## Doctor of Philosophy (PhD)

## Master of Science (MS)

## Master of Science in Operations Research (MSOR)

## Graduate Certificate

## Mathematics Courses

**MATH 5001. Accelerated Linear Algebra. (2 Hours)**

Offers an intensive exploration of linear algebra. Covers linear algebra concepts at an accelerated pace including vector spaces, linear transformations, matrix operations, eigenvalues and eigenspaces, inner product spaces, orthogonal basis, least squares, positive semidefinite matrices, and singular value decompositions. Includes applications to statistics, dynamical systems, and machine learning.

**MATH 5002. Accelerated Multivariable Calculus. (2 Hours)**

Offers an intensive exploration of multivariable calculus. Covers vector calculus concepts at an accelerated pace including partial derivatives, level sets, gradient vectors, optimizations, Lagrange multipliers, multiple integrals, vector fields, line integrals, and surface integrals. Includes applications to statistics, engineering, and machine learning.

**MATH 5003. Accelerated Probability and Statistics. (2 Hours)**

Offers an intensive exploration of probability and statistics including both theoretical concepts and practical applications. Covers discrete and continuous random variables, probability distributions, expectation and variance, central limit theorem, estimation theory, maximum likelihood estimation, and hypothesis testing. Includes applications to machine learning and solving real-world problems.

**MATH 5010. Foundations of Statistical Theory and Probability. (4 Hours)**

Presents a comprehensive foundational overview of the probability theory and mathematical statistics critical to various methodologies needed to perform and interpret a statistical analysis. Explores probability distributions and the theoretical justifications around common statistical measures such as means, variances, and expected values. Introduces likelihood and Bayesian theory as tools to derive parameter estimates and begin to make inferences. Culminates with formal procedures for statistical inference including hypothesis testing, the construction of confidence intervals and credible intervals, and calculations of power and sample size. All topics are grounded in theory, with appropriate practice using a programming language and real-world datasets to communicate and visualize data and results.

**MATH 5101. Analysis 1: Functions of One Variable. (4 Hours)**

Offers a rigorous, proof-based introduction to mathematical analysis and its applications. Topics include metric spaces, convergence, compactness, and connectedness; continuous and uniformly continuous functions; derivatives, the mean value theorem, and Taylor series; Riemann integration and the fundamental theorem of calculus; interchanging limit operations; sequences of functions and uniform convergence; Arzelà-Ascoli and Stone-Weierstrass theorems; inverse and implicit function theorems; successive approximations and existence/uniqueness for ordinary differential equations; linear operators on finite-dimensional vector spaces and applications to systems of ordinary differential equations. Provides a series of computer projects that further develop the connections between theory and applications. Requires permission of instructor and head advisor for undergraduate students.

**MATH 5102. Analysis 2: Functions of Several Variables. (4 Hours)**

Continues MATH 5101. Studies basics of analysis in several variables. Topics include derivative and partial derivatives; the contraction principle; the inverse function and implicit function theorems; derivatives of higher order; Taylor formula in several variables; differentiation of integrals depending on parameters; integration of functions of several variables; change of variables in integrals; differential forms and their integration over simplexes and chains; external multiplication of forms; differential of forms; Stokes’ formula; set functions; Lebesgue measure; measure spaces; measurable functions; integration; comparison with the Riemann integral; L2 as a Hilbert space; and Parseval theorem and Riesz-Fischer theorem. Requires permission of instructor and head advisor for undergraduate students.

**MATH 5110. Applied Linear Algebra and Matrix Analysis. (4 Hours)**

Offers a robust introduction to the basic results of linear algebra on real and complex vector spaces with applications to differential equations and Markov chains. Introduces theoretical results along the way, along with matrix analysis, eigenvalue analysis, and spectral decomposition. Includes a significant computational component, focused on applications of linear algebra to mathematical modeling.

**MATH 5111. Algebra 1. (4 Hours)**

Discusses fundamentals of the theory of groups and some applications in Galois theory. Topics may include quotient groups and isomorphism theorems; group actions and Sylow theory; simplicity and solvability; permutations and the simplicity of the higher alternating groups; fields and polynomial rings; splitting fields; the Galois correspondence; computations of Galois groups; and applications of Galois theory, including non-solvability of polynomials by radicals.

**MATH 5112. Algebra 2. (4 Hours)**

Provides a comprehensive introduction to commutative algebra: rings and modules. Topics in commutative ring theory include ideals, prime and maximal ideals, ring homomorphisms, Euclidean domains, principal ideal domains, unique factorization domains, fields of quotients, polynomial rings, irreducibility criteria, and the Chinese Remainder Theorem. Topics in module theory include module homomorphisms, the structure theorem for modules over a PID and applications, and the Jordan and rational canonical forms.

**MATH 5121. Topology 1. (4 Hours)**

Provides an introduction to topology, starting with the basics of point set topology (topological space, continuous maps, homeomorphisms, compactness and connectedness, and identification spaces). Moves on to the basic notions of algebraic and combinatorial topology, such as homotopy equivalences, fundamental group, Seifert-VanKampen theorem, simplicial complexes, classification of surfaces, and covering space theory. Ends with a brief introduction to simplicial homology and knot theory. Requires permission of instructor and head advisor for undergraduate students.

**Prerequisite(s): **MATH 5111 with a minimum grade of C-

**MATH 5122. Geometry 1. (4 Hours)**

Covers differentiable manifolds, such as tangent bundles, tensor bundles, vector fields, Frobenius integrability theorem, differential forms, Stokes’ theorem, and de Rham cohomology; and curves and surfaces, such as elementary theory of curves and surfaces in R3, fundamental theorem of surfaces in R3, surfaces with constant Gauss or mean curvature, and Gauss-Bonnet theorem for surfaces. Requires permission of instructor and head advisor for undergraduate students.

**Prerequisite(s): **MATH 5101 with a minimum grade of C- ; MATH 5111 with a minimum grade of C-

**MATH 5131. Introduction to Mathematical Methods and Modeling. (4 Hours)**

Presents mathematical methods emphasizing applications. Uses ordinary and partial differential equations to model the evolution of real-world processes. Topics chosen illustrate the power and versatility of mathematical methods in a variety of applied fields and include population dynamics, drug assimilation, epidemics, spread of pollutants in environmental systems, competing and cooperating species, and heat conduction. Requires students to complete a math-modeling project. Requires undergraduate-level course work in ordinary and partial differential equations.

**Attribute(s): ** NUpath Capstone Experience, NUpath Writing Intensive

**MATH 5352. Quantum Computation and Information. (4 Hours)**

Introduces the foundations of quantum computation and information, including finite dimensional quantum mechanics, gates and circuits, quantum algorithms, quantum noise, and error-correcting codes. Assumes a working knowledge of linear algebra and matrix analysis, but no prior experience with quantum theory or algorithms is required.

**MATH 6000. Professional Development for Co-op. (0 Hours)**

Introduces the cooperative education program. Offers students an opportunity to develop job-search and career-management skills; to assess their workplace skills, interests, and values and to discuss how they impact personal career choices; to prepare a professional resumé; and to learn proper interviewing techniques. Explores career paths, choices, professional behaviors, work culture, and career decision making.

**MATH 6241. Stochastic Processes. (2 Hours)**

Reviews probability theory; conditional probability and expectations; discrete-time and continuous-time Markov chains; Poisson process and exponential distributions (renewal theory and its applications, queueing theory); Brownian motion; and stochastic systems. Examines applications to Monte Carlo simulation and various real-world scenarios. Involves practical applications through computer simulations by a language programming and/or project in real-world problems such as finance, biostatistics, machine learning, etc.

**Prerequisite(s): **MATH 5010 with a minimum grade of C-

**MATH 6243. Statistical Learning. (4 Hours)**

Presents fundamental principles of machine learning from a statistical standpoint. Focuses on supervised learning, including regression and classification methods. Topics include linear and polynomial regression, logistic regression, and linear discriminant analysis; cross-validation and the bootstrap, model selection, and regularization methods (ridge and lasso); nonlinear models, splines, and generalized additive models; tree-based methods, random forests, and boosting; and support-vector machines. Examines methods used to learn from data taking a deeper look at algorithms that may appear to be different but actually use similar principles. Discusses some unsupervised learning methods principal components and clustering. Uses a programming language to fit these models using modern techniques for visualization and reporting.

**Prerequisite(s): **MATH 5010 with a minimum grade of C-

**MATH 6910. Master's Project. (1 Hour)**

Offers the opportunity to conduct an experimental capstone project or a theoretical work project under faculty supervision. Students identify skills learned in the degree program and articulate their skills, knowledge, and abilities as a part of the course, as they transition to the field or advance in the field. May be repeated once.

**MATH 6954. Co-op Work Experience - Half-Time. (0 Hours)**

Provides eligible students with an opportunity for work experience. May be repeatedwithout limit.

**MATH 6955. Co-op Work Experience Abroad - Half Time. (0 Hours)**

Provides eligible students with an opportunity for work experience abroad.

**MATH 6961. Internship. (1-4 Hours)**

Offers students an opportunity for internship work. May be repeated without limit.

**MATH 6962. Elective. (1-4 Hours)**

Offers elective credit for courses taken at other academic institutions. May be repeated without limit.

**MATH 6964. Co-op Work Experience. (0 Hours)**

Provides eligible students with an opportunity for work experience. May be repeated without limit.

**MATH 6965. Co-op Work Experience Abroad. (0 Hours)**

Provides eligible students with an opportunity for work experience abroad. May be repeated without limit.

**MATH 7202. Partial Differential Equations 1. (4 Hours)**

Introduces partial differential equations, their theoretical foundations, and their applications, which include optics, propagation of waves (light, sound, and water), electric field theory, and diffusion. Topics include first-order equations by the method of characteristics; linear, quasilinear, and nonlinear equations; applications to traffic flow and geometrical optics; principles for higher-order equations; power series and Cauchy-Kowalevski theorem; classification of second-order equations; linear equations and generalized solutions; wave equations in various space dimensions; domain of dependence and range of influence; Huygens’ principle; conservation of energy, dispersion, and dissipation; Laplace’s equation; mean values and the maximum principle; the fundamental solution, Green’s functions, and Poisson kernels; applications to physics; properties of harmonic functions; the heat equation; eigenfunction expansions; the maximum principle; Fourier transform and the Gaussian kernel; regularity of solutions; scale invariance and the similarity method; Sobolev spaces; and elliptic regularity.

**MATH 7203. Numerical Analysis 1. (4 Hours)**

Introduces methods and techniques used in contemporary number crunching. Covers floating-point computations involving scalars, vectors, and matrices; solvers for sparse and dense linear systems; matrix decompositions; integration of functions and solutions of ordinary differential equations (ODEs); and Fast Fourier transform. Focuses on finding solutions to practical, real-world problems. Knowledge of programming in Matlab is assumed. Knowledge of other programming languages would be good but not required.

**MATH 7205. Numerical Analysis 2. (4 Hours)**

Covers numerical analysis and scientific computation. Topics include numerical solutions of ordinary differential equations (ODEs) and one-dimensional boundary value problems; solving partial differential equations (PDEs) using modal expansions, finite-difference, and finite-element methods; stability of PDE algorithms; elementary computational geometry and mesh generation; unconstrained optimization with application to data modeling; and constrained optimization of convex functions: linear and quadratic programming. Focuses on techniques commonly used for data fitting and solving problems from engineering and physical science. Knowledge of programming in MATLAB is assumed. Knowledge of other programming languages beneficial but not required.

**MATH 7207. Algorithms for Optimization. (4 Hours)**

Covers algorithms used to solve optimization problems, specifically algorithms used to optimize a set of continuous design variables. Presents classic algorithms such as gradient descent, Newton’s method, simulated annealing, the simplex method, and others. Studies algorithms for both unconstrained and constrained optimization. Discusses optimization of both convex and nonconvex objective functions. Emphasizes hands-on and practical implementation of the algorithms presented by writing computer programs. Uses applications from business and industry to illustrate how the algorithms work and when they should be applied.

**MATH 7221. Topology 2. (4 Hours)**

Continues MATH 5121. Introduces homology and cohomology theory. Studies singular homology, homological algebra (exact sequences, axioms), Mayer-Vietoris sequence, CW-complexes and cellular homology, calculation of homology of cellular spaces, and homology with coefficients. Moves on to cohomology theory, universal coefficients theorems, Bockstein homomorphism, Knnneth formula, cup and cap products, Hopf invariant, Borsuk-Ulam theorem, and Brouwer and Lefschetz-Hopf fixed-point theorems. Ends with a study of duality in manifolds including orientation bundle, Poincaré duality, Lefschetz duality, Alexander duality, Euler class, Lefschetz numbers, Gysin sequence, intersection form, and signature.

**MATH 7223. Riemannian Optimization. (4 Hours)**

Offers a self-contained introduction to optimization on smooth manifolds. Covers both theoretical foundations and practical computational methods that students can apply to their own work. Introduces the theory of Riemannian geometry. Emphasizes those elements that are relevant for the construction of optimization algorithms (tensor fields, metrics, connections, geodesics, retractions, and transporters). Applies this geometric machinery to devise first- and second-order smooth optimization methods on generic Riemannian manifolds. Focuses on the development of practical computational techniques, with applications to robotics, computer vision, and machine learning.

**MATH 7233. Graph Theory. (4 Hours)**

Covers fundamental concepts in graph theory. Topics include adjacency and incidence matrices, paths and connectedness, and vertex degrees and counting; trees and distance including properties of trees, distance in graphs, spanning trees, minimum spanning trees, and shortest paths; matchings and factors including matchings in bipartite graphs, Hall’s matching condition, and min-max theorems; connectivity, such as vertex connectivity, edge connectivity, k-connected graphs, and Menger’s theorem; network flows including maximum network flow, and integral flows; vertex colorings, such as upper bounds, Brooks, theorem, graphs with large chromatic number, and critical graphs; Eulerian circuits and Hamiltonian cycles including Euler’s theorem, necessary conditions for Hamiltonian cycles, and sufficient conditions; planar graphs including embeddings and Euler’s formula, characterization of planar graphs (Kuratowski’s theorem); and Ramsey theory including Ramsey’s theorem, Ramsey numbers, and graph Ramsey theory.

**MATH 7234. Optimization and Complexity. (4 Hours)**

Offers theory and methods of maximizing and minimizing solutions to various types of problems. Studies combinatorial problems including mixed integer programming problems (MIP); pure integer programming problems (IP); Boolean programming problems; and linear programming problems (LP). Topics include convex subsets and polyhedral subsets of n-space; relationship between an LP problem and its dual LP problem, and the duality theorem; simplex algorithm, and Kuhn-Tucker conditions for optimality for nonlinear functions; and network problems, such as minimum cost and maximum flow-minimum cut. Also may cover complexity of algorithms; problem classes P (problems with polynomial-time algorithms) and NP (problems with nondeterministic polynomial-time algorithms); Turing machines; and NP-completeness of traveling salesman problem and other well-known problems.

**MATH 7235. Discrete Geometry 1. (4 Hours)**

Discusses basic concepts in discrete and combinatorial geometry. Topics may include convex sets and their basic properties; theorems of Helly, Radon, and Carathéodory; separation theorems for convex bodies; convex polytopes; face vectors; Euler’s theorem and Dehn-Sommerville equations; upper bound theorem; symmetry groups; regular polytopes and tessellations; reflection groups and Coxeter groups; regular tessellations on surfaces; abstract regular and chiral polytopes; and other topics at instructor’s discretion.

**MATH 7241. Probability 1. (4 Hours)**

Offers an introductory course in probability theory, with an emphasis on problem solving and modeling. Starts with basic concepts of probability spaces and random variables, and moves on to the classification of Markov chains with applications. Other topics include the law of large numbers and the central limit theorem, with applications to the theory of random walks and Brownian motion.

**MATH 7243. Machine Learning and Statistical Learning Theory 1. (4 Hours)**

Introduces both the mathematical theory of learning and the implementation of modern machine-learning algorithms appropriate for data science. Modeling everything from social organization to financial predictions, machine-learning algorithms allow us to discover information about complex systems, even when the underlying probability distributions are unknown. Algorithms discussed include regression, decision trees, clustering, and dimensionality reduction. Offers students an opportunity to learn the implications of the mathematical choices underpinning the use of each algorithm, how the results can be interpreted in actionable ways, and how to apply their knowledge through the analysis of a variety of data sets and models.

**MATH 7301. Functional Analysis. (4 Hours)**

Provides an introduction to essential results of functional analysis and some of its applications. The main abstract facts can be understood independently. Proof of some important basic theorems about Hilbert and Banach spaces (Hahn-Banach theorem, open mapping theorem) are omitted, in order to allow more time for applications of the abstract techniques, such as compact operators; Peter-Weyl theorem for compact groups; spectral theory; Gelfand’s theory of commutative C*-algebras; mean ergodic theorem; Fourier transforms and Sobolev embedding theorems; and distributions and elliptic operators.

**Prerequisite(s): **MATH 5102 with a minimum grade of C- or MATH 5102 with a minimum grade of C-

**MATH 7311. Commutative Algebra. (4 Hours)**

Introduces some of the main tools of commutative algebra, particularly those tools related to algebraic geometry. Topics include prime ideals, localization, and integral extensions; primary decomposition; Krull dimension; chain conditions, and Noetherian and Artinian modules; and additional topics from ring and module theory as time permits.

**Prerequisite(s): **MATH 5111 with a minimum grade of C-

**MATH 7314. Algebraic Geometry 1. (4 Hours)**

Concentrates on the techniques of algebraic geometry arising from commutative and homological algebra, beginning with a discussion of the basic results for general algebraic varieties, and developing the necessary commutative algebra as needed. Considers affine and projective varieties, morphisms of algebraic varieties, regular and singular points, and normality. Discusses algebraic curves, with a closer look at the relations between the geometry, algebra, and function theories. Examines the Riemann-Roch theorem with its many applications to the study of the geometry of curves. Studies the singularities of curves.

**Prerequisite(s): **MATH 5112 with a minimum grade of C-

**MATH 7315. Algebraic Number Theory. (4 Hours)**

Covers rings of integers, Dedekind domains, factorization of ideals, ramification, and the decomposition and inertia subgroups; units in rings of integers, Minkowski’s geometry of numbers, and Dirichlet’s unit theorem; and class groups, zeta functions, and density sets of primes.

**Prerequisite(s): **MATH 5111 with a minimum grade of C-

**MATH 7317. Modern Representation Theory. (4 Hours)**

Introduces students to modern techniques of representation theory, including those coming from geometry and mathematical physics. Covers applications of geometry to the representation theory of semisimple Lie algebras, algebraic groups and related algebraic objects, questions related to the representation theory of infinite dimensional Lie algebras, quantum groups, and p-adic groups, as well as category theory methods in representation theory.

**Prerequisite(s): **MATH 5111 with a minimum grade of C-

**MATH 7320. Modern Algebraic Geometry. (4 Hours)**

Introduces students to modern techniques of algebraic geometry, including those coming from Lie theory, symplectic and differential geometry, complex analysis, and number theory. Covers subjects related to invariant theory, homological algebra questions of algebraic geometry, including derived categories and complex analytic, differential geometric, and arithmetic aspects of the geometry of algebraic varieties. Students not meeting course prerequisites or restrictions may seek permission of instructor.

**Prerequisite(s): **MATH 7314 with a minimum grade of C- or MATH 7361 with a minimum grade of C-

**MATH 7321. Topology 3. (4 Hours)**

Continues MATH 7221 and studies classical algebraic topology and its applications. Introduces homotopy theory. Topics include higher homotopy groups, cofibrations, fibrations, homotopy sequences, homotopy groups of Lie groups and homogeneous spaces, Hurewicz theorem, Whitehead theorem, Eilenberg-MacLane spaces, obstruction theory, Postnikov towers, and spectral sequences.

**Prerequisite(s): **MATH 7221 with a minimum grade of C-

**MATH 7338. Probability and Measure. (4 Hours)**

Focuses on measure-theoretic probability theory, with an emphasis on proofs. Introduces measure spaces and the different notions of convergence that are relevant for random variables. Studies the classical limit theorems about sums of independent, identically distributed random variables. Proves both the law of large numbers and central limit theorem under optimal assumptions. Covers the theory of martingales, with applications toward random walks, Markov chains, and continuous time diffusion processes.

**Prerequisite(s): **MATH 5102 with a minimum grade of C-

**MATH 7339. Machine Learning and Statistical Learning Theory 2. (4 Hours)**

Continues MATH 7243. Further covers theory and methods for regression and classification, along with more advanced topics in machine learning, statistical learning, and deep learning. Reviews the basics of machine learning in a broader and deeper way. Additional topics are drawn from smoothing methods, clustering, latent variable models, mixture models, Markov decision process and reinforcement learning, and neural networks. Discusses recent research papers on image classification and segmentation, generative adversarial network, neural style transfer, natural language processing, and topological data analysis. Uses theory, models, and algorithms to analyze a variety of datasets.

**Prerequisite(s): **CS 6140 with a minimum grade of C- or DS 5220 with a minimum grade of C- or EECE 5644 with a minimum grade of C- or MATH 7243 with a minimum grade of C-

**MATH 7340. Statistics for Bioinformatics. (4 Hours)**

Introduces the concepts of probability and statistics used in bioinformatics applications, particularly the analysis of microarray data. Uses statistical computation using the open-source R program. Topics include maximum likelihood; Monte Carlo simulations; false discovery rate adjustment; nonparametric methods, including bootstrap and permutation tests; correlation, regression, ANOVA, and generalized linear models; preprocessing of microarray data and gene filtering; visualization of multivariate data; and machine-learning techniques, such as clustering, principal components analysis, support vector machine, neural networks, and regression tree.

**MATH 7341. Probability 2. (4 Hours)**

Continues MATH 7241. Studies probability theory, with an emphasis on its use in modeling and queueing theory. Starts with basic properties of exponential random variables, and then applies this to the study of the Poisson process. Queueing theory forms the bulk of the course, with analysis of single-server queues, multiserver queues, and networks of queues. Also includes material on continuous-time Markov processes, renewal theory, and Brownian motion.

**Prerequisite(s): **MATH 7241 with a minimum grade of C- or IE 6200 with a minimum grade of C-

**MATH 7342. Mathematical Statistics. (4 Hours)**

Introduces mathematical statistics, emphasizing theory of point estimations. Topics include parametric estimations, minimum variance unbiased estimators, sufficiency and completeness, and Rao-Blackwell theorem; asymptotic (large sample) theory, maximum likelihood estimator (MLE), consistency of MLE, asymptotic theory of MLE, and Cramer-Rao bound; and hypothesis testing, Neyman-Pearson fundamental lemma, and likelihood ratio test.

**MATH 7343. Applied Statistics. (4 Hours)**

Designed as a basic introductory course in statistical methods for graduate students in mathematics as well as various applied sciences. Topics include descriptive statistics, inference for population means, analysis of variance, nonparametric methods, and linear regression. Studies how to use the computer package SPSS, doing statistical analysis and interpreting computer outputs.

**MATH 7344. Regression, ANOVA, and Design. (4 Hours)**

Discusses one-sample and two-sample tests; one-way ANOVA; factorial and nested designs; Cochran’s theorem; linear and nonlinear regression analysis and corresponding experimental design; analysis of covariance; and simultaneous confidence intervals.

**MATH 7351. Mathematical Methods of Classical Mechanics. (4 Hours)**

Overviews the mathematical formulation of classical mechanics. Topics include Hamilton’s principle and Lagrange’s equations; solution of the two-body central force problem; rigid body rotation and Euler’s equations; the spinning top; Hamilton’s equations; the Poisson bracket; Liouville’s theorem; and canonical transformations.

**MATH 7359. Elliptic Curves and Modular Forms. (4 Hours)**

Introduces elliptic curves and modular forms. Examines elliptic curves as algebraic varieties over the complex numbers, finite fields, and local and global fields. Related topics include the j-invariant, the Tate module and the Weil pairing, zeta functions and the Weil conjectures, and the Mordell-Weil theorem. Modular forms are defined on moduli spaces of elliptic curves. Related topics include Eisenstein series, cusp forms, congruence subgroups of SL2(Z), modularity and the Taniyama-Shimura conjecture, and Fermat’s Last Theorem.

**Prerequisite(s): **MATH 5111 with a minimum grade of C- or MATH 5112 with a minimum grade of C-

**MATH 7362. Topics in Algebra. (4 Hours)**

Focuses on various advanced topics in algebra, the specific subject matter depending on the interests of the instructor and of the students. Topics may include homological algebra, commutative algebra, representation theory, or combinatorial aspects of commutative algebra. May be repeated without limit.

**MATH 7363. Topics in Algebraic Geometry. (4 Hours)**

Focuses on various advanced topics in algebraic geometry, the specific subject matter depending on the interests of the instructor and of the students. Topics may include integrable systems, cohomology theory of algebraic schemes, study of singularities, geometric invariant theory, and flag varieties and Schubert varieties. May be repeated without limit.

**MATH 7364. Topics in Representation Theory. (4 Hours)**

Offers topics in the representation theory of the classical groups, topics vary according to the interest of the instructor and students. Topics may include root systems, highest weight modules, Verma modules, Weyl character formula, Schur commutator lemma, Schur functors and symmetric functions, and Littlewood-Richardson rule. May be repeated up to five times.

**MATH 7371. Morse Theory. (4 Hours)**

Covers basic Morse theory for nondegenerate smooth functions, and applications to geodesics, Lie groups and symmetric spaces, Bott periodicity, Morse inequalities, and Witten deformation.

**Prerequisite(s): **(MATH 5122 with a minimum grade of C- or MATH 5122 with a minimum grade of D- ); MATH 7221 with a minimum grade of C- ; MATH 7301 with a minimum grade of C-

**MATH 7375. Topics in Topology. (4 Hours)**

Offers various advanced topics in algebraic and geometric topology, the subject matter depending on the instructor and the students. Topics may include Morse theory, fiber bundles and characteristic classes, topology of complex hypersurfaces, knot theory and low-dimensional topology, K-theory, and rational homotopy theory. May be repeated without limit.

**Prerequisite(s): **MATH 5121 (may be taken concurrently) with a minimum grade of C-

**MATH 7381. Topics in Combinatorics. (4 Hours)**

Offers various advanced topics in combinatorics, the subject matter depending on the instructor and the students. May be repeated without limit.

**Prerequisite(s): **(MATH 5122 with a minimum grade of C- or MATH 5122 with a minimum grade of D- ); MATH 7222 with a minimum grade of C-

**MATH 7382. Topics in Probability. (4 Hours)**

Offers various advanced topics in probability and related areas. The specific subject matter depends on the interest of the instructor and students. May be repeated up to five times.

**MATH 7721. Readings in Topology. (4 Hours)**

Offers a reading course to be arranged between an individual student and instructor on a topic of their mutual choice. May be repeated without limit.

**MATH 7733. Readings in Graph Theory. (4 Hours)**

Offers a reading course to be arranged between an individual student and instructor on a topic of their mutual choice. May be repeated without limit.

**MATH 7734. Readings in Algebra. (4 Hours)**

Offers a reading course to be arranged between an individual student and instructor on a topic of their mutual choice. May be repeated without limit.

**MATH 7735. Readings in Algebraic Geometry. (4 Hours)**

**MATH 7736. Readings in Discrete Geometry. (4 Hours)**

**MATH 7741. Readings in Probability and Statistics. (4 Hours)**

**MATH 7755. Readings in Partial Differential Equations. (4 Hours)**

Offers a reading course to be arranged between an individual student and instructor on a topic of their mutual choice. May be repeated up to five times.

**Prerequisite(s): **MATH 7202 with a minimum grade of C-

**MATH 7771. Readings in Geometry. (4 Hours)**

Offers topics in geometry that are beyond the ordinary undergraduate topics. Topics include the regular polytopes in dimensions greater than three, straight-edge and compass constructions in hyperbolic geometry, Penrose tilings, the geometry and algebra of the wallpaper, and three-dimensional Euclidean groups. May be repeated without limit.

**MATH 7962. Elective. (1-4 Hours)**

Offers elective credit for courses taken at other academic institutions. May be repeated without limit.

**MATH 8450. Research Seminar in Mathematics. (4 Hours)**

Introduces graduate students to current research in geometry, topology, mathematical physics, and in other areas of mathematics. Requires permission of instructor for undergraduate mathematics students. May be repeated without limit.

**MATH 8984. Research. (1-4 Hours)**

Offers an opportunity to conduct research under faculty supervision. May be repeated without limit.

**MATH 8986. Research. (0 Hours)**

Offers an opportunity to conduct full-time research under faculty supervision. May be repeated without limit.

**MATH 9000. PhD Candidacy Achieved. (0 Hours)**

Indicates successful completion of the doctoral comprehensive exam.

**MATH 9984. Research. (1-4 Hours)**

Offers an opportunity to conduct research under faculty supervision. May be repeated without limit.

**MATH 9990. Dissertation Term 1. (0 Hours)**

Offers dissertation supervision by members of the department.

**Prerequisite(s): **MATH 9000 with a minimum grade of S

**MATH 9991. Dissertation Term 2. (0 Hours)**

Offers dissertation supervision by members of the department.

**Prerequisite(s): **MATH 9990 with a minimum grade of S

**MATH 9996. Dissertation Continuation. (0 Hours)**

Offers dissertation supervision by members of the department.

**Prerequisite(s): **MATH 9991 with a minimum grade of S or Dissertation Check with a score of REQ