**Mathematics Courses**

**MST 605. Applied Multivariable Mathematics. **(3) Introduction to several topics in applied mathematics including complex numbers, probability, matrix algebra, multivariable calculus, and ordinary differential equations. May not be used toward any graduate degree offered by the department.

**MST 606. Advanced Mathematics for the Physical Sciences. **(3) Advanced topics in linear algebra, special functions, integral transforms, and partial differential equations. May not be used toward any graduate degree offered by the department. *P—MST 605*

**MST 611, 612. Introductory Real Analysis I, II.** (3, 3) Limits and continuity in metric spaces, sequences and series, differentiation and Riemann-Stieltjes integration, uniform convergence, power series and Fourier series, differentiation of vector functions, implicit and inverse function theorems.

**MST 617. Complex Analysis I. **(3) Analytic functions Cauchy’s theorem and its consequences, power series, and residue calculus.

**MST 622. Modern Algebra II. **(3) A continuation of modern abstract algebra through the study of additional properties of groups, rings, and fields.

**MST 624. Linear Algebra II.** (3) A thorough treatment of vector spaces and linear transformations over an arbitrary field, canonical forms, inner product spaces, and linear groups.

**MST 626. Numerical Linear Algebra.** (3) An introduction to numerical methods for solving matrix and related problems in science and engineering using a high-level matrix-oriented language such as MATLAB. Topics include systems of linear equations, least squares methods, and eigenvalue computations. Special emphasis is given to applications. Also listed as CSC 652.

**MST 631. Geometry.** (3) An introduction to axiomatic geometry including a comparison of Euclidean and non-Euclidean geometries.

**MST 634. Differential Geometry.** (3) Introduction to the theory of curves and surfaces in two and three dimensional space including such topics as curvature, geodesics, and minimal surfaces.

**MST 645. Elementary Number Theory.** (3) Course topics include properties of integers, congruences, and prime numbers, with additional topics chosen from arithmetic functions, primitive roots, quadratic residues, Pythagorean triples, and sums of squares.

**MST 646. Modern Number Theory** (3) Course topics include a selection of number-theory topics of recent Interest. Some examples include elliptic curves, partitions, modular forms, the Riemann zeta function, and algebraic number theory.

**MST 647. Graph Theory. **(3) Paths, circuits, trees, planar graphs, spanning trees, graph coloring, perfect graphs, Ramsey theory, directed graphs, enumeration of graphs and graph theoretic algorithms.

**MST 648, 649. Combinatorial Analysis I, II.** (3, 3) Enumeration techniques, generating functions, recurrence formulas, the principle of inclusion and exclusion, Polya theory, graph theory, combinatorial algorithms, partially ordered sets, designs, Ramsey theory, symmetric functions, and Schur functions.

**MST 652. Partial Differential Equations. **(3) Detailed study of partial differential equations, including the heat, wave, and Laplace equations, using methods such as separation of variables, characteristics, Green’s functions, and the maximum principle.

**MST 653. Probability Models. **(3) Course topics include an introduction to probability models, Markov chains, Poisson process and Markov decision processes. Applications will emphasize problems in business and management science.

**MST 654. Discrete Dynamical Systems.** (3) Introduction to the theory of discrete dynamical systems as applied to disciplines such as biology and economics. Includes methods for finding explicit solutions, equilibrium and stability analysis, phase plane analysis, analysis of Markov chains and bifurcation theory.

**MST 655. Introduction to Numerical Methods.** (3) An introduction to numerical computations on modern computer architectures; floating point arithmetic and round-off error including programming in a scientific/engineering language such as MATLAB, C or Fortran. Topics include algorithms and computer techniques for the solution of problems such as roots of functions, approximations, integration, systems of linear equations and least squares methods. Also listed as CSC 655.

**MST 657. Probability. **(3) Course topics include probability distributions, mathematical expectation, and sampling distributions. MST 657 covers much of the material on the syllabus for the first actuarial exam.

**MST 681. Individual Study. **(1 or 2) A course of independent study directed by a faculty adviser. By prearrangement. May be repeated for credit.

**MST 682. Reading in Mathematics. **(1, 2, or 3) Reading in mathematical topics to provide a foundational basis for more advanced study in a particular mathematical area. Topics vary and may include material from algebra, analysis, combinatorics, computational or applied mathematics, number theory, topology, or statistics. May not be used to satisfy any requirement in the mathematics MA degree with thesis. No more than three hours may be applied to the requirements for the mathematics MA degree without thesis. May be repeated for credit for a total of 3 hours.

**MST 683. Advanced Topics in Mathematics. **(1, 2 or 3) Topics in mathematics that are not considered in regular courses. Content varies.

**MST 711, 712. Real Analysis. **(3, 3) Measure and integration theory, elementary functional analysis, selected advanced topics in analysis.

**MST 715, 716. Seminar in Analysis. **(1, 1)

**MST 717. Optimization in Banach Spaces.** (3) Banach and Hilbert spaces, best approximations, linear operators and adjoints, Frechet derivatives and nonlinear optimization, fixed points and iterative methods. Applications to control theory, mathematical programming, and numerical analysis.

**MST 718. Topics in Analysis.** (3) Selected topics from functional analysis or analytic function theory.

**MST 721, 722. Abstract Algebra.** (3, 3) Groups, rings, fields, extensions, Euclidean domains, polynomials, vector spaces, and Galois theory.

**MST 723, 724. Seminar on Theory of Matrices. **(1, 1)

**MST 725, 726. Seminar in Algebra**. (1, 1)

**MST 728. Topics in Algebra.** (3) Topics vary and may include algebraic coding theory, algebraic number theory, matrix theory, representation theory, non-commutative ring theory.

**MST 731. Topology. **(3) Point-set topology including topological spaces, continuity, connectedness, compactness, and metric spaces. Addition al topics in topology may include classification of surfaces, algebraic topology, and knot theory.

**MST 732. Topics in Topology and Geometry.** (3) Topics vary and may include knot theory, algebraic topology, differential topology, manifolds, and Riemannian geometry. May be repeated for credit. *P—731 or POI*

**MST 735, 736. Seminar on Topology.** (1, 1)

**MST 737, 738. Seminar on Geometry. **(1, 1)

**MST 744. Topics in Number Theory.** (3) Topics vary and are chosen from the areas of analytic, algebraic, and elementary number theory. Topics may include Farey fractions, the theory of partitions, Waring’s problem, prime number theorem, and Dirichlet’s problem.

**MST 745, 746. Seminar on Number Theory.** (1, 1)

**MST 747. Topics in Discrete Mathematics.** (3) Topics vary and may include enumerative combinatorics, graph theory, algebraic combinatorics, combinatorial optimization, coding theory, experimental designs, Ramsey theory, Polya theory, representation theory, set theory and mathematical logic.

**MST 748, 749. Seminar on Combinatorial Analysis. **(1, 1)

**MST 750. Dynamical Systems. **(3) Introduction to modern theory of dynamical systems. Linear and nonlinear autonomous differential equations, invariant sets, closed orbits, Poincare maps, structural stability, center manifolds, normal forms, local bifurcations of equilibria, linear and non-linear maps, hyperbolic sets, attractors, symbolic representation, fractal dimensions. *P—MST 611*

**MST 752. Topics in Applied Mathematics.** (3) Topics vary and may include computational methods in differential equations, optimization methods, approximation techniques, eigenvalue problems. May be repeated for credit.

**MST 753. Nonlinear Optimization.** (3) The problem of finding global minimums of functions is addressed in the context of problems in which many local minima exist. Numerical techniques are emphasized, including gradient descent and quasi-Newton methods. Current literature is examined and a comparison made of various techniques for both unconstrained and constrained optimization problems. Credit not allowed for both MST 753 and CSC 753. *P—MST 655 or CSC 655.*

**MST 754. Numerical Methods for Partial Differential Equations**. (3) Numerical techniques for solving partial differential equations (including elliptic, parabolic and hyperbolic) are studied along with applications to science and engineering. Theoretical foundations are described and emphasis is placed on algorithm design and implementation using either C, FORTRAN or MATLAB. Credit not allowed for both MST 754 and CSC 754. *P—MST 655 or CSC 655*

**MST 791, 792. Thesis Research.** (1-9). May be repeated for credit. *Satisfactory/Unsatisfactory*

**Statistics Courses **

**STA 610. Probability. (3h) **Distributions of discrete and continuous random variables, sampling distributions. Covers much of the material on the syllabus for the first actuarial exam.

**STA 611. Statistical Inference. (3h) **Derivation of point estimators, hypothesis testing, and confidence intervals, using both frequentist and Bayesian approaches. P – STA 610 or MST 657 or POI.

**STA 612. Linear Models. (3h) **Theory of estimation and testing in linear models. Topics include least squares and the normal equations, the Gauss-Markov Theorem, testing general linear hypotheses, model selection, and applications. P — STA 610.

**STA 652. Networks: Models and Analysis. (3h) **A course in fundamental network theory concepts, including measures of network structure, community detection, clustering, and network modelling and inference. Topics also draw from recent advances in the analysis of networks and network data, as well as applications in economics, sociology, biology, computer science, and other areas.

**STA 662. Multivariate Statistics. (3h) **Multivariate and linear methods for classification, visualization, discrimination, and analysis of high dimensional data.

**STA 663. Introduction to Statistical Learning. (3h)** An introduction to supervised learning from a statistical perspective. Topics may include lasso and ridge regression, splines, generalized additive models, random forests, and support vector machines. Requires prior experience with R programming.

**STA 664. Computational and Nonparametric Statistics. (3h) **Computationally intensive statistical methods. Topics include simulation, Monte Carlo integration and Markov Chain Monte Carlo, sub-sampling, and non-parametric estimation and regression. Students will make extensive use of statistical software throughout the course. P – STA 610 or MST 657, or POI.

**STA 668. Time Series and Forecasting. (3h) **Methods and models for time series processes and autocorrelated data. Topics include model diagnostics, ARMA models, spectral methods, computational considerations, and forecasting error. P – STA 610 or MST 657, or POI.

**STA 679. Advanced Topics in Statistics. (1-3h) **Topics in statistics not considered in regular courses or which continue study begun in regular courses. Content varies.

**STA 682. Readings in Statistics. (1-3h) **Reading in statistical topics to provide a foundational basis for more advanced study in a particular mathematical area. May not be used to satisfy any requirement in the MA degree with thesis. No more than three hours may be applied to the requirements for the MA degree without thesis. May be repeated for credit for a total of 3 hours.

**STA 683. Individual Study. (1-3h) **A course of independent study directed by a faculty adviser. By prearrangement.

**STA 710. Stochastic Processes and Applications. (3)** This course includes the axiomatic foundations of probability theory and an introduction to stochastic processes. Applications may include Markov chains, Markov Chain Monte Carlo with Metropolis-Hastings, Gibbs sampling, Brownian motion, and related topics, with an emphasis on modern developments. P — STA 610 and MST 611, or POI.

**STA 711. Advanced Statistical Inference. (3)** Advanced mathematical treatment of point estimators, hypothesis testing, and confidence intervals, using both frequentist and Bayesian approaches. P — STA 610 or POI.

**STA 712. Generalized Linear Models. (3)** Extensions of the classical linear model to cover models for binary and count data, ordinal and nominal categorical data, and time-to-event data, along with numerical maximization techniques needed to fit such models. Additional topics may include longitudinal data, the Expectation-Maximization algorithm, non-linear models, or related topics. P—STA 612 or POI

**STA 7****20. Bayesian Analysis (3). **Fundamental concepts, theory, and computational methods for Bayesian inference. Topics may include decision theory, evaluating Bayesian estimators, Bayesian testing and credible intervals, Markov chain Monte Carlo methods, and hierarchical models. P – STA 610 or POI.

**STA 779. Topics in Statistics. (3)** Topics vary by instructor. May be repeated for credit.

**STA 791/792 Thesis Research.** (1-9). May be repeated for credit. *Satisfactory/Unsatisfactory*