Exploration of a topic (to be determined) not covered by the standard curriculum but of interest to faculty and students in a particular semester.

Matrix methods required for data analysis with an emphasis on applications and using software. Matrix norms, orthogonality, eigendecomposition, SVD, LS, QRD, LDA, PCA. Not for math majors or minors. Additional work required of graduate students.

Partial derivatives, implicit functions, integration in several variables, line and surface integrals.

Real-number system, set theory and elementary topological properties of the real line, continuity and differentiability, sequences and series, uniform convergence, Riemann integration, and improper integrals.

Complex number system and its arithmetic, geometric representation. Linear transformations. Analytic functions and the Cauchy-Riemann equations. Integration and Cauchy's theorem, Taylor and Laurent series, singularities, poles, and residues. Applications.

Partial differential equations, boundary-value problems, Fourier series and orthogonal expansions, Bessel functions, and Legendre polynomials.

Orthogonal functions, Fourier series, Fourier transforms-continuous and discrete, Haar wavelets and multiresolution analysis, applications to signal processing. Additional work required of graduate students.

Algebra of sets. Probability in finite sample spaces. Binomial and multinomial coefficients. Random variables. Expected value and standard deviation. Density functions. Statistical applications.

Statistical methods (such as hypothesis testing, parameter estimation, regression, ANOVA, sampling, experimental design) required for data science. Emphasis on applications and using software. Additional work required for graduate students.

Concept, theory, methods, and applications of simple linear regression, multiple linear regression, and logistic regression models; Parameter estimation and testing, Prediction, Regression diagnostics and model adequacy checking, Variable selection and model building.

Estimation and confidence intervals. Normal distribution and central limit theorem. Testing hypotheses, chi-square, t, and F distributions. Least squares, regression, and correlation.

Discrete time Markov chains, Poisson process, continuous time Markov chains and other selected stochastic processes.

One-way, two-way, and multi-way analysis of variance, analysis of covariance, random/mixed effect models, repeated measure analysis, and special designs such as a randomized block design, an incomplete block design, Latin square, and a nested design.

Applied probability focusing on distributions for actuarial applications. Conditional expectation. Moment generating functions. Limit theorems. Loss and Survival models. Parametric and Non-Parametric estimation. Model assessment. Benefit reserves and risk measures. Additional work required of graduate students.

Concepts of prior and posterior, Bayesian inference for binomial and Poisson distributions, Monte Carlo approximation, Bayesian inference for normal distribution, Gibbs sampling, hierarchical models, multivariate normal distribution, Metropolis-Hastings algorithms, linear regression models.

Abstract vector spaces and inner product spaces, linear transformations and linear operators, eigenvalues and diagonalization. Primarily for mathematics majors.

Factorization of matrices, eigenvalues and eigenvectors, orthogonality. Applications of matrices to such topics as least-squares approximation, fast Fourier transform, difference and differential equations, linear programming, networks, game theory.

Theory of groups, rings, and fields, including the integers and polynomial rings.

Prime numbers, greatest common divisors, congruences. Euler's function, Fermat's theorem, primitive roots, indices, quadratic residues, Legendre and Jacobi symbols, and the quadratic reciprocity law.

Permutations, combinations, recurrence relations, generating functions, inclusion-exclusion and applications, introductory graph theory.

Synthetic projective geometries. Coordinate systems for projective spaces. Algebraic representation of projective transformations; euclidean, non-euclidean, and affine geometries as real cases of projective geometry.

Theory of curves in three-dimensional space, including Frenet's formula, Gaussian and mean curvature, geodesics, developable surfaces, special conformal mappings.

Metrics and metric spaces, topologies and topological spaces, separation properties, compactness, connectedness, and continuity.

Offered through SUAbroad by educational institution outside the United States. Student registers for the course at the foreign institution and is graded according to that institution's practice. SUAbroad works with the S.U. academic department to assign the appropriate course level, title, and grade for the student's transcript.

Approximation methods for solution of nonlinear equations. Interpolation problems. Numerical integration. Solution of ordinary differential equations. Error analysis and writing computer programs. Primarily for mathematics and engineering students.

Exploration of a problem, or problems, in depth. Individual independent study upon a plan submitted by the student. Admission by consent of supervising instructor(s) and the department.

Mathematical concepts in their historical perspective. Character and contributions of the great mathematicians and relation of mathematics to other sciences.

In-depth investigation of one or more statistical topics, applications of statistical methods and tools, real-world data analysis project using software, comprehensive presentation of project findings.

Topic Chosen by the instructor. Permission of department.

Exploration of a topic (to be determined) not covered by the standard curriculum but of interest to faculty and students in a particular semester.

Real and complex numbers, elementary point set topology, sequences and series, continuity, differentiation.

Riemann-Stieltjes integration, functional sequences and series, functions of several variables.

Linear algebra, linear transformations, eigenvectors, diagonalization, inner product spaces, groups, quotient groups, group actions, Sylow theorems, finitely generated Abelian groups, rings, unique factorization domains, finitely generated modules over principal ideal domains, fields, Galois theory.

Continuation of MAT 631.

Fundamentals of graph theory and special topics including networks, matching, connectivity, planarity, and automorphism groups.

Generating functions, Polya enumeration, set systems, design parameters, finite projective planes, matroids.

Calculus of probabilities, univariate and multivariate random variables and distribution functions, expectations and variance, conditional distributions, transformations of random variables, characteristic functions, basic limit theorems including Borel-Cantelli, Khinchin, Lindeberg- Feller.

Point and interval estimation, consistent, efficient, and sufficient statistics, Rao-Blackwellization, hypothesis testing, brief treatment of ranking and selection, decision theory.

Simulation and Monte Carlo techniques appropriate where statistical theory does not yet provide a solution. Design and analysis of experiments under nonstandard conditions.

Point estimation by least squares, regression, curve fitting, testing a linear hypothesis, analysis of variance, simple experimental designs.

Topological spaces, continuous mappings, compactness, connectedness, path connectedness, separation axioms, metric spaces, quotient spaces, CW complexes, the fundamental group, and the classification of 2-dimensional manifolds.

Participation in a discipline or subject related experience. Student must be evaluated by written or oral reports or an examination. Permission in advance with the consent of the department chairperson, instructor, and dean. Limited to those in good academic standing.

Solution of linear equations. Norms and conditioning. Calculation of eigenvalues and eigenvectors. Least squares approximation and orthogonal functions. Error analysis and writing computer programs.

Numerical methods of interpolation, approximation, integration, and differentiation, solutions of nonlinear equations.

Analysis of numerical methods for approximating solutions of ordinary and partial differential equations.

Mathematical model building, dimensional analysis, scaling, and perturbation theory. Models selected from the natural and social sciences according to the interests of instructor and students. Examples are: planetary orbits, fluid flow, isomers in organic chemistry, biological competition, biochemical kinetics, and physiological flow.

In-depth exploration of a problem or problems.  Individual independent study upon a plan submitted by the student. Admission by consent of supervising instructor or instructors and the department.

Fundamental methods for data science, such as regression, linear discriminant analysis, k-nearest neighbors, support vector machine, k-means, principal component analysis, and nonlinear dimension reduction. Performance evaluation and model selection. Additional work required of graduate students.

Exploration of a topic (to be determined) not covered by the standard curriculum but of interest to faculty and students in a particular semester.

Measure and integration, including basic theorems on integration and differentiation of sequences of functions; modes of convergence, product measures.

Norms, seminorms, and inner products on linear spaces. Standard theorems on linear functionals and operations. Dual spaces and weak topologies, classical spaces and their duals. Applications.

Existence theorems for ordinary differential equations, linear differential equations and systems, Euler variational equations, typical Cauchy and boundary-value problems for partial differential equations.

Differentiable manifolds, differential forms, exterior calculus, integration over manifolds, Stokes' Theorem, other selected topics.

Cauchy theory, power series, analytic continuation, entire functions, the residue theorem, contour integration, maximum modulus theorem and applications, conformal representation. Dirichlet series, special functions.

Measure and integration. Random variables, their distributions and transforms. Modes of convergence. Classical limit laws. Markov chains.

Conditional expectation. Martingales. Brownian motion. Ergodic theorem. Random walks.

Submodules, factor modules, chain conditions, Hilbert basis theorem, division rings, Schur's lemma, Jacobson density theorem, semi-simple modules, socles, Jacobson radical, semi primitive rings, Artin-Wedderburn theorem, integral extensions, completions, localization.

Projective and injective resolutions, Tor and Ext, flatness, homology, derived categories, spectral sequences.

Localization, primary decomposition, and dimension theory; Nullstellensatz; Artin-Rees lemma and completion; integral and flat extensions; Koszul complex, Cohen-Macaulay and regular rings.

The course covers representations of finite groups and finite-dimensional algebras. Topics will come from: ordinary and modular representations of finite groups, Auslander-Reiten theory, representations of quivers, Koszul algebras, Hopf algebras and Frobenius algebras.

The study of the zeros of polynomials. Classical algebraic varieties in affine and projective space, followed by introduction to modern theory of sheaves, schemes, and cohomology.

Working with real data taken from case studies, published papers, and current projects in the statistical laboratory.

Statistical selection of the best category or population. Preference-zone and subset formulations. Multivariate preferences and populations. Applications. Recent developments, including Multiple Comparisons with the Best (MCB) and the Heteroscedastic Method (HM).

Minimax theorems, completeness of the class of Bayes procedures. Invariance. Criteria for admissibility.

General sequential decision problems, sequential probability ratio test, sequential test among three hypotheses, sequential estimation, optimal stopping, Wald's identity. Generalized SPRT's, Cox's theorem, sequential regression, functional equations, dynamic programming, sequential choice of experiments.

Multivariate normal distribution, conditional densities, partial correlation, multiple correlation, regression coefficients, maximum likelihood estimates, Hotelling's statistic, Wishart distribution, tests of hypotheses, and linear discriminant functions.

Overview of nonparametric statistics, empirical CDF, statistical functionals, density and probability function estimation, nonparametric kernel methods, nonparametric regression, conditional density estimation, bootstrap, partial linear models, single index models, and other advanced topics.

Topics in machine learning from statistical and probabilistic perspective: prediction errors and cross validation, bias-variance trade-off, regularized regression, variable selection, nonparametric regression and classification, bootstrap, trees-based methods, neural network, and other advanced topics.

Fundamental group covering spaces, chain complexes, simplicial or singular homology and cohomology theory, exact sequences, and the Eilenberg-Steenrod axioms.

Homology, cohomology ring, universal coefficient theorem, duality, homotopy, theory, selected topics.

Differential manifolds, tensor fields and mappings, differential forms and Stokes's theorem, affine connections, exponential mapping, covariant differentiation, torsion and curvature tensors, Riemannian connections, complete Riemannian manifolds, other modern topics.

Newton methods, interior point methods, proximal point algorithms, alternating direction method of multipliers, coordinate descent method, and stochastic/randomized algorithms.

Continuation of MAT 781.

Exploration of a topic (to be determined) not covered by the standard curriculum but of interest to faculty and students in a particular semester.

Topics in real variables and measure theory, such as differentiation theory in euclidean and abstract spaces, generalized derivatives and integrals, ergodic theory, martingales, surface area.

Abstract integration, Radon-Nikodym theorem. Representation of set functions by integrals. Ergodic theorems. Duality. Weak topologies, convex sets, and extreme points. Elements of spectral theory.

First-order linear equations. Hamilton, Jacobi, and Lie transformations. Classifications of second-order linear equations. Boundary- and initial-value problems. Sturm-Liouville problems and connections with integral equations. Nonlinear equations.

Classification and examples of regular integral equations. Fredholm's theorems, Hilbert-Schmidt theory. Applications to differential equations. Nonlinear integral equations. Connections with general functional analysis.

Continuation of MAT 712

Contents vary from semester to semester. May be repeated for credit with permission.

For advanced graduate students and staff members; credit determined by extent of participation in the seminar.

Exploration of a topic (to be determined) not covered by the standard curriculum but of interest to faculty and students in a particular semester.

Exploration of a problem, or problems, in depth. Individual independent study upon a plan submitted by the student. Admission by consent of supervising instructor(s) and the department.

Research work on a doctoral dissertation, under supervision of some member of the graduate staff. Credit depends on amount of time devoted to the work; course may be repeated up to a maximum of 30 credits.