This course covers the measurement of interest, certain annuities, yield rates, amortization and sinking funds, bonds and other securities and application of interest theory.

Prerequisite: Permission of instructor

Survival distributions and life tables, life insurance, life annuities, benefit premiums and reserves and multiple life functions are some topics covered in this course.

Prerequisite: MATH 5680

Continuation of Actuarial Mathematics I. Multiple decrement models, collective risk models, applications of risk theory and application of these models to insurance problems.

Prerequisite: MATH 5260

Theory of vector spaces and linear transformations, including such topics as matrices, determinants, inner products, eigenvalues and eigenvectors, and rational and Jordan canonical forms. MATH 5310 LINEAR ALGEBRA II [3 hours] Hermitian and normal operators, multilinear forms, spectral theorem and other topics.

Prerequisite: MATH 5300

Arithmetic of the integers, unique factorization and modular arithmetic; group theory including normal subgroups, factor groups, cyclic groups, permutations, homomorphisms, the isomorphism theorems, abelian groups and p-groups.

Prerequisite: MATH 3190

Ring theory including integral domains, field of quotients, homomorphisms, ideals, Euclidean domains, polynomial rings, vector spaces, roots of polynomials and field extensions.

Prerequisite: MATH 5330

Matrices, systems of equations, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, generalized inverses, rank, numerical methods and applications to various areas of science.

Prerequisite: MATH 1890

Discrete mathematical structures for applications in computer science such as graph theory, combinatorics, groups theory, asymptotics, recurrence relations and analysis of algorithms.

Prerequisite: MATH 3320 or 5330

Theory of automata and formal languages, computability by Turing machines and recursive functions, uncomputability, NP-Hard and NP-Complete problems.

Prerequisite: MATH 5380

Metric spaces, topological spaces, continuous maps, bases and sub-bases, closure and interior operators, products, subspaces, sums, quotients, separation axioms, compactness and local compactness.

Prerequisite: MATH 3190

Connectedness and local connectedness, convergence, metrization, function spaces. The fundamental groups and its properties, covering spaces, classical applications, e.g. Jordan Curve Theorem, Fundamental Theorem of Algebra, Brouwer ’s Fixed Point Theorem.

Prerequisite: MATH 5450 Corequisite: MATH 3320 or 5330

This course presents an introduction to the logical foundations of mathematics. The geometry of two, three and four dimensional space is examined starting from the axioms of Euclid. The course culminates with an investigation of hyperbolic and projective geometries. Of interest to high school teachers.

Prerequisite: MATH 5050 or 5330

Smooth curves in Euclidean space including the Frenet formulae. Immersed surfaces with the Gauss map, principal curvatures and the fundamental forms. Special surfaces including ruled surfaces and minimal surfaces. Intrinsic Geometry including the Gauss Theorem Egregium.

Prerequisite: MATH 3860

Tensors, vector fields and the Cartan approach to surface theory, Bonnet’s Theorem and the construction of surfaces via solutions of the Gauss Equation. Geodesics, parallel transport and Jacobi Fields. Theorems of a global nature such as Hilbert’s Theorem or the Theorem of Hopf-Rinow.

Prerequisite: MATH 5540

Geometrical and topological aspects of curves and surfaces in Euclidean space. The concepts are to be highlighted by the study of specific examples as in minimal surface theory and in “soap bubble” geometry.

Prerequisite: MATH 5030 and 5050

Real data applications of statistical methods. Emphasis is placed on exploratory data analysis and the use of computing facilities to analyze data and produce statistical reports. Statistical packages used include MINITAB, SAS and S-Plus.

Prerequisite: Permission of instructor

Continuation of Applications of Statistics II.

Prerequisite: MATH 5600

Multiple regression, analysis of variance and covariance, general linear models and model building for linear models. Experimental designs include one-way, randomized block, Latin square, factorial and nested designs.

Prerequisite: MATH 6650 or permission of instructor

The mathematical basis to estimation in various sampling contexts, including probability proportional to size sampling, stratified sampling, two-stage cluster sampling and double sampling, is developed.

Prerequisite: MATH 5680 or permission of instructor Corequisite: MATH 5690 or 6650

Error analysis of statistical algorithms. Numerical linear algebra for linear models. Approximation methods for distribution function probabilities and quantiles. Uniform and non-uniform random number generation. Introduction to simulation methods.

Prerequisite: Permission of instructor

The basic probability models of applied mathematics and physics, including random walks, Markov chains, branching processes, renewal processes, random graphs and queuing.

Prerequisite: MATH 5680 and 5300 or 5350

Confounding, fractional replication, complex designs, response surface designs.

Prerequisite: MATH 5620

Probability spaces, random variables, probability distributions, moments and moment generating functions, limit theorems, transformations and sampling distributions.

Prerequisite: MATH 3190 or permission of instructor and MATH 5350

Sampling distributions, point estimation, interval estimation, hypothesis testing, regression and analysis of variance.

Prerequisite: MATH 5680

Floating point arithmetic; polynomial interpolation; numerical solution of nonlinear equations; Newton’s method. Likely topics include: numerical differentiation and integration; solving systems of linear equations; Gaussian elimination; LU decomposition; Gauss-Seidel method.

Prerequisite: MATH 3860 and a computer programming course or permission of instructor

Likely topics include: Computation of eigenvalues and eigenvectors; solving systems of nonlinear equations; least squares approximations; rational approximations; cubic splines; fast Fourier transforms; numerical solutions to initial value problems; ordinary and partial differential equations.

Prerequisite: MATH 5710

Series and numerical solutions to ordinary differential equations, special functions, orthogonal functions, Sturm-Liouville Problems, self-adjointness, vector analysis.

Prerequisite: MATH 3860

Continuation of vector analysis, introduction to complex analysis, partial differential equations, Fourier series and integrals.

Prerequisite: MATH 5740

Introduction to contemporary ideas of dynamics and the chaotic behavior that occurs when a simple function of one variable is iterated.

Prerequisite: MATH 5030 or 5820

This course is a supplement to MATH 5760 and may be taken concurrently with or after completion of that course. Students will demonstrate the theory in a number of computer-based experiments.

Prerequisite: MATH 5760

Extrema for functions of one or more variables, Lagrange multipliers, indeterminate forms, inverse and implicit function theorems, uniform convergences, power series, transformations, Jacobians, multiple integrals.

Prerequisite: MATH 2850

An introduction to finite-dimensional combined optimization as it relates to business and economics. Linear and non-linear programming.

Prerequisite: MATH 3860 and 1890

Modern theory of differential equations; transforms and matrix methods; existence theorems and series solutions; and other selected topics.

Prerequisite: MATH 3860

First and second order equations; numerical methods; separation of variables; solutions of heat and wave equations using eigenfunction techniques; and other selected topics.

Prerequisite: MATH 3860 and permission of instructor

A rigorous treatment of the Calculus in one and several variables. Topics to include: the real number system; sequences and series; elementary metric space theory including compactness, connectedness and completeness; the Riemann Integral.

Prerequisite: MATH 3190

Differentiable functions on Rn; the Implicit and Inverse Function Theorems; sequences and series of continuous functions; Stone-Weierstrass Theorem; Arsela-Ascoli Theorem; introduction to measure theory; Lebesgue integration; the Lebesgue Dominated Convergence Theorem.

Prerequisite: MATH 5820

Theory of Laplace, Fourier and other transforms; use of complex variable theory for inversions; applications.

Prerequisite: MATH 5880

Conditions for an extreme (Euler ’s equations, Erdman corner conditions, conditions of Legendre, Jacobi and Weierstrass, fields of extremals, Hilbert’s invariant integral); ); Raleigh-Ritz method; isoperimetric problems; Lagrange, Mayer-Bolza problems.

Prerequisite: MATH 1890 Recommended: MATH 5820.

Pontryagin’s maximum principle; necessary and sufficient conditions for optimal control, controllability, time optimal control, existence of optimal controls, relationship to the calculus of variations.

Prerequisite: MATH 5860

Analytic functions; Cauchy’s theorem; Taylor and Laurent series; residues; contour integrals; conformal mappings, analytic continuation and applications.

Prerequisite: MATH 3860

The primary activity will be student solution and presentation of problems of a type given on actuarial exams to be run as a problem seminar.

Students must submit for approval by their adviser a report on the solution of a practical problem involving mathematics. The problem must be drawn from a company, university department of government unit.

Prerequisite: Admission into program

Special topics in mathematics.

Students will complete a project devised jointly by student and adviser. The project will have significant mathematical content and might originate from courses taken in the program or some aspect of lesson preparation, in the case of practicing high school teachers. Required in the M.A. technology track

Prerequisite: MATH 5020, 5030 and 5050

Normed linear spaces, Banach and Hilbert spaces, linear operators and their spectrum, spectral analysis, illustrative examples from science and engineering.

Prerequisite: MATH 5300

Simplex algorithm, ellipsoidal algorithm, Karmarkar’s method, interior point methods, elementary convex analysis, optimality conditions and duality for smooth problems, convex programming, algorithms and their convergence.

Prerequisite: MATH 5820

Introduction to nonlinear analysis, abstract optimization problems on abstract spaces, applications to calculus of variations, optimal control theory and game theory.

Prerequisite: MATH 6150 or 6810 or equivalent

Group actions, Sylow’s theorems, permutation groups, nilpotent and solvable groups, abelian groups, rings, unique factorization domains, fields.

Prerequisite: MATH 5340 or equivalent

Field extensions, Galois theory, modules, Noetherian and Artinian rings, tensor products, primitive rings, semisimple rings and modules, the Wedderburn-Artin theorem.

Prerequisite: MATH 6300

Radical theory, rings of quotients, Goldie’s Theorem, chain conditions, dimensions of rings, module theory, topics in commutative rings.

Prerequisite: MATH 6310

Advanced topics in ring theory. Possible topics include group rings, enveloping algebras, almost split sequences, PI-rings, division rings, selfinjective rings, and ordered rings.

Prerequisite: MATH 6310

Fundamental topics in group theory. Possible topics include free groups, presentations, free products and amalgams, permutation groups, abelian groups, nilpotent and solvable groups, subnormality, extensions, the Schur-Zassenhaus theorem, the transfer homomorphism, linear methods, local analysis.

Prerequisite: MATH 6310

Advanced topics in group theory. Possible topics include eohomology of groups, locally, finite groups, character theory, modular, representation theory, representation theory of symmetic and classical groups, finite simple groups, geometric group theory.

Prerequisite: MATH 6310

Topological spaces, continuous functions, compactness, product spaces, Tychonov’s theorem, quotient spaces, local compactness, homotopy theory, the fundamental group, covering spaces.

Prerequisite: MATH 5450 or equivalent

Homology theory, excision, homological algebra, the Brouwer fixed point theorem, cohomology, differential manifolds, orientation, tangent bundles, Sard’s theorem, degree theory.

Prerequisite: MATH 6400

Categorical properties of and constructions in topological spaces, compactness, connectedness, dimension theory, metrization.

Prerequisite: MATH 6400

Compactification and proximity spaces, uniform spaces, completeness and completions, rings of continuous functions.

Prerequisite: MATH 6420

Introduction to differential geometry. Topics include differentiable manifolds, vector fields, tensor bundles, the Frobenius theorem, Stokes’ theorem, Lie groups.

Prerequisite: MATH 6410

Topics include connections on manifolds, Riemannian geometry, the Gauss-Bonnet theorem. Further topics may include: homogeneous and symmetric spaces, minimal surfaces, Morse theory, comparison theory, vector and principal bundles.

Prerequisite: MATH 6440

Simplicial and cellular complexes, simplicial and cellular homology, universal coefficient theorem, Kunneth theorem, cohomology theories, cohomology operations, duality on manifolds.

Prerequisite: MATH 6410

General homotopy theory, fibrations and cofibrations, higher homotopy groups, weak homotopy equivalence, Hurewicz’s theorem, Eilenberg-MacLane spaces, classifying spaces, spectral sequences.

Prerequisite: MATH 6460

Existence, uniqueness and dependence on initial conditions and parameter, nonlinear planar systems, linear systems, Floquet theory, second order equations, Sturm-Liouville theory. MATH 5830 or equivalent

Prerequisite: presence of contaminated data or error distribution misspecification. Prerequisite: MATH 5680 Corequisite: MATH 5690 or 6650

Topics selected from an array of modern statistical methods such as survival analysis, nonlinear regression, Monte Carlo methods, etc. MATH 6650 STATISTICAL INFERENCE [3 hours] Estimation, hypothesis testing, prediction, sufficient statistics, theory of estimation and hypothesis testing, simultaneous inference, decision theoretic models.

Prerequisite: MATH 5680

Real analysis, probability spaces and measures, random variables and distribution functions, independence, expectation, law of large numbers, central limit theorem, zero-one laws, characteristic functions, conditional expectations given a s-algebra, martingales.

Prerequisite: MATH 5680 Corequisite: MATH 6800 recommended

Exponential families, sufficiency, completeness, optimality, equivariance, efficiency. Bayesian and minimax estimation. Unbiased and invariant tests, uniformly most powerful tests. Asymptotic properties for estimation and testing. Most accurate confidence intervals.

Prerequisite: MATH 5960 or 6650 and 6670

Multivariate normal sampling distributions, T tests and MANOVA, tests on covariance matrices, simultaneous inference, discriminant analysis, principal components, cluster analysis and factor analysis.

Prerequisite: MATH 5690 or 6650

Analytic functions, residues, method of steepest descent, complex differential equations, regular singularities, integral representation, real and complex vector spaces, matrix groups, Hilbert spaces, coordinate transformations.

Self-adjoint operators, special functions, orthogonal polynomials, partial differential equations and separation of variables, boundary value problems, Green’s functions, integral equations, tensor analysis, metrics and curvature, calculus of variations, finite groups and group representations.

Prerequisite: MATH 6720.

Completeness, connectedness and compactness in metric spaces, continuity and convergence, the Stone-Weierstrass Theorem, Lebesgue measure and integration on the real line, convergence theorems, Egorov’s and Lusin’s theorems, derivatives, functions of bounded variation.

Prerequisite: MATH 4830 and 5830

The Vitali covering theorem, absolutely continuous functions, Lebesgue-Stieltjes integration, the Riesz representation theorem , Banach spaces, Lp-spaces, abstract measures, the Radon-Nikodym theorem, measures on locally compact Hausdorff spaces.

Prerequisite: MATH 6800.

Topics include Topological vector spaces, Banach spaces, convexity, the Hahn-Banch theorem, weak and strong topologies, Lp spaces and duality.

Prerequisite: MATH 6810

Topics include the Mackey-Ahrens Theorem, Banach algebras, spectra in Banach algebras, commutative Banach algebras, unbounded operators, the spectral theorem, topics in functional analysis.

Prerequisite: MATH 6820

Elementary analytic functions, complex integration, the residue theorem, infinite sequences of analytic functions, Laurent expansions, entire functions.

Prerequisite: MATH 6800

Meromorphic functions, conformal mapping, harmonic functions and the dirichlet problem, the Riemann mapping theorem, monodromy, algebraic functions, Riemann surfaces, elliptic functions and the modular function.

Prerequisite: MATH 6840

Practicum in solving problems in graduate algebra, topology and analysis. Supplements 6300-10, 6400-10 and 6800-10 and prepares students for doctoral qualifying examination.

Lectures by visiting mathematicians and staff members on areas of current interest in mathematics.

Problems and techniques of teaching elementary college mathematics, supervised teaching, seminar in preparation methods.

Special topics in mathematics or statistics.

Readings in areas of mathematics of mutual interest to the student and the professor.

Theory of vector spaces and linear transformations, including such topics as matrices, determinants, inner products, eigenvalues and eigenvectors, and rational and Jordan canonical forms.

First order quasi-linear systems of partial differential equations, boundary value problems for the heat and wave equation, Dirichlet problem for Laplace equation, fundamental solutions for Laplace, heat and wave equations.

Prerequisite: MATH 5830 or equivalent

Topic include the flow-box theorem, Poincare maps, attractors, w limit sets, Lyapunov stability, invariant submanifolds, Hamiltonian systems and symplectic manifolds.

Prerequisite: MATH 6500

Topics may include local bifurcations of vector fields, global stability, ergodic theorems, integrable systems, symbolic dynamics, chaos theory.

Prerequisite: MATH 6520

Possible topics may include: the CauchyKovalevskaya Theorem, nonlinear partial differential equations of the first order, theory of Sobolev spaces, linear second order PDE’s of elliptic, hyperbolic and parabolic type.

Prerequisite: MATH 6510

Selected topics in Partial Differential Equations of current interest emphasizing nonlinear theory. Possible topics may include: Minimal surfaces, applications of the Hopf maximum principle, free boundary value problems, harmonic maps, geometric evoluation equations and the Navier-Stokes equation.

Prerequisite: MATH 6540.

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Prerequisite: Permission of instructor

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Prerequisite: Permission of instructor

Important methods and modeling techniques using generalized linear models and emphasizing loglinear and logit modeling.

Prerequisite: MATH 5680 Corequisite: MATH 6650

Statistical methods based on counts and ranks; methods designed to be effective in the presence of contaminated data or error distribution misspecification.

Prerequisite: MATH 5680 Corequisite: MATH 5690 or 6650

Hermitian and normal operators, multilinear forms, spectral theorem and other topics.

Prerequisite: MATH 5300

Arithmetic of the integers, unique factorization and modular arithmetic; group theory including normal subgroups, factor groups, cyclic groups, permutations, homomorphisms, the isomorphism theorems, abelian groups and p-groups.

Prerequisite: MATH 3190

Ring theory including integral domains, field of quotients, homomorphisms, ideals, Euclidean domains, polynomial rings, vector spaces, roots of polynomials and field extensions.

Prerequisite: MATH 5330

Matrices, systems of equations, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, generalized inverses, rank, numerical methods and applications to various areas of science.

Prerequisite: MATH 1890

Discrete mathematical structures for applications in computer science such as graph theory, combinatorics, groups theory, asymptotics, recurrence relations and analysis of algorithms.

Prerequisite: MATH 3320 or 5330

Theory of automata and formal languages, computability by Turing machines and recursive functions, uncomputability, NP-Hard and NP-Complete problems.

Prerequisite: MATH 5380

Metric spaces, topological spaces, continuous maps, bases and sub-bases, closure and interior operators, products, subspaces, sums, quotients, separation axioms, compactness and local compactness.

Prerequisite: MATH 3190

Connectedness and local connectedness, convergence, metrization, function spaces. The fundamental groups and its properties, covering spaces, classical applications, e.g. Jordan Curve Theorem, Fundamental Theorem of Algebra, Brouwer ’s Fixed Point Theorem.

Prerequisite: MATH 5450 Corequisite: MATH 3320 or 5330

Smooth curves in Euclidean space including the Frenet formulae. Immersed surfaces with the Gauss map, principal curvatures and the fundamental forms. Special surfaces including ruled surfaces and minimal surfaces. Intrinsic Geometry including the Gauss Theorem Egregium.

Prerequisite: MATH 3860

Tensors, vector fields and the Cartan approach to surface theory, Bonnet’s Theorem and the construction of surfaces via solutions of the Gauss Equation. Geodesics, parallel transport and Jacobi Fields. Theorems of a global nature such as Hilbert’s Theorem or the Theorem of Hopf-Rinow.

Prerequisite: MATH 5540

Real data applications of statistical methods. Emphasis is placed on exploratory data analysis and the use of computing facilities to analyze data and produce statistical reports. Statistical packages used include MINITAB, SAS and S-Plus.

Prerequisite: Permission of instructor

Continuation of Applications of Statistics II.

Prerequisite: MATH 5600

Multiple regression, analysis of variance and covariance, general linear models and model building for linear models. Experimental designs include one-way, randomized block, Latin square, factorial and nested designs.

Prerequisite: MATH 6650 or permission of instructor

The mathematical basis to estimation in various sampling contexts, including probability proportional to size sampling, stratified sampling, two-stage cluster sampling and double sampling, is developed.

Prerequisite: MATH 5680 or permission of instructor Corequisite: MATH 5690 or 6650

Error analysis of statistical algorithms. Numerical linear algebra for linear models. Approximation methods for distribution function probabilities and quantiles. Uniform and non-uniform random number generation. Introduction to simulation methods.

Prerequisite: Permission of instructor

The basic probability models of applied mathematics and physics, including random walks, Markov chains, branching processes, renewal processes, random graphs and queuing.

Prerequisite: MATH 5680 and 5300 or 5350

Confounding, fractional replication, complex designs, response surface designs.

Prerequisite: MATH 5620

Probability spaces, random variables, probability distributions, moments and moment generating functions, limit theorems, transformations and sampling distributions.

Prerequisite: MATH 3190 or permission of instructor and MATH 5350

Sampling distributions, point estimation, interval estimation, hypothesis testing, regression and analysis of variance.

Prerequisite: MATH 5680

Floating point arithmetic; polynomial interpolation; numerical solution of nonlinear equations; Newton’s method. Likely topics include: numerical differentiation and integration; solving systems of linear equations; Gaussian elimination; LU decomposition; Gauss-Seidel method.

Prerequisite: MATH 3860 and a computer programming course or permission of instructor

Likely topics include: Computation of eigenvalues and eigenvectors; solving systems of nonlinear equations; least squares approximations; rational approximations; cubic splines; fast Fourier transforms; numerical solutions to initial value problems; ordinary and partial differential equations.

Prerequisite: MATH 5710

Series and numerical solutions to ordinary differential equations, special functions, orthogonal functions, Sturm-Liouville Problems, self-adjointness, vector analysis.

Prerequisite: MATH 3860

Continuation of vector analysis, introduction to complex analysis, partial differential equations, Fourier series and integrals.

Prerequisite: MATH 5740

Extrema for functions of one or more variables, Lagrange multipliers, indeterminate forms, inverse and implicit function theorems, uniform convergences, power series, transformations, Jacobians, multiple integrals.

Prerequisite: MATH 2850

An introduction to finite-dimensional combined optimization as it relates to business and economics. Linear and non-linear programming.

Prerequisite: MATH 3860 and 1890

Modern theory of differential equations; transforms and matrix methods; existence theorems and series solutions; and other selected topics.

Prerequisite: MATH 3860

First and second order equations; numerical methods; separation of variables; solutions of heat and wave equations using eigenfunction techniques; and other selected topics.

Prerequisite: MATH 3860 and permission of instructor

A rigorous treatment of the Calculus in one and several variables. Topics to include: the real number system; sequences and series; elementary metric space theory including compactness, connectedness and completeness; the Riemann Integral.

Prerequisite: MATH 3190

Differentiable functions on Rn; the Implicit and Inverse Function Theorems; sequences and series of continuous functions; Stone-Weierstrass Theorem; Arsela-Ascoli Theorem; introduction to measure theory; Lebesgue integration; the Lebesgue Dominated Convergence Theorem.

Prerequisite: MATH 5820

Theory of Laplace, Fourier and other transforms; use of complex variable theory for inversions; applications.

Prerequisite: MATH 5880

Conditions for an extreme (Euler’s equations, Erdman corner conditions, conditions of Legendre, Jacobi and Weierstrass, fields of extremals, Hilbert’s invariant integral); Raleigh-Ritz method; isoperimetric problems; Lagrange, Mayer-Bolza problems.

Prerequisite: MATH 1890 Recommended: MATH 5820

Pontryagin’s maximum principle; necessary and sufficient conditions for optimal control, controllability, time optimal control, existence of optimal controls, relationship to the calculus of variations.

Prerequisite: MATH 5860

Analytic functions; Cauchy’s theorem; Taylor and Laurent series; residues; contour integrals; conformal mappings, analytic continuation and applications.

Prerequisite: MATH 3860

Special topics in mathematics.

Normed linear spaces, Banach and Hilbert spaces, linear operators and their spectrum, spectral analysis, illustrative examples from science and engineering.

Prerequisite: MATH 5300

Simplex algorithm, ellipsoidal algorithm, Karmarkar’s method, interior point methods, elementary convex analysis, optimality conditions and duality for smooth problems, convex programming, algorithms and their convergence.

Prerequisite: MATH 5820

Introduction to nonlinear analysis, abstract optimization problems on abstract spaces, applications to calculus of variations, optimal control theory and game theory.

Prerequisite: MATH 6150 or 6810 or equivalent

Group actions, Sylow’s theorems, permutation groups, nelpotent and solvable groups, abelian groups, rings, unique factorization domains, fields.

Prerequisite: MATH 5340 or equivalent

Field extensions, Galois theory, modules, Noetherian and Artinian rings, tensor products, primitive rings, semisimple rings, and modules, the Wedderburn-Artin theorem.

Prerequisite: MATH 6300

Radical theory, rings of quotients, Goldie’s Theorem, chain conditions, dimensions of rings, module theory, topics in commutative rings.

Prerequisite: MATH 6310

Advanced topics in ring theory. Possible topics include group rings, enveloping algebras, almost split sequences, PI-rings, division rings, selfinjective rings, and ordered rings.

Prerequisite: MATH 6310

Fundamental topics in group theory. Possible topics include free groups, presentations, free products and amalgams, permutation groups, abelian groups, nilpotent and solvable groups, subnormality, extensions, the Schur-Zassenhaus theorem, the transfer homomorphism, linear methods, local analysis.

Prerequisite: MATH 6310

Advanced topics in group theory. Possible topics include cohomolgy of groups, locally finite groups, character theory, modular representation theory, representation theory of symmetric and classical groups, finite simple groups, geometric group theory.

Prerequisite: MATH 6310

Topological spaces, continuous functions, compactness, product spaces, Tychonov’s theorem, quotient spaces, local compactness, homotopy theory, the fundamental group, covering spaces.

Prerequisite: MATH 5450 or equivalent

Homology theory, excision, homological algebra, the Brouwer fixed point theorem, cohomology, differential manifolds, orientation, tangent bundles, Sard theorem, degree theory.

Prerequisite: MATH 6400.

Categorical properties of and constructions in topological spaces, compactness, connectedness, dimension theory, metrization.

Prerequisite: MATH 6400

Compactification and proximity spaces, uniform spaces, completeness and completions, rings of continuous functions.

Prerequisite: MATH 6420

Introduction to differential geometry. Topics include differentiable manifolds, vector fields, tensor bundles, the Frobenius theorem, Stokes’ theorem, Lie groups.

Prerequisite: MATH 6410

Topics include connections on manifolds, Riemannian geometry, the Gauss-Bonnet theorem. Further topics may include homogeneous and symmetric spaces, minimal surfaces. Morse theory, comparison theory, vector and principal bundles.

Prerequisite: MATH 6410

Simplicial and cellular complexes, simplicial and cellular homology, universal coefficient theorem, Kunneth theorem, cohomology theories, cohomology operations, duality on manifolds.

Prerequisite: MATH 6410

General homotopy theory, fibrations and cofibrations, higher homotopy groups, weak homotopy equivalence, Hurewicz’s theorem, Eilenberg-MacLane spaces, classifying spaces, spectral sequences.

Prerequisite: MATH 6460

Existence, uniqueness and dependence on initial conditions and parameter, nonlinear planar systems, linear systems, Floquet theory, second order equations, Sturm-Liouville theory.

Prerequisite: MATH 5830 or equivalent

First order quasi-linear systems of partial differential equations, boundary value problems for the heat and wave equation, Dirichlet problem for Laplace equation, fundamental solutions for Laplace, heat and wave equations.

Prerequisite: MATH 5830 or equivalent

Topic include the flow-box theorem, Poincare maps, attractors, w-limit sets, Lyapunov stability, invariant submanifolds, Hamiltonian systems and symplectic manifolds.

Prerequisite: MATH 6500

Possible topics may include: the CauchyKovalevskaya Theorem, nonlinear partial differential equations of the first order, theory of Sobolev spaces, linear second order PDE’s of elliptic, hyperbolic and parabolic type.

Prerequisite: MATH 6510

Selected topics in Partial Differential Equations of current interest emphasizing nonlinear theory. Possible topics may include: Minimal surfaces, applications of the Hopf maximum principle, free boundary value problems, harmonic maps, geometric evolution equations and the Navier-Stokes equation.

Prerequisite: MATH 6540

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Prerequisite: Permission of instructor

Real data applications of various statistical methods, project design and analysis including statistical consulting experience.

Prerequisite: Permission of instructor

Important methods and modeling techniques using generalized linear models and emphasizing loglinear and logit modeling.

Prerequisite: MATH 5680 Corequisite: MATH 6650

Statistical methods based on counts and ranks; methods designed to be effective in the presence of contaminated data or error distribution misspecification.

Prerequisite: MATH 5680 Corequisite: MATH 5690 or 6650

Topics selected from an array of modern statistical methods such as survival analysis, nonlinear regression, Monte Carlo methods, etc.

Estimation, hypothesis testing, prediction, sufficient statistics, theory of estimation and hypothesis testing, simultaneous inference, decision theoretic models.

Prerequisite: MATH 5680

Real analysis, probability spaces and measures, random variables and distribution functions, independence, expectation, law of large numbers, central limit theorem, zero-one laws, characteristic functions, conditional expectations given a s-algebra, martingales.

Prerequisite: MATH 5680 Corequisite: MATH 6800 recommended

Exponential families, sufficiency, completeness, optimality, equivariance, efficiency. Bayesian and minimax estimation. Unbiased and invariant tests, uniformly most powerful tests. Asymptotic properties for estimation and testing. Most accurate confidence intervals.

Prerequisite: MATH 5960 or 6650 and 6670

Multivariate normal sampling distributions, T tests and MANOVA, tests on covariance matrices, simultaneous inference, discriminant analysis, principal components, cluster analysis and factor analysis.

Prerequisite: MATH 5690 or 6650

Analytic functions, residues, method of steepest descent, complex differential equations, regular singularities, integral representation, real and complex vector spaces, matrix groups, Hilbert spaces, coordinate transformations.

Self-adjoint operators, special functions, orthogonal polynomials, partial differential equations and separation of variables, boundary value problems, Green’s functions, integral equations, tensor analysis, metrics and curvature, calculus of variations, finite groups and group representations.

Prerequisite: MATH 6720

Completeness, connectedness and compactness in metric spaces, continuity and convergence, Stone-Weierstrass Theorem, Lebesgue measure and integration on the real line, convergence theorems, Egorov’s and Lusin’s theorems, derivatives, functions of bounded variation.

Prerequisite: MATH 4830, 5830 and 7830

The Vitali covering theorem, absolutely continuous functions, Lebesgue-Stieltjes integration, the Reisz representation theorem, Banach spaces, Lp-spaces, abstract measures, the Radon-Nikodym theorem, measures on locally compact Hausdorff spaces.

Prerequisite: MATH 6800

Topics include Topological vector spaces, Banach spaces, convexity, the Hahn-Banach theorem, weak and strong topologies, Lp spaces and duality.

Prerequisite: MATH 6810

Topics include the Mackey-Ahrens Theorem, Banach algebras, spectra in Banach algebras, commutative Banach algebras, unbounded operators, the spectral theorem, topics in functional analysis.

Prerequisite: MATH 6820

Elementary analytic functions, complex integration, the residue theorem, infinite sequences of analytic functions, Laurent expansions, entire functions.

Prerequisite: MATH 6800

Meromorphic functions, conformal mapping, harmonic functions and the Dirichlet problem, the Riemann mapping theorem, monodromy, algebraic functions, Riemann surfaces, elliptic functions and the modular function.

Prerequisite: MATH 6840

Practicum in solving problems in graduate algebra, topology and analysis. Supplements 6300-10, 6400-10 and 6800-10 and prepares students for doctoral qualifying examination.

Lectures by visiting mathematicians and staff members on areas of current interest in mathematics.

Problems and techniques of teaching elementary college mathematics, supervised teaching, seminar in preparation methods.

Special topics in mathematics or statistics.

Readings in areas of mathematics of mutual interest to the student and the professor.