shallow water theory, Poincare, Kelvin, and Rossby waves; boundary layer theory; wind-driven ocean circulation models; quasigeostrophic motion on a sphere, thermocline problem; stability theories. Also offered by the departments of Oceanography and Meteorology.
MAA 6416r. Advanced Topics in Analysis (3). May be repeated to a maximum of twelve (12) semester hours.
MAA 6526r. Advanced Topics in Functional Analysis (3). May be repeated to a maximum of twelve (12) semester hours.
MAA 6939r. Advanced Seminar in Analysis (1). (S/U grade only.) May be repeated to a maximum of eight (8) semester hours.
MAD 6408r. Advanced Topics in Numerical Analysis (3). May be repeated to a maximum of twelve (12) semester hours.
MAD 6939r. Advanced Seminar in Scientific Computing (1). (S/U grade only.) May be repeated to a maximum of eight (8) semester hours.
MAP 6316r. Advanced Topics in Differential Equations (3). May be repeated to a maximum of twelve (12) semester hours.
MAP 6434r. Advanced Topics in Hydrodynamics (3). May be repeated to a maximum of eighteen (18) semester hours.
MAP 6437r. Advanced Topics in Applied Mathematics (3). May be repeated to a maximum of twelve (12) semester hours.
MAP 6939r. Advanced Seminar in Applied Mathematics (1). (S/U grade only.) May be repeated to a maximum of eight (8) semester hours.
MAS 6396r, 6397r. Advanced Topics in Algebra I, II (3, 3). Each course may be repeated to a maximum of six (6) semester hours.
MAS 6939r. Advanced Seminar in Algebra (1). (S/U grade only.) May be repeated to a maximum of eight (8) semester hours.
MAT 6908r. Directed Individual Study (1 - 4). (S/U grade only.) May be repeated to a maximum of twelve (12) semester hours.
MAT 6980r. Dissertation (1 - 12). (S/U grade only.)
MTG 6396r. Advanced Topics in Topology (3). May be repeated to a maximum of six (6) semester hours.
MTG 6939r. Advanced Seminar in Topology (1). (S/U grade only.) May be repeated to a maximum of eight (8) semester hours.
MAT 8964. Doctoral Preliminary Examination (0).
MAT 8966. Master's Comprehensive Examination (0).
MAT 8976. Master's Thesis Defense (0).
MAT 8985r. Defense of Dissertation (0).
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