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Mathematical Pedagogy

MATH 5000 - Mathematical Pedagogy

The theory and practice of teaching mathematics at the college level. Basic skills, grading methods, cooperative learning, active learning, use of technology, classroom problems, history of learning theory, reflective practice. Open to graduate students in Mathematics, others with consent of instructor. May not be used to satisfy degree requirements in mathematics.

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MATH 5010 - Topics in Analysis I

Advanced topics in analysis. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5011 - Topics in Analysis II

Advanced topics in analysis. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5016 - Topics in Probability

Advanced topics in probability theory, theory of random processes, mathematical statistics, and related fields. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5020 - Topics in Algebra

Advanced topics chosen from group theory, ring theory, number theory, Lie theory, combinatorics, commutative algebra, algebraic geometry, homological algebra, and representation theory.

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MATH 5026 - Topics in Mathematical Logic

Topics include, but are not restricted to, Computability Theory, Model Theory, and Set Theory.

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MATH 5030 - Topics in Geometry and Topology I

Advanced topics in Geometry and Topology. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5031 - Topics in Geometry and Topology II

Advanced topics in Geometry and Topology. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5040 - Topics in Applied Analysis I

Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.

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MATH 5041 - Topics in Applied Analysis II

Advanced topics from the theory of ordinary or partial differential equations. Other possible topics: integral equations, optimization theory, the calculus of variations, advanced approximation theory.

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MATH 5046 - Introduction to Complex Variables

Functions of a complex variable, integration in the complex plane, conformal mapping. Not open to students who have passed MATH 3146. Open for master’s credit but not doctoral credit toward degree in Mathematics.

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MATH 5050 - Analysis

Introduction to the theory of functions of a real variable. Not open to students who have passed MATH 3150. Open for masters credit but not doctoral credit toward degree in Mathematics.

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MATH 5070 - Topics in Scientific Computation

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MATH 5110 - Introduction to Modern Analysis

Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stielties integral, functions of several variables.

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MATH 5111 - Measure and Integration

General theory of measure and Lebesgue integration, L^p-spaces.

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MATH 5120 - Complex Function Theory I

An introduction to the theory of analytic functions, with emphasis on modern points of view.

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MATH 5121 - Topics in Complex Function Theory

Advanced topics of contemporary interest. These include Riemann surfaces, Kleinian groups, entire functions, conformal mapping, several complex variables, and automorphic functions, among others. May be repeated for credit with a change in content and consent of the instructor.

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MATH 5130 - Functional Analysis I

Normed linear spaces and algebras, the theory of linear operators, spectral analysis.

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MATH 5131 - Functional Analysis II

Normed linear spaces and algebras, the theory of linear operators, spectral analysis. With a change of content, this course is repeatable to a maximum ofsix credits.

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MATH 5140 - Fourier Analysis

Foundations of harmonic analysis developed through the study of Fourier series and Fourier transforms.

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MATH 5141 - Fourier Analysis on Groups

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MATH 5160 - Probability Theory and Stochastic Processes I

Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion.

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MATH 5161 - Probability Theory and Stochastic Processes II

Contemporary theory of stochastic processes, including stopping times, stochastic integration, stochastic differential equations and Markov processes, Gaussian processes, and empirical and related processes with applications in asymptotic statistics.

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MATH 5210 - Abstract Algebra I

Group theory, ring theory and modules, and universal mapping properties.

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MATH 5211 - Abstract Algebra II

Linear and multilinear algebra, Galois theory, category theory, and commutative algebra.

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MATH 5220 - Introduction to Representation Theory

Semi-simple rings, Jacobson radical, density theory, Wedderburn’s Theorem, representations and characters of groups, orthogonality relations, Burnside’s theorem.

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MATH 5230 - Algebraic Number Theory

Algebraic integers, ideal class group, ramification, Frobenius elements in Galois groups, Dirichlet’s unit theorem, localization, and completion. Further topics (zeta-functions, function fields, non-maximal orders) as time permits.

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MATH 5250 - Modern Matrix Theory and Linear Algebra

The LU, QR, symmetric, polar, and singular value matrix decompositions. Schur and Jordan normal forms. Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius theory and graph criteria in the theory of non-negative matrices.

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MATH 5260 - Mathematical Logic I

Predicate calculus, completeness, compactness, Lowenheim-Skolem theorems, formal theories with applications to algebra, Godel’s incompleteness theorem. Further topics chosen from: axiomatic set theory, model theory, recursion theory, computational complexity, automata theory and formal languages.

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MATH 5310 - Introduction to Geometry and Topology I

Topological spaces, maps, induced topologies, separation axioms, compactness, connectedness, classification of surfaces, the fundamental group and its applications, covering spaces.

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MATH 5311 - Introduction to Geometry and Topology II

Smooth manifolds, vector fields, differential forms, de Rham cohomology, homology theory, singular (co)homology, Poincare duality. With a change of content, this course is repeatable to a maximum of twelve credits.

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MATH 5320 - Algebraic Geometry I

This course is an introduction to algebraic varieties: affine and projective varieties, dimension of varieties and subvarieties, algebraic curves, singular points, divisors and line bundles, differentials, intersections.

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MATH 5321 - Algebraic Geometry II

This course introduces further concepts and methods of modern algebraic geometry, including schemes and cohomology.

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MATH 5360 - Differential Geometry

This course is an introduction to the study of differentiable manifolds on which various differential and integral calculi are developed. The topics include covariant derivatives and connections, geodesics and exponential map, Riemannian metrics, curvature tensor, Ricci and scalar curvature.

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MATH 5410 - Introduction to Applied Mathematics I

Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.

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MATH 5411 - Introduction to Applied Mathematics II

Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.

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MATH 5420 - Ordinary Differential Equations

Existence and uniqueness of solutions, stability and asymptotic behavior. If time permits: eigenvalue problems, dynamical systems, existence and stability of periodic solutions.

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MATH 5430 - Applied Analysis

Convergence of Fourier Series, Legendre and Hermite polynomials, existence and uniqueness theorems, two-point boundary value problems and Green’s functions. Not open for graduate credit toward degrees in Mathematics.

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MATH 5435 - Introduction to Partial Differential Equations

Solution of first and second order partial differential equations with applications to engineering and science. Not open to students who have passed MATH 3435. Not open for graduate credit toward degrees in Mathematics.

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MATH 5440 - Partial Differential Equations

Cauchy Kowalewsky Theorem, classification of second-order equations, systems of hyperbolic equations, the wave equation, the potential equation, the heat equation in Rn.

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MATH 5510 - Numerical Analysis and Approximation Theory I

The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

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MATH 5511 - Numerical Analysis and Approximation Theory II

The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.

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MATH 5520 - Finite Element Solution Methods I

Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

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MATH 5521 - Finite Element Solution Methods II

Numerical solution of elliptic, parabolic and hyperbolic partial differential equations by finite element solution methods. Applications.

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MATH 5530 - Mathematical Modeling

Development of mathematical models emphasizing linear algebra, differential equations, graph theory and probability. In-depth study of the model to derive information about phenomena in applied work.

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MATH 5540 - Computerized Modeling in Science

Development and computer-assisted analysis of mathematical models in chemistry, physics, and engineering. Topics include chemical equilibrium, reaction rates, particle scattering, vibrating systems, least squares analysis, quantum chemistry and physics.

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MATH 5580 - Optimization

Theory of linear programming: convexity, bases, simplex method, dual and integer programming, assignment, transportation, and flow problems. Theory of nonlinear programming: unconstrained local optimization, Lagrange multipliers, Kuhn-Tucker conditions, computational algorithms. Concrete applications.

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MATH 5620 - Financial Mathematics I

The mathematics of measurement of interest, accumulation and discount, present value, annuities, loans, bonds, and other securities.

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MATH 5621 - Financial Mathematics II

The continuation of MATH 5620. Theory and practice of mathematical models applied to corporate finance. Satisfies the Society of Actuaries’ learning objectives for Validation by Educational Experience for Corporate Finance.

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MATH 5630 - Actuarial Mathematics I

Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models.

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MATH 5631 - Actuarial Mathematics II

Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models.

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MATH 5633 - Survival Models

Analysis, estimation, and validation of lifetime tables.

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MATH 5635 - Introduction to Operations Research

Introduction to the use of mathematical and statistical techniques to solve a wide variety of organizational problems. Topics include linear programming, project scheduling, queuing theory, decision analysis, dynamic and integer programming and computer simulation.

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MATH 5637 - Risk Theory

Individual risk theory, distribution theory, ruin theory, stoploss, reinsurance and Monte Carlo methods. Emphasis is on problems in insurance.

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MATH 5640 - Advanced Topics in Actuarial Mathematics I

Survival models, mathematical graduation, or demography.

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MATH 5641 - Advanced Topics in Actuarial Mathematics II

Credibility theory or advanced theory of interest.

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MATH 5660 - Advanced Financial Mathematics

An introduction to the standard models of modern financial mathematics including martingales, the binomial asset pricing model, Brownian motion, stochastic integrals, stochastic differential equations, continuous time financial models, completeness of the financial market, the Black-Scholes formula, the fundamental theorem of finance, American options, and term structure models.

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MATH 5710 - Tensor Calculus I

An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.

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MATH 5711 - Tensor Calculus II

An introduction to tensor algebra and tensor calculus with applications chosen from the fields of the physical sciences and mathematics.

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MATH 5720 - Vector Field Theory I

Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.

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MATH 5721 - Vector Field Theory II

Vector algebra and vector calculus with particular emphasis on invariance. Classification of vector fields. Solution of the partial differential equations of field theory.

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MATH 5800 - Investigation of Special Topics

Students who have well defined mathematical problems worthy of investigation and advanced reading should submit to the department a semester work plan.

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MATH 5850 - Graduate Field Study Internship

Participation in internship and paper describing experiences.

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MATH 6000 - Seminar in Current Mathematical Literature

Participation and presentation of mathematical papers in joint student faculty seminars. Variable topics.

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MATH 6010 - Seminar in Analysis

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6020 - Seminar in Algebra

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6026 - Seminar in Mathematical Logic

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MATH 6027 - Seminar in Set Theory

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6030 - Seminar in Topology

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6036 - Seminar in Geometry

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6040 - Seminar in Applied Mathematics

Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatistactory).

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MATH 6060 - Computers in Mathematical Research

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