Mathematics Courses

MATH 7001 Topics in Mathematics-General (cr.arr.).

Organized study of selected topics. Subjects and earnable credit may vary from semester to semester. Prerequisites: graduate standing and MATH 2300 and instructor's consent. Departmental consent for repetition.

MATH 7060 Connecting Geometry to Middle and Secondary Schools (3).

Euclidian foundations, basic concepts of symbolic logic, polyhedra, Euler Characteristic, congruence, area, Picks Theorem, volume, Cavalier's Principles, surface area, similarity, reflections, translations, rotations, symmetry, vectors, general transformations, determinants, matricies, transformations using matrices, brief introduction to spherical geometry. Prerequisite: MATH 1360 or 1500, enrollment in restricted to Math Education majors.

MATH 7070 Connecting Algebra to Middle and Secondary Schools (3).

A detailed study of integer and rational arithmetic and algebra. Topics include: Binomial Theorem, induction, division algorithm, Euclid's Algorithm, Fundamental Theorem of Arithmetic, Pythagorean triples, modular arithmetic and generalizations to polynomials, matrices and other axiomatic structures. Prerequisite: MATH 1320, enrollment is restricted to Math Education majors

MATH 7080 Connect Calculus to Middle and Secondary Schools (3).

Course topics include: sequences, series functions, limits, continuity, differentiation, optimization, curve sketching, antidifferentiation, area of plane regions, lengths of plane curves, areas of surfaces of revolution, and volumes of solids, Prerequisite: MATH 1160, enrollment is restricted to Math Education majors.

MATH 7100 Differential Equations (3).

Traditional introductory course in ordinary differential equations. Includes 1st and 2nd order linear differential equations with numerous applications; Laplace transforms; power series solutions; numerical methods, linear systems. Prerequisite: graduate standing and MATH 2300.

MATH 7110 Advanced Calculus With Applications (3).

Linear mappings, Jacobi matrices and determinants, change of variables, vector fields, line and surface integrals, theorems of Green, Gauss and Stokes, sequences and series of functions, uniform convergence, special functions. Prerequisite: graduate standing and MATH 2300.

MATH 7120 Combinatorics (3).

Study of a variety of topics from combinatorial mathematics, especially graph theory and enumerative combinatorics. Topics include graph coloring, matchings and coverings, generating functions, recurrence relations, Polya's Enumeration Theorem, introduction to Ramsey theory. Prerequisites: graduate standing and MATH 2320 or instructor's consent.

MATH 7130 Theory of Equations (3).

Study of polynomials and their zeros and elementary determinant and matrix theory. Prerequisites: graduate standing and MATH 2300 or 2320.

MATH 7140 Matrix Theory (3).

Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs. Prerequisite: graduate standing and one of MATH 2300, 2320, 2120 or 2340.

MATH 7150 History of Mathematics (3).

This is a history course with mathematics as its subject. Includes topics in the history of mathematics from early civilizations onwards. The growth of mathematics, both as an abstract discipline and as a subject which interacts with others and with practical concerns, is explored. Pre- or Co-requisite: MATH 2300 or 2340 and graduate standing.

MATH 7160 Mathematical Logic (3).

Introduction to classical modern logics as deductive systems; applications to foundations of mathematics. Prerequisites: graduate standing and interest in Mathematics or Philosophy.

MATH 7300 Numerical Analysis (3).

Machine arithmetic, approximation and interpolation, numerical differentiation and integration, nonlinear equations, linear systems, differential equations, error analysis. Selected algorithms will be programmed for solution on computers. Prerequisites: graduate standing and MATH 2300 and familiarity with software such as Mathematica or MatLab or Maple, etc.

MATH 7310 Numerical Linear Algebra (3).

Solution of linear systems of equations by direct and iterative methods. Calculation of eigenvalues and eigenvectors of matrices. Selected algorithms programmed for solution on computers. Prerequisites: graduate standing and MATH 2300 and prior experience writing programs in Mathematica and /or in a computer language such as Fortran, Pascal, or C. Recommended: MATH 4140.

MATH 7315 Introduction to Mathematical Statistics (3).

(same as Statistics 7710). Introduction to theory of probability and statistics using concepts and methods of calculus. Prerequisites: graduate standing and MATH 2300 or instructor's consent.

MATH 7320 Introduction to Probability Theory (3).

(same as Statistics 7750). Probability spaces; random variables and their distributions; repeated trials; probability limit theorems. Prerequisites: graduate standing and MATH 2300 or instructor's consent.

MATH 7325 Linear Programming (3).

Linear dependence and rank in vector spaces in Rn, Farkas' Lemma, Polyhedral Decomposition. Strong duality and complementary theorems. The simplex method, revised simplex, and sensitivity analysis. Primal Dual simplex method and network simplex methods. Computational Complexity and Karmarkar's Algorithm. Prerequisites: graduate standing and MATH 4140 or instructor's consent.

MATH 7330 Theory of Numbers (3).

Divisibility, factorization, arithmetic functions, means value theorems, distribution of prime numbers, congruences, primitive roots, character theory, Riemann zeta function, and Dirichlet L-functions. Prerequisites: graduate standing and MATH 2300; recommended 2320 or 2340, and 4940/7940.

MATH 7335 College Geometry (3).

Euclidean geometry from an advanced viewpoint. Synthetic and coordinate methods will be used. The Euclidean group of transformations will be studied. Prerequisite: graduate standing and MATH 2300.

MATH 7340 Projective Geometry (3).

Basic ideas and methods of projective geometry built around the concept of geometry as the study of invariants of a group. Extensive treatment of collineations. Prerequisite: graduate standing and MATH 2300.

MATH 7345 Foundations of Geometry (3).

Coordination of affine, projective planes by means of various kinds of algebraic structures: planar ternary rings, Veblen-Wedderburn systems, divisions rings, skew fields, and fields. Prerequisite: graduate standing and MATH 2300.

MATH 7350 Introduction to Non-Euclidean Geometry (3).

Account of rise, development of non-Euclidean geometries. Intensive study of plane hyperbolic geometry. Prerequisite: graduate standing and MATH 2300.

MATH 7355 Investment Science I (3).

Deterministic cash flow streams. The present value. Bonds, bonds' yield, duration. The term structure of interest rates. Single-period random yield analysis. Random returns. Portfolio mean-variance analysis. Markovitz model. Prerequisites: graduate standing and MATH 2300 and STAT 2500 or instructor's consent.

MATH 7360 Actuarial Mathematics (3).

Basic actuarial methods, mathematical population studies and and models of population growth. Compound interest and annuities certain. Values of endowment and annuities, calculation of premiums, surrender values. Stochastic models of populations growth. Prerequisite: graduate standing and MATH 2300 and STAT 2500, or instructor's consent. No variable credit.

MATH 7370 Actuarial Modeling I (3).

This course covers the main probability tools applied to financial risk modeling, and the financial mathematics concepts used in calculating present and accumulated values for various cash flows. It is a helpful tool in preparing for the Society of Actuaries exams P (Probability) and FM (Financial Mathematics), and it is oriented towards problem solving techniques illustrated with previous exam problems. Prerequisites: MATH 2300, MATH 4320/STAT 4750. Students are encouraged to take MATH 4355 prior to this course

MATH 7371 Actuarial Modeling II (3).

This course covers the actuarial models and their applications to insurance and other business decisions. It is a helpful tool in preparing for the Society of Actuaries exam (M) (Actuarial Models), and it is oriented towards problem solving techniques illustrated with previous exam problems. Prerequisites: MATH 2300, MATH 4320/STAT 4750. Student are encouraged to take MATH 4355 prior to this course.

MATH 7400 Introduction to Topology (3).

Topics from topology of Euclidean spaces, generalizations to metric spaces and topological spaces. Fundamentals of point set topology. Prerequisite: graduate standing and MATH 2300.

MATH 7500 Applied Analysis (3).

Solution of the standard partial differential equations (wave, heat, Laplace's eq.) by separation of variables and transform methods; including eigenfunction expansions, Fourier and Laplace transform. Boundary value problems, Sturm-Liouville theory, orthogonality, Fourier, Bessel, and Legendre series, spherical harmonics. Prerequisite: graduate standing and MATH 4100.

MATH 7510 Higher Algebra (3).

Introduction to rings, integral domains, fields, groups. Prerequisites: graduate standing and MATH 2300 OR 2320.

MATH 7520 Statistical Inference I (3).

(same as Statistics 7760). Sampling; point estimation; sampling distribution; tests of hypotheses; regression and linear hypotheses. Prerequisite: graduate standing and MATH 4320.

MATH 7530 Applied Modern Algebra (3).

Introduction to modern algebra; emphasis on applications to computer science, engineering, related subjects. Basic concepts of modern algebra applied to computer design. Prerequisites: graduate standing and MATH 2300 or 2320 and the ability to program in a high level language such as Fortran, Pascal or C.

MATH 7540 Mathematical Modeling I (3).

Solution of problems from industry, physical, social and life sciences, economics, and engineering using mathematical models. Prerequisites: graduate standing and 3 semesters of calculus and some exposure to ordinary differential equations or instructor's consent.

MATH 7560 Nonlinear Dynamics, Chaos and Fractals (3).

Conceptual introduction to nonlinear dynamics, bifurcation and stability of steady states, chaos in nonlinear differential equations and maps, fractal dimension, strange attractors, and applications to physical science. Prerequisite: graduate standing and MATH 4100/7100, 4140/7140, and familiarity with software such as MATHEMATICA, MATLAB, or MAPLE.

MATH 7570 Fluid Dynamics and Geophysical Applications (3).

Mathematical theory of fluid dynamics and applications to meteorology and oceanography. Prerequisites: graduate standing and MATH 2300 and instructor's consent.

MATH 7580 Mathematical Modeling II (3).

Solution of problems from industry, physical, social and life sciences, economics, and engineering using mathematical models. More general classes of problems than in Mathematics 7540 will be considered. Prerequisites: 3 semesters of calculus and some exposure to ordinary differential equations or instructor's consent. MATH 7540 is not a prerequisite.

MATH 7590 Investment Science II (3).

Derivative securities, forward and future contracts, forward prices, hedging. Mean-variance hedging. Stochastic models of asset dynamics, random walks and binomial models. Capital budgeting, optimal portfolios. Basic option theory, put-call parity. Prerequisite: graduate standing and MATH 2300 and STAT 2500 or instructor's consent. Recommended: MATH 4355. No variable credit.

MATH 7620 Differential Geometry I (3).

Metric properties of restricted portions of curves and surfaces in three-dimensional Euclidean space. Prerequisite: MATH 2300.

MATH 7700 Advanced Calculus I (3).

Basic topology of the real line, numerical sequences and series, continuity, differentiability, Riemann integration, uniform convergence, power series. Prerequisite: graduate standing and MATH 2300. Recommended 4140 and one other mathematics course number above 2300.

MATH 7720 Introduction to Abstract Algebra I (3).

Basic properties of integers, fundamental theorem of arithmetic, introduction to groups, rings and fields. Prerequisite: MATH 2300. Recommended: MATH 4140 and one other Mathematics course numbered above 2300. .

MATH 7900 Advanced Calculus II (3).

This is a course in calculus in several variables. The following is core material: Basic topology of n-dimensional Euclidian space; limits and continuity of functions; the derivative as a linear transformation; Taylor's formula with remainder; the Inverse and Implicit Function Theorems, change of coordinates; integration (including transformation of integrals under changes of coordinates); Green's Theorem. Additional material from the calculus of several variables may be included, such as Lagrange multipliers, differential forms, etc. Prerequisite: MATH 4700.

MATH 7920 Introduction to Abstract Linear Algebra (3).

Study of vector spaces over arbitrary fields: topics include linear maps on finite dimensional vector spaces, bilinear and multi-linear forms, invariant subspaces and canonical forms. Prerequisite: MATH 4720.

MATH 7940 Introduction to Complex Variables (3).

Complex functions, contour integration, power series, residues and poles, conformal mapping. Prerequisites: graduate standing and MATH 4110 OR 4700.

MATH 7960 Special Readings in Mathematics (1-3).

Prerequisites: graduate standing and MATH 2300 and instructor's consent.

MATH 7970 Senior Seminar in Mathematics (3).

Seminar with student presentations, written projects, and problem solving. May be used for the capstone requirement. Prerequisite: 12 hours of mathematics courses numbered 4000 or above.

MATH 7980 Mathematics Problem Solving (3).

Creative advanced problem solving bringing together methods such as integration, probability and Euclidean geometry. Prerequisite: graduate standing and MATH 4140 and another 4000 level Mathematics course, or instructor's consent.

MATH 8085 Problems in Mathematics (1-3).

MATH 8090 Master's Thesis Research in Mathematics (3).

Students will be required to complete an independent project. Topics are chosen in consultation with a faculty advisor and are subject to departmental consent. Graded on S/U basis only.

MATH 8102 Topics in Algebra (3).

Advanced topics in algebra. Prerequisite: MATH 8410.

MATH 8190 Masters Project in Mathematics (3).

MATH 8202 Topics in Functional Analysis (3).

Prerequisite: graduate standing required.

MATH 8210 Basic Algebra (3).

Accelerated problem solving course in linear and abstract algebra. Will prepare students for the algebra qualifying exam. Prerequisites: MATH 4720, 4920, instructor's consent or equivalent. Corequisites: MATH 8220 and 8250.

MATH 8220 Basic Analysis (3).

Accelerated problem-solving course in advanced calculus and complex analysis. Will prepare students for the analysis qualifying exam. Prerequisites: MATH 4700, 4900, 4940, instructor's consent or equivalent. Corequisites: MATH 8210 and 8250.

MATH 8250 Basic Topology and Geometry (3).

Topological spaces, differential manifolds, differential forms, integration of vector fields. Prerequisites: MATH 4700, 4900, 4140, instructor's consent or equivalent. Corequisites: MATH 8210 and 8220.

MATH 8302 Topics in Harmonic Analysis (cr.arr.).

Prerequisite: graduate standing.

MATH 8402 Topics in Mathematical Physics (cr.arr.).

Prerequisite: graduate standing.

MATH 8410 Algebra I (3).

Theory of algebraic structures--groups, rings, fields, algebraic and transcendental extensions of fields. Prerequisites: MATH 4720 and 4920, or equivalent.

MATH 8411 Algebra II (3).

Theory of modules, Galois theory and additional topics to be selected by the instructor. Prerequisite: MATH 8410 or equivalent.

MATH 8420 Theory of Functions of Real Variables I (3).

Properties of functions of one real variable. Lebesgue measure and integration on the line. Prerequisites: MATH 4700 and 4900, or equivalent.

MATH 8421 Theory of Functions of Real Variables II (3).

Continuation of Mathematics 8420. General measure and integration theory. Elements of the theory of Hilbert and Banach spaces, linear functions and linear operators. Prerequisite: MATH 8420.

MATH 8425 Complex Analysis I (3).

Rigorous introduction to the theory of functions of a complex variable. Prerequisite: MATH 4900 or equivalent.

MATH 8426 Complex Analysis II (3).

Analytic continuation, Riemann surfaces, entire and meromorphic functions, selected topics. Prerequisites: MATH 8425.

MATH 8440 Advanced Ordinary Differential Equations I (3).

Topics from existence and uniqueness theorems, plane autonomous systems, periodicity and boundedness of solutions of second order nonlinear equations, perturbation theory, Sturm-Liouville systems, behavior of solutions at singularities. Prerequisite: MATH 4700 or equivalent.

MATH 8441 Advanced Ordinary Differential Equations II (3).

Continuation of Mathematics 8440.

MATH 8442 Calculus of Variations I (3).

Development of necessary conditions and of sufficient conditions for nonparametric and parametric problems. Hamilton's principle, related topics. Prerequisite: instructor's consent.

MATH 8445 Partial Differential Equations I (3).

Fourier and integral transforms, first and second order partial differential equations, methods of characteristics, Laplace's equation, Direchlet and Neumann problems, Green's functions and maximum principles. Prerequisite: MATH 4500 or instructor's consent.

MATH 8446 Partial Differential Equations II (3).

The Cauchy-Kovalevski theorem, the Lewy example, the heat operator, the wave operator, Sobolev spaces, local regularity of elliptic boundary value problems. Prerequisite: MATH 8445, 8420 recommended.

MATH 8450 Differential Geometry for Scientists and Engineers (3).

Tensors and multilinear forms. Connections, covariant differentiation, geodesics and curvature on Riemannian and pseudo Riemannian manifolds. Applications to special relativity and general relativity. Prerequisites: MATH 4110 and some knowledge of Matrix Theory.

MATH 8460 Mathematical Finance I (3).

(same as Finance 8340). Financial instruments and derivative: stocks, bonds, futures option prices on interest rates, swaps, etc. Mathematical models of stock price fluctuations. Interest rates and options on interest rates. Swaps. Open markets and properties of stock option prices. Stochastic models. Bionomial trees. Continuous time stochastic modeling. No arbitrage modeling. European and American options. Black- Scholes model and differential equation, for the price of European option. Exotic options. Interest rate models. Prerequisite: graduate standing in Mathematics. Knowledge of elementary probability or instructor's consent.

MATH 8461 Mathematical Finance II (3).

Diffusion Processes as models for stock price fluctuations. Contingent claims and arbitrage. Mathematical analysis of risk neutral valuation of contingent claims. Self-financing portfolios and hedging. Hedging contingent claims. Partial differential equations for valuation of derivative securities. Completeness of the markets and hedging. Parity relations and delta hedging. Several underlying assets. Prerequisites: knowledge of advance probability/stochastic processes or instructor's consent; recommended MATH 8460.

MATH 8465 Mathematical Methods of Risk Theory (3).

Probability aspects of Risk. Claim number processes. Accumulated claim number processes. Retentions and Reserves. Mathematics of reinsurance. Ruin probability calculations. Stability and dividends policy. Utility as criterion of stability. The problem of risk exchange. Prerequisite: knowledge of elementary probability or the instructor's consent; graduate standing.

MATH 8470 Advanced Numerical Analysis (3).

Analysis and implementation of numerical algorithms selected from approximation theory, splines, quadrature, nonlinear systems, ordinary differential equations, and optimization. Prerequisites: MATH 4700, 4300 or equivalent, and 4140.

MATH 8480 Advanced Probability (3).

(same as Statistics 9810). Measure theoretic probability theory. Characteristic functions; conditional probability and expectation; sums of independent random variables including strong law of large numbers and central limit problem. Prerequisites: MATH 4320 or 8220; or instructor's consent.

MATH 8502 Topics of Geometry (cr.arr.).

Prerequisite: instructor's consent.

MATH 8587 Topology Seminar (cr.arr.).

MATH 8602 Topics in Financial Mathematics and Insurance (cr.arr.).

Prerequisite: graduate standing.

MATH 8615 Algebraic Geometry I (3).

Affine and projective varieties and schemes; nullstellensatz; Zariski topology, morphisms and rational maps; divisors and linear systems; topics from curves, surfaces, Grassmann varieties; commutative algebra and homological algebra as needed. Prerequisite: MATH 8410.

MATH 8616 Algebraic Geometry II (3).

Continuation of Mathematics 8615. Prerequisite: MATH 8615.

MATH 8618 Introduction to Algebraic Topology (3).

Development of singular homology theory; reference to other homology and cohomology theories. Introduction to homological algebra. Prerequisite: MATH 8250.

MATH 8628 Functional Analysis I (3).

Linear topological spaces, Banach spaces, Hilbert spaces. Operator theory, including the Hahn-Banach, uniform boundedness and closed graph theorems. Prerequisite: MATH 4900 and instructor's consent or MATH 8420.

MATH 8629 Functional Analysis II (3).

Topological vector spaces, duality theory, Banach algebras. Prerequisite: MATH 8628.

MATH 8630 Harmonic Analysis I (3).

An introduction to Fourier Analysis in one and higher Dimensions. Topics include Fourier Series, conjugate functions, Fourier transforms, distributions, interpolation, and maximal functions. Prerequisite: MATH 8420.

MATH 8631 Harmonic Analysis II (3).

Singular integrals, Littlewood-Paley theory, Hardy spaces, bounded mean oscillation, weighted norm inequalities, boundary value problems, and analysis on groups. Prerequisite: MATH 8630.

MATH 8642 Nonlinear Differential Equations (3).

Existence theorems; criteria for periodic solutions; boundedness of solutions; perturbation theory. Emphasizes second order equations. Prerequisites: MATH 4100 and 4110 or 4700.

MATH 8644 Topological Dynamics (3).

Periodicity and its generalizations in dynamical systems. Prerequisite: MATH 8420.

MATH 8648 Advanced Mathematics for the Physical Sciences (3).

Study of selected topics in quantum and statistical mechanics. Schrodinger operators and their self-adjointness. Semi-classical methods and their application to estimation of eignevalues. Partition functions in many body problems and methods ofestimation. Prerequisites: instructor's consent, MATH 4110, 4700, PHYSCS 8660 recommended.

MATH 8650 Differentiable Manifolds and Riemannian Geometry (3).

Tensor product spaces and tensor fields on manifolds. Differentiation and integration of differential forms. Riemannian geometry and applications. Prerequisites: MATH 4700 or 4400.

MATH 8655 General Topology I (3).

Introduction to axiomatic theory of general topology. Continuous functions and homeomorphisms. Convergence in abstract topological spaces. Compact and locally compact spaces. Connectedness. Metrizable spaces. Prerequisites: MATH 4900, 4400 or instructor's consent.

MATH 8656 General Topology II (3).

Continuation of Mathematics 8655. Product spaces and Tychonoff's theorem. Introduction to homotopy theory. Fixed point theorems. Prerequisite: MATH 8655.

MATH 8670 Advanced Numerical Linear Algebra (3).

Advanced techniques for solving systems of linear equations, least squares problems, and eigenvalue problems. Analysis of stability of algorithms. Discussion of both full and sparce matrices. Prerequisites: MATH 4700, 4300 or 4310, and 4140 or instructor's consent.

MATH 8675 Numerical Solution of Partial Differential Equations (3).

Study of finite difference and finite element methods for solving partial differential equations. Prerequisites: MATH 4140, 4110, 4700 or instructor's consent.

MATH 8680 Stochastic Processes (3).

(same as Statistics 9820). Markov processes, martingales, orthogonal sequences, processes with independent and orthogona increments, stationarity, linear prediction. Prerequisite: MATH 8480.

MATH 8702 Topics in Applied Mathematics (cr.arr.).

Selected topics in applied mathematics drawn from variety of areas: partial differential equations, tensor analysis, calculus of variations, asymptotic methods, integral equations, advanced theory of transforms and distributions, numerical analysis.

MATH 8787 Numerical Mathematics Seminar (cr.arr.).

MATH 9090 Doctoral Dissertation Research in Mathematics (cr.arr.).

Graded on a S/U basis only.

MATH 9187 Algebra Seminar (cr.arr.).

MATH 9287 Functional Analysis Seminar (cr.arr.).

Prerequisite: graduate standing.

MATH 9387 Harmonic Analysis Seminar (cr.arr.).

Prerequisite: graduate standing.

MATH 9487 Mathematical Physics Seminar (cr.arr.).

Prerequisite: graduate standing.

MATH 9502 Topics in Topology (cr.arr.).

Advanced topics in topology or topological algebra.

MATH 9587 Geometry Seminar (cr.arr.).

MATH 9687 Financial Mathematics Seminar (cr.arr.).

Prerequisite: graduate standing.

MATH 9702 Topics in Numerical Mathematics (cr.arr.).

Prerequisite: instructor's consent.

MATH 9787 Applied Mathematics Seminar (cr.arr.).

MATH 9887 Analysis Seminar (cr.arr.).