## Graduate Department of Mathematics

### College of Arts and Sciences

**Website**: https://math.fsu.edu

**Chair:** Washington Mio; **Associate Chair for Academic Affairs:** Hurdal; **Associate Chair for Graduate Studies:** Ökten; **Associate Chair for Undergraduate Studies: **Kercheval; **Director of Pure Mathematics:** Ballas; **Director of Applied and Computational Mathematics:** Musslimani; **Director of Financial Mathematics:** Zhu; **Director of Biomathematics:** Bertram; **Coordinator of Graduate Teaching Assistants:** Kirby; **Coordinator of Actuarial Science:** Paris; **Professors:** Aldrovandi, Aluffi, Bertram, P. Bowers, Cogan, Fenley, Gallivan, Heil, Huckaba, Hurdal, Kercheval, Kim, Klassen, Mio, Musslimani, Nolder, Ökten, Sussman, Tam, van Hoeij; **Associate Professors:** Agashe, Ballas, Bao, Bauer, Fahim, Lee, R. Oberlin, Reznikov, Zhu; **Assistant Professors:** Asllani, Banerjee, Farhat, Feng, Karamched, Morsky, Needham, Nyuyen, Ozanski, Usatine, Zhang; **Teaching Professors:** Kirby, Paris; **Research Associate in Mathematics:** Boyd; **Teaching Faculty III**: Ewald, Harris; **Teaching Faculty II:** K. Bowers, Hollingsworth; **Teaching Faculty I:** Acar, Budkie, Simmons, Valdes, Wilk; **Professors Emeriti:** Bellenot, Blumsack, Bryant, Case, Hironaka, Kopriva, Kreimer, Mesterton-Gibbons, Mott, Nichols, D. Oberlin, Quine, Sumners, Wright; **Courtesy Professors:** Absil, Goldberg, Henry De Frahan, Hironaka, Huang, Marchand, Marcolli, van Dooren

The Department of Mathematics is strongly committed to graduate education and research and offers programs of study leading to both the master's (MA and MS) and the doctoral (PhD) degrees. Its programs are designed to prepare students for mathematical careers in the academic, corporate, and governmental sectors. PhD and master's degrees are offered with concentrations in five areas: Pure Mathematics, Applied and Computational Mathematics, Financial Mathematics, Biomathematics, and Interdisciplinary Data Science. For more information, please visit https://www.math.fsu.edu.

The department has cooperative relationships with science, social science, business, and engineering departments; the College of Medicine; and many institutes and laboratories on campus including: the Geophysical Fluid Dynamics Institute, the Laboratory of Imaging Studies, the Institute for Molecular Biophysics, the National High Magnetic Field Laboratory, the Program in Neuroscience, and the Department of Scientific Computing. Aside from a wide array of beginning and advanced courses in graduate mathematics, students may take advantage of approved courses outside the department. These include courses in biochemistry, computer science, economics, engineering, finance, molecular biology and biophysics, physics, and statistics. Financial Mathematics students may broaden their employment opportunities by pursuing a concentration in actuarial science. Students participate in the weekly colloquia; they also rotate responsibility for running a graduate-student seminar, where they discuss and critique their work and invite speakers to broadly address professional development. They may attend any subset of over a dozen seminar series whose topics vary according to the current research interests of the department.

The faculty of the department includes two Robert O. Lawton Distinguished Professors, the Carol M. Brennan Professor, the Christopher Hunter Professor, the Dwight B. Goodner Professor, the Tam Family Professor, the Roger W. Roberts Professor, two Distinguished Research Professors, four recipients of Developing Scholar Award, and more than a dozen recipients of University Teaching and Advising Awards.

The five study areas give opportunities for graduate student and faculty interaction. The resulting research, publication, and recognition is in a variety of specializations including: algebraic geometry, arithmetic geometry, biofilms, biomathematics, collegiate mathematics education, complex analysis, computational anatomy and pattern analysis, complex dynamical systems, computational acoustics, computational neuroscience, conformal mapping, cryptography, data science, dynamical systems, financial mathematics and computational finance, fluid dynamics, game theory, geometric topology, harmonic analysis, high performance computing, homological algebra, homotopy theory, human brain mapping, knotting of DNA, mathematical economics, mathematical physics, mathematics history and biography, number theory, numerical analysis, partial differential equations, pattern recognition, physiology, protein geometry, shape theory, stochastic analysis, and symbolic computation. Faculty and graduate students are supported in their work by FSU research initiatives and by outside agencies including: Air Force Office of Scientific Research, American Heart Association, the Boeing Company, Goodrich Aerostructures, the Institute for Applied Mathematics (Minnesota), Mathematical Biosciences Institute (Ohio State), National Aeronautics and Space Administration, National Institutes of Health, National Mathematics and Science Initiative, National Security Agency, National Science Foundation, Ohio Aerospace Institute, Simons Foundation, U.S. Department of Education, and the U.S. Department of Energy.

The Department of Mathematics has a full range of computing facilities available for a variety of instructional and research needs. Faculty and graduate students share high-performance workstations and file and computer servers. Across the university, students and faculty have access to a variety of state-of-the-art machines, including supercomputers and computer clusters. Florida State University provides a nearly campus-wide outdoor wi-fi network as well as indoor wireless in the libraries, the union, and the university student computer labs. As a member of the Florida Lambda Rail, FSU has multiple high-capacity backbones to other research universities and laboratories. The Library provides access to a number of databases (including Mathematical Reviews, MathSciNet, and JSTOR), an increasing number of eJournals (such as SIAM Journals and Springer LINK), as well as books, journals, and carrels for study.

## Graduate Requirements

There are both University- and college-wide degree requirements that apply to all graduate students; these are summarized in the appropriate chapters of this* Graduate Bulletin. *Post-publication revisions to the degree guidelines and the course information listed below are available at https://math.fsu.edu, or at the Department's main office; students are alerted to changes or modifications by e-mail.

A number of graduate students receive support through fellowships or by working as teaching or research assistants. Graduate students in mathematics are strongly encouraged to include teaching skills as part of their professional-development activities. The department's recognized orientation and training programs accompany practice in several instructional delivery modes. Teaching Assistants participate in lecture-recitation delivery in computer classrooms and progress to full classroom responsibility. They are encouraged to investigate academic and research careers and are well prepared for teaching employment at various types of colleges and universities.

In order to obtain final graduation clearance from the Department of Mathematics, all MS and PhD candidates must complete an exit survey in their final semester. Additionally, PhD candidates must complete the information required for the national “Doctorates Granted” survey.

### Master's (MA or MS) Degree

The department offers master's degrees in Pure Mathematics, Applied and Computational Mathematics, Financial Mathematics, Biomathematics, and Interdisciplinary Data Science. Each area has its own required and approved elective courses and seminars. No 4000-level course in this department may count toward the master's degree. The student should consult the graduate programs' websites to learn more about the specific requirements for each area.

A course-based master's degree is available in all five areas. Pure Mathematics, Applied and Computational Mathematics, and Biomathematics, require thirty-six hours of graduate courses, of which at least thirty hours must be letter-graded. Financial Mathematics requires thirty-five hours of graduate courses, of which at least thirty-three hours must be letter-graded. Data Science requires thirty hours of letter-graded graduate courses. Certain seminars must be taken in Financial Mathematics and Biomathematics; consult the area websites for details.

In Pure Mathematics and Applied and Computational Mathematics, a thesis-based master's degree is also available. The thesis-based master's degree requires at least thirty hours of letter-graded graduate courses, and six semester hours in MAT 5971r and appropriate thesis defense.

**Pure Mathematics.**The pure mathematics option gives the student a well-rounded exposure to the foundations of modern mathematics. Coursework includes graduate sequences in algebra, real and complex analysis, and topology. Electives include more advanced courses in these disciplines as well as applied topics such as symbolic computation and statistics. The master's degree in pure mathematics provides excellent preparation for many careers in education, industry, and government. It is also an appropriate first step for those students who wish to pursue a PhD, either in some mathematical field or in another discipline that uses mathematics or rigorous logical thinking.**Applied and Computational Mathematics.**This option provides students with extensive research and educational experiences in modeling, analysis, algorithm development, and simulation for problems arising throughout mathematics, sciences, and engineering. After completing this master's degree, students may choose to pursue a doctoral degree in the area of Applied and Computational Mathematics or related areas, or pursue educational, financial, industrial, or governmental jobs involving applications of mathematical and computational skills.**Financial Mathematics.**This interdisciplinary degree program includes coursework in several departments and prepares students for work as quantitative analysts in financial firms, as actuaries, or for further doctoral study in quantitative finance. The degree includes a professional component oriented to skills needed for financial industry employment. Students are encouraged to pursue summer internships in the financial industry.**Biomathematics.**Studies in this interdisciplinary program include courses in biomathematics and various biomathematics seminars. It also includes supporting courses from statistics, biological science, chemistry, computer science, and scientific computing. This course of study prepares students for careers in computational biology and the biological applications of mathematics.**Interdisciplinary Data Science.**This is an interdisciplinary degree jointly offered with Departments of Computer Science, Mathematics, Scientific Computing, and Statistics. The master's degree with major in Mathematics emphasizes the mathematical underpinnings of data science along with data analysis, computation and visualization. Students will learn the mathematical background necessary to understand how and why many machine learning and data analysis algorithms work. For more details, see the Bulletin entry for Interdisciplinary Master's Degree in Data Science.

### Doctor of Philosophy (PhD) Degree

The PhD degree indicates knowledge of mathematics and a demonstrated capacity to do original, independent scholarly investigation. Early in the doctoral program, the student will complete major concentration-area course requirements or their equivalents (including courses required for the area MS degree) and will arrange a major professor or co-director within the department to direct the doctoral research. Three to six additional members complete the supervisory committee so that it is mutually agreeable to the student, the major professor or co-director, and the department chair. The supervisory committee must include three or more graduate faculty members of the department as well as a University Representative appropriately drawn from outside the department. The student then satisfies the area, department, and university requirements for doctoral candidacy (MAT 8964) and writes and defends a dissertation of original and independent research. All committee members and the student must attend the entire defense in real time, either by being physically present or participating via distance technology. A grade of PASS for the defense of dissertation requires at least a majority approval of the committee.

Studies leading to the PhD are available in Pure Mathematics, Applied and Computational Mathematics, Financial Mathematics, and Biomathematics. Each area of study specifies its own course requirements. The PhD qualification and candidacy examinations together comprise the preliminary examination, MAT 8964. Course requirements are chosen to provide the student with a strong basis for research. Standard foundational material is covered in the 5000-level courses with more advanced material in topics courses and seminars. Some of the required courses may be offered by other departments. The student will be expected to actively participate in at least one of the seminar series offered by the department and to regularly attend the weekly mathematics colloquium.

The doctoral student in mathematics can be required by his/her supervisory committee to demonstrate proficiency in a minor; normally this is accomplished by completing six or more semester hours in an approved mathematics-related subject with a grade point average (GPA) of at least 3.0. At the discretion of the student's supervisory committee, the student may be required to demonstrate competence in research tools appropriate to the student's program of studies. Such tools may include a reading knowledge of one or more foreign languages, technological skills, a minor, or other competencies.

After the student is admitted to doctoral candidacy, the writing of a dissertation becomes the major concern, although further coursework is usually required. The University's residency requirement must be satisfied. After admission to candidacy the student must register for at least twenty-four hours of dissertation credit (MAT 6980) and also register and participate in the appropriate research seminar for a minimum of three semesters, as well as the mathematics colloquium for a minimum of two semesters. It is a University requirement that the defense of dissertation must be held within five years from the time the student is admitted to doctoral candidacy; if this time limit is not met, the student may be required to repeat the qualifying or candidacy examination.

## Definition of Prefixes

**MAA**—Mathematics: Analysis

**MAD**—Mathematics: Discrete

**MAP**—Mathematics Applied

**MAS**—Mathematics: Algebraic Structures

**MAT**—Mathematics

**MHF**—Mathematics: History and Foundations

**MTG**—Mathematics: Topology and Geometry

**OCP**—Physical Oceanography

## Prerequisite Courses

**Note:** Please refer to the *General Bulletin* for full course descriptions.

**MAA ****4226 **Advanced Calculus I (3)

**MAA ****4227 **Advanced Calculus II (3)

**MAA ****4402 **Complex Variables (3)

**MAC ****2312 **Calculus with Analytic Geometry II (4)

**MAC ****2313 **Calculus with Analytic Geometry III (5)

**MAD ****3703 **Numerical Analysis I (3)

**MAP ****2302 **Ordinary Differential Equations (3)

**MAP ****3305 **Engineering Mathematics I (3)

**MAP ****3306 **Engineering Mathematics II (3)

**MAP ****4153 **Vector Calculus with Introduction to Tensors (3)

**MAP ****4170 **Introduction to Actuarial Mathematics (4)

**MAP ****4341 **Elementary Partial Differential Equations I (3)

**MAP ****4342 **Elementary Partial Differential Equations II (3)

**MAS ****3105 **Applied Linear Algebra I (4)

**MAS ****4302 **Introduction to Abstract Algebra I (3)

**MAS ****4303 **Introduction to Abstract Algebra II (3)

**PHY ****2048C **General Physics [for Physical Sciences] (5)

**STA ****4321 **Introduction to Mathematical Statistics (3)

## Graduate Courses

**Note:** Prerequisites are stated by number from the above list of FSU courses. The equivalent course at another institution as agreed by or consent of the instructor is sufficient.

**MAA ****5306. ****Advanced Calculus I (3)**. Prerequisites: MAC 2313; MAS 3105. Functions, sequences, limits, continuity, uniform continuity; differentiation; integration; convergence, uniform convergence.

**MAA ****5307. ****Advanced Calculus II (3)**. Prerequisite: MAA 5306. Continuation of MAA 5306.

**MAA ****5406. ****Theory of Functions of a Complex Variable I (3)**. Prerequisite: MAA 4227 or 5307; alternatively MAA 4226 and 4402. Algebra and geometry of complex numbers; elementary functions and their mappings. Analytic functions; integration in the complex plane; Cauchy's integral theorem and related theorems. Representation theorems including the Taylor and Laurent expansions. Calculus of residues. Entire and meromorphic functions.

**MAA ****5407. ****Theory of Functions of a Complex Variable II (3)**. Prerequisite: MAA 5406. Continuation of MAA 5406.

**MAA ****5616. ****Measure and Integration I (3)**. Prerequisite: MAA 4227 or 5307. Lebesgue measure and integration; Banach spaces of integrable functions; abstract measure and integration.

**MAA ****5617. ****Measure and Integration II (3)**. Prerequisite: MAA 5616. Continuation of MAA 5616.

**MAA ****5932r. ****Topics in Analysis (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MAA ****6416r. ****Advanced Topics in Analysis (3)**. May be repeated to a maximum of twelve semester hours.

**MAA ****6939r. ****Advanced Seminar in Analysis (1)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAD ****5305. ****Graph Theory (3)**. Prerequisite: Graduate standing (for majors) or department approval (for non-majors). 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 ****5306. ****Graph Theory and Networks (3)**. Prerequisite: MAS 3105. This course provides the mathematical tools necessary to analyze abstract and real-life networks. Topics include the mathematical representation of networks, various forms of network centrality, the structure of real-life networks, and random networks.

**MAD ****5403. ****Foundations of Computational Mathematics I (3)**. Prerequisites: MAS 3105; competence in a programming language suitable for numeric computation. Analysis and implementation of numerical algorithms. Matrix analysis, conditioning, errors, direct and iterative solution of linear systems, rootfinding, systems of nonlinear equations, numerical optimization.

**MAD ****5404. ****Foundations of Computational Mathematics II (3)**. Prerequisite: MAD 5403. Interpolation, quadrature, approximation theory, numerical methods for ordinary differential equations and partial differential equations.

**MAD ****5420. ****Numerical Optimization (3)**. Prerequisites: MAC 2313; MAS 3105; C, C++, or 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 ****5738. ****Numerical Solution of Partial Differential Equations I (3)**. Prerequisites: MAD 5404; MAP 4342 or 5346. Finite difference methods for parabolic, elliptic, and hyperbolic problems; consistency, convergence, stability.

**MAD ****5739. ****Numerical Solution of Partial Differential Equations II (3)**. Prerequisite: MAD 5738. Continuation of MAD 5738.

**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 ****5932r. ****Topics in Computational Mathematics (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MAD ****6408r. ****Advanced Topics in Numerical Analysis (3)**. May be repeated to a maximum of twelve semester hours.

**MAD ****6939r. ****Advanced Seminar in Scientific Computing (1)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAP ****5107. ****Mathematical Modeling (3)**. Prerequisites: MAD 5404; MAP 5431, 5345. Formulation and application of mathematical models for problems arising in the natural sciences, engineering, economics, and industry. Related mathematical topics, including dimensional analysis and scaling, role of dimensionless numbers, perturbation methods, self-similar solutions, traveling waves and solitons, symmetry and symmetry breaking, bifurcations, inverse problems and regularization techniques.

**MAP ****5165. ****Methods of Applied Mathematics I (3)**. Prerequisites: MAP 2302, MAC 2313, and MAS 3105. Continuous and discrete models from physics, chemistry, biology, and engineering are analyzed using perturbation methods, analytical and geometrical tools and dynamical systems theory.

**MAP ****5177. ****Actuarial Models (3)**. Prerequisites: MAP 4170; STA 4321. Survival models; life probabilities; tables, mortality laws; contingent payment models; life annuities; premium principles and net premium reserves for continuous, discrete 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 ****5178. ****Advanced Actuarial Models, Credibility, and Simulation (3)**. Prerequisite: MAP 5177. This course examines claim frequency models, individual loss models, aggregate loss models, multiple-life and multiple-decrement survival models, multiple-state transition models, credibility theory, and simulation.

**MAP ****5196. ****Mathematics for Data Science (3)**. Prerequisite: MAP 2312. This course covers the core mathematics of data science: linear algebra, vector calculus and optimization, probability and graph theory. Applications of these topics will be presented and used as motivation.

**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 Pontryagin's maximum principle.

**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 5345; alternatively MAP 4341 and 4342 or instructor permission. 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: MAD 5738 and, C++ or 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: MAP 4153; MAP 4341 or Corequisite MAP 5345; PHY 2048C. 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. ****Computational Methods in Biology (3)**. This course introduces biological topics where mathematical and computational methods are applicable, including discrete and continuous models of biological systems, numerical methods for differential equations, nonlinear differential equations, and stochastic methods.

**MAP ****5513. ****Wave Propagation Theory (3)**. Prerequisites: MAP 4342 or 5346; MAP 5431. 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 or 3305; MAS 3105; STA 4321. Partial differential equations, Brownian motion, Black-Scholes analysis, introduction to measure and probability; financial applications.

**MAP ****5611. ****Introduction to Computational Finance (3)**. Prerequisites: MAP 5601; C, C++ or appropriate computer language. Computational methods for solving mathematical problems in finance: basic numerical methods, numerical solution of parabolic partial differential equations, including convergence and stability, solution of the Black-Scholes equation, boundary conditions for American options and binomial and random walk methods.

**MAP ****5615. ****Monte Carlo Methods in Financial Mathematics (3)**. Prerequisites: MAP 5601 and competence in a programming language for scientific computing. This course examines how the theory of Monte Carlo Methods is developed in the context of topics selected from computational finance, such as pricing exotic derivatives, American option pricing, and estimating sensitivities. The theory includes pseudorandom numbers, generation of random variables, variance reduction techniques, low-discrepancy sequences, and randomized quasi-Monte Carlo methods.

**MAP ****5932r. ****Topics in Applied Mathematics (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MAP ****6356.**** ****Advanced Partial Differential Equations I (3)****.** Prerequisites: MAA 4226 or equivalent. This course introduces the general classical theory of ordinary differential equations, hyperbolic, parabolic, and elliptic partial differential equations. With each case, some fundamental analytical tools are utilized to probe the nature of the corresponding solutions. Main themes of the course include: the existence and uniqueness of solutions for various initial and boundary value problems, and the properties and solutions of the wave equation, the heat equation, and the Laplace and Poisson equations.

**MAP ****6357.**** ****Advanced Partial Differential Equations II (3)****.** Prerequisites: MAA 4226 or equivalent, MAP 6356. This course is an introduction to functional analytic methods and modern theory in PDEs. Main themes are weak derivatives and Sobolev spaces, some embedding theorems and fundamental inequalities used in the analysis of most PDEs, second order linear elliptic problems (existence of weak solutions, regularity, and eigenfunction expansion), Fourier series and the Galerkin method, applications to linear parabolic equations of second order, distributions, Fourier transform and its applications.

**MAP ****6437r. ****Advanced Topics in Applied Mathematics (3)**. May be repeated to a maximum of twelve semester hours.

**MAP ****6621. ****Financial Engineering I (3)**. Prerequisites: FIN 5515, MAP 5601, 5611 (Recommended: STA 5807). A quantitative treatment of core problems in the investment industry. Topics include an analysis of active portfolio management including risk factor models and mean-variance optimization, the Martingale approach to derivative pricing for both discrete and continuous models, applied stochastic calculus, and stochastic interest rate models.

**MAP ****6939r. ****Advanced Seminar in Applied Mathematics (1)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAS ****5307. ****Groups, Rings, and Vector Spaces I (3)**. Prerequisites: MAS 3105, 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 ****5308. ****Groups, Rings, and Vector Spaces II (3)**. Prerequisite: MAS 5307. Continuation of MAS 5307.

**MAS ****5311. ****Abstract Algebra I (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 ****5312. ****Abstract Algebra II (3)**. Prerequisite: MAS 5311. Continuation of MAS 5311.

**MAS ****5731. ****Computer Algebra (3)**. Prerequisite: MAS 4302. Corequisite: MAS 5307. Factorization of polynomials; decomposition of polynomials; the method of Groebner bases, applications; computing with algebraic numbers.

**MAS ****5932r. ****Topics in Algebra (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MAS ****6939r. ****Advanced Seminar in Algebra (1)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAT ****5907r. ****Directed Individual Study (1–4)**. (S/U grade only). May be repeated to a maximum of eighteen semester hours.

**MAT ****5911r. ****Supervised Research (1–5)**. (S/U grade only). Cannot be applied to the master's degree. May be repeated to a maximum of five semester hours.

**MAT ****5921r. ****Graduate Mathematics Colloquium (1)**. (S/U grade only). Prerequisite: Graduate standing. Speakers drawn from within the department, the wider mathematical community, and from colleagues in fields with related interests; descriptions of timely, cutting edge research in and utilizing mathematics; a full range of current mathematical research including the following: geometry and algebra, classical applied mathematics, computational techniques, biomedical applications, financial economics, mathematical aspects of cryptography and computer security. May be repeated to a maximum of eighteen semester hours.

**MAT ****5932r. ****Selected Advanced Topics (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MAT ****5933r. ****Special Topics in Mathematics (1–3)**. (S/U grade only). Prerequisite: Graduate standing. May be repeated to a maximum of twelve semester hours.

**MAT ****5939r. ****Graduate Seminar (1)**. (S/U grade only). Prerequisite: Instructor permission. May be repeated within the same term to a maximum of twelve semester hours.

**MAT ****5941. ****Internship in College Teaching (1–3)**. (S/U grade only).

**MAT ****5945r. ****Graduate Professional Internship (1–3)**. (S/U grade only). Prerequisite: Instructor permission. Supervised internship individually arranged to accommodate professional development in an area of application. May be repeated to a maximum of three semester hours.

**MAT ****5946r. ****Supervised Teaching (1–5)**. (S/U grade only). May be repeated to a maximum of five semester hours.

**MAT ****5971r. ****Thesis (3–6)**. (S/U grade only). A minimum of six semester hours credit is required for a thesis plan.

**MAT ****6908r. ****Directed Individual Study (1–4)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAT ****6932r. ****Advanced Topics in Mathematics (1–3)**. May be repeated to a maximum of twelve semester hours.

**MAT ****6933r. ****Selected Advanced Topics (1–3)**. (S/U grade only). May be repeated to a maximum of twelve semester hours.

**MAT ****6939r. ****Advanced Graduate Seminar (1)**. (S/U grade only). Prerequisite: Graduate standing. Each specialized seminar introduces students to new aspects of a theoretical or application area. May be repeated to a maximum of twelve semester hours.

**MAT ****6980r. ****Dissertation (1–12)**. (S/U grade only).

**MAT ****8964. ****Doctoral Preliminary Examination (0)**. (P/F grade only.)

**MAT ****8966. ****Master's Comprehensive Examination (0)**. (P/F grade only.)

**MAT ****8968r. ****Doctoral Qualifying Examination (0)**. (P/F grade only.)

**MAT ****8985r. ****Defense of Dissertation (0)**. (P/F grade only.)

**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 4302. Propositional and predicate logic, models. Godels completeness theorem and related theorems. Applications to modern algebra. Non-standard analysis.

**MTG ****5326. ****Topology I (3)**. Prerequisite: Graduate standing. This course examines fundamental group and covering spaces, simplicial and CW complexes, elementary homotopy theory, elementary homology theory, and point set topology.

**MTG ****5327. ****Topology II (3)**. Prerequisite: MTG 5326. Continuation of MTG 5326.

**MTG ****5346. ****Algebraic Topology I (3)**. Prerequisite: MTG 5327. Singular homology and cohomology, orientation of manifolds, cup and cap products, Poincare and Lefschetz duality, acyclic models.

**MTG ****5347. ****Algebraic Topology II (3)**. Prerequisite: MTG 5346. This course examines singular homology and cohomology, orientation of manifolds, cup and cap products, Poincare and Lefschetz duality, and acyclic models.

**MTG ****5356. **** Topological Data Analysis (3)**. Prerequisites: MAA 4224 and MAS 3105. This introductory course applies ideas from algebraic topology to data science, focusing on persistent homology and presenting a balance of high-level theory and real-world applications to data analysis.

**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 semester hours.

**MTG ****5932r. ****Topics in Geometry (1–3)**. Prerequisite: Instructor permission. May be repeated to a maximum of twelve semester hours.

**MTG ****6396r. ****Advanced Topics in Topology (3)**. May be repeated to a maximum of twelve semester hours.

**MTG ****6939r. ****Advanced Seminar in Topology (1)**. (S/U grade only). May be repeated to a maximum of eight semester hours.

**OCP ****5256. ****Fluid Dynamics: Geophysical Applications (3)**. Prerequisites: MAP 5431, 5346; or instructor permission. 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.