Graduate Studies |
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• alumni directory • application information • contact information • graduate degrees • financial support • mathematics courses • program of study • qualifying exam • welcome |
Graduate Studies |
|
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• alumni directory • application information • contact information • graduate degrees • financial support • mathematics courses • program of study • qualifying exam • welcome |
Starting Fall 2004 courses are renumbered using four digits instead of three. The old course number is listed in [ ] (brackets) after the new. Please note that 7xyz (graduate credit) and 4xyz (undergraduate credit) refer to the same course.
4001/7001 [301] Topics (cr. arr.). Organized study of selected topics. Subjects and earnable credit may vary from semester to semester. Prerequisites: 2300 and instructor’s consent. Departmental consent for repetition.
4100/7100 [304] 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 and linear systems. Prerequisite: 2300.
4110/7110 [302] 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, and special functions. Prerequisite: 2300.
4120/7120 [327] 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 and introduction to Ramsey theory. Prerequisite: 2320 or 4140 or instructor’s consent.
4130/7130 [330] Theory of Equations (3). Study of polynomials and their zeros, elementary determinant and matrix theory. Prerequisite: 2300 or 2320.
4140/7140 [331] Matrix Theory (3). Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs. Prerequisite: either 2300, 2320, 2120, or 2340.
4150/7150 [355] 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. Prerequisite or co-requisite: 2300 or 2340.
4160/7160 [358] Mathematical Logic (3). Introduction to classical modern logics as deductive systems; applications to foundations of mathematics. Prerequisites: junior or senior standing and interest and background in Mathematics or Philosophy.
4300/7300 [307] Numerical Analysis (3) (same as CECS 4300/7300). Machine arithmetic, approximation and interpolation, numerical differentiation and integration, nonlinear equations, linear systems, differential equations, error analysis. Selected algorithms programmed for solution on computers. Prerequisites: 2300, and the ability to program in high-level language such as Fortran, Pascal, or C.
4310/7310 [308] Numerical Linear Algebra (3) (same as CECS 4310/7310). 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: 2300 and the ability to program in a high-level language such as Fortran, Pascal, or C. Recommended: 4140.
4315/7315 [320] Introduction to Mathematical Statistics (3) (same as STAT 4710). Introduction to theory of probability and statistics using concepts and methods of calculus. Prerequisite: 2300 or instructor’s consent.
4320/7320 [325] Introduction to Probability Theory (3) (same as STAT 4750). Probability spaces, random variables and their distributions, repeated trials, probability limit theorems. Prerequisite: 2300 or instructor’s consent.
4325/7325 [332] Linear Programming (3). Linear dependence and rank in vector spaces in Rn, Farkas’ Lemma, Polyhedral Decomposition, strong duality and complementarity theorems, the simplex method, revised simplex, sensitivity analysis, primal dual simplex method, network simplex methods, computational complexity and Karmarkar’s algorithm. Prerequisite: 4140 or instructor’s consent.
4330/7330 [335] Theory of Numbers (3). Divisibility, factorization, arithmetic functions, mean value theorems, distribution of prime numbers, congruences, primitive roots, character theory, Riemann zeta function, and Dirichlet L-functions. Prerequisites: 2300; recommended: 2340 or 2320, and 4940.
4335/7335 [360] 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: 2300.
4340/7340 [362] 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: 2300.
4345/7345 [366] 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: 2300.
4350/7350 [367] Introduction to Non-Euclidean Geometry (3). Account of rise, development of non-Euclidean geometries. Intensive study of plane hyperbolic geometry. Prerequisite: 2300.
4355/7355 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: 2300 and Stat 2500, or instructor’s consent.
4360/7360 Actuarial Mathematics (3). Basic actuarial methods, mathematical population studies and models of population growth, compound interest and annuities, certain values of endowment and annuities, calculation of premiums, surrender values, and stochastic models of population growth. Prerequisites: 2300 and Stat 2500, or instructor’s consent No variable credit.
4400/7400 [372] Introduction to Topology (3). Topics from topology of Euclidean spaces, generalizations to metric spaces and topological spaces, and fundamentals of point set topology. Prerequisite: 2300.
4500/7500 [309] Applied Analysis (3). Solution of the standard partial differential equations (wave, heat, and Laplace’s eq.) by separation of variables in various coordinate systems and by transform methods; including eigenfunction expansions, and Fourier and Laplace transforms. Fourier series, boundary value problems, Sturm-Liouville theory, orthogonality, Bessel and Legendre series, spherical harmonics. Prerequisite: 4100.
4510/7510 [333] Higher Algebra (3). Introduction to rings, integral domains, fields, and groups. Prerequisite: 2300 or 2320.
4520/7520 [326] Statistical Inference I (3) (same as STAT 4751). Sampling, point estimation, sampling distribution, tests of hypotheses, regression and linear hypotheses. Prerequisite: 4320.
4530/7530 [349] Applied Modern Algebra (3). Introduction to modern algebra with emphasis on applications to computer science, engineering and related subjects. Basic concepts of modern algebra applied to computer design. Prerequisites: 2300 or 2320 and the ability to program in a high-level language such as Fortran, Pascal, or C.
4540/7540 [314] Mathematical Modeling I (3). Solution of problems from industry, physical, social and life sciences, economics, and engineering using mathematical models. Prerequisites: 3 semesters of calculus and some exposure to ordinary differential equations or instructor’s consent.
4560/7560 [316] Nonlinear Dynamics, Fractals and
Chaos (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 physicsl science. Prerequisite: MATH 4100/ 7100,
4140/7140, and familiarity with software such as MATHEMATICA,
MATLAB, or MAPLE.
4580/7580 [315] 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 Math 4540 will be considered. Prerequisites: 3 semesters of calculus and some exposure to ordinary differential equations or instructor’s consent. Math 4540 is not a prerequisite.
4590/7590 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. Prerequisites: 2300 and Stat 2500, or instructor’s consent. Recommended: 4355. No variable credit.
4700/7700 [310] Advanced Calculus I (3). Basic topology of the real line, numerical sequences and series, properties of continuous functions, differentiation, Riemann integration and uniform convergence, power series. Prerequisite: 2300. Recommended: 4140, and one other mathematics course numbered above 2300.
4720/7720 [340] Introduction to Abstract Algebra (3). Basic properties of integers, fundamental theorem of arithmetic, introduction to groups, rings and fields. Prerequisite: 2300. Recommended: 4140 and one other mathematics course numbered above 4000.
4900/7900 [311] Advanced Calculus II (3). Calculus in several variables. Core material: Basic topology of n-dimensional Euclidean 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; Green’s Theorem. Additional material from the calculus of several variables may be included, such as Lagrange multipliers, differential forms, etc. Prerequisite: 4700 or equivalent.
4920/7920 [341] 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. Prerequisites: Math 4720.
4940/7940 [305] Introduction to Complex Variables (3). Complex functions, contour integration, power series, residues and poles, conformal mapping. Prerequisite: 4110 or 4700.
4960/7960 [350] Special Readings (1-3). Prerequisites: 2300 and instructor’s consent.
4970/7970 [390] Senior Seminar (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.
4980/7980 [395] Mathematics Problem Solving (3). Creative advanced problem solving bringing together methods such as integration, probability and Euclidean geometry. Prerequisite: 4140 and another 4000 level Mathematics course, or instructor’s consent.
7620 [458] Differential Geometry I (3). Metric properties of restricted portions of curves and surfaces in three-dimensional Euclidean space. Prerequisite: 2300.
8085 [400] Problems (cr. arr.).
8090 Master’s Thesis (3). Students will be required to complete an independent thesis. Topics are chosen in consultation with a faculty advisor and are subject to departmental approval.
8102 [434] Topics in Algebra (3). Advanced topics in field theory and commutative and non-commutative ring theory. Prerequisite: 8410.
8202 Topics in Functional Analysis (3).
8190 [489] Master's Project in Mathematics (3). Students will be required to complete and present a project. Topics are chosen in consultation with a faculty advisor and are subject to departmental consent.
8210 [402] Basic Algebra (3). The major topics covered in
this course will be groups, rings, fields, vector spaces, linear
mappings, rational and Jordan canonical forms. The topics will be
presented in an accelerated fashion. This course should help prepare
students for the algebra qualifying exam, however, it will also be of
interest and value to any mathematically inclined undergraduate or
graduate student. Prerequisite: Math 7720, 7920 or equivalent, or
instructor’s consent.
8220 [401] Basic Analysis (3). The major topics covered in
this course include sequences and series of functions, elementary
Fourier series, integration, and differentiation in one and several
real variables, the Stone-Weierstrass theorem, the inverse function and
the implicit function theorem, elementary properties of holomorphic
functions, Cauchy’s theorem, power series representation, calculus of
residues, the maximum modulus principle and conformal mappings. This
course should help prepare students for the analysis qualifying exam,
however, it will also be of interest and value to any graduate student.
Prerequisite: Math 7700, 7900, 7940 or equivalent, or instructor’s
consent.
8250 [403] Basic Topology and Geometry (3). Basics of
topological spaces and continuous functions. Differential manifolds,
differential forms, integration of vector fields. Additional topics to
be selected by the instructor. Prerequisite: Math 7700, 7900, 7140 or
equivalent, or instructor’s consent.
8302 Topics in Harmonic Analysis (3).
8402 Topics in Mathematical Physics (3).
8410 [432] Algebra I (3). Theory of algebraic structures-groups, rings, fields, algebraic and transcendental extensions of fields. Prerequisites: 7720 and 7920, or equivalent.
8411 [433] Algebra II (3). Theory of modules, Galois theory and additional topics to be selected by the instructor. Prerequisite: 8411 or equivalent.
8420 [404] Theory of Functions of Real Variables I (3). Properties of functions of one real variable, lebesgue measure and integration on the line. Prerequisites: 7700 and 7900, or equivalent.
8421 [405] Theory of Functions of Real Variables II (3). Continuation of 8420. General measure and integration theory, elements of the theory of Hilbert and Banach spaces, linear functionals and linear operators. Prerequisite: 8420.
8425 [413] Complex Analysis I (3). Rigorous introduction to the theory of functions of a complex variable. Prerequisite: 7900 or equivalent.
8426 [414] Complex Analysis II (3). Analytic continuation, Riemann surfaces, entire and meromorphic functions and selected topics. Prerequisite: 8425.
8440 [426] 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: 7700 or equivalent.
8441 [427] Advanced Ordinary Differential Equations II (3). Continuation of 8440.
8442 [412] Calculus of Variations I (3). Development of necessary conditions and of sufficient conditions for nonparametric and parametric problems. Hamilton’s principle and related topics. Prerequisite: instructor’s consent.
8445 [407] Partial Differential Equations I (3). Fourier and integral transforms, first and second order partial differential equations, method of characteristics, Laplace’s equation, Dirichlet and Neumann problems, Green’s functions and maximum principles. Prerequisites: 7500 or consent of instructor.
8446 [408] Partial Differential Equations II (3). The Cauchy-Kovalevski theorem, the Lewy example, the heat operator, the wave operator, Sobolev spaces, local regularity of elliptic operators, elliptic boundary value problems. Prerequisite: 8445, and 8420 recommended.
8450 [457] 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: 7110 and some knowledge of matrix theory.
8460 Mathematical Finance I (3). Financial instruments and derivatives: stocks, bonds, futures options, options on interest rates, swaps, etc; mathematical models of stock price fluctuations; interest rates and options on interest rates; swaps; option markets and properties of stock option prices; stochastic models; binomial 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. Prerequisites: Graduate standing in Mathematics. Knowledge of elementary probability or the consent of the instructor.
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. Prerequisities. Knowledge of advance probability/stochastic processes, or the consent of the instructor. Recommended: 8460.
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. Prerequisites. Knowledge of elementary probability or the consent of the instructor.
8470 [423] Advanced Numerical Analysis (3). Analysis and implementation of numerical algorithms selected from approximation theory, splines, quadrature, nonlinear systems, ordinary differential equations, and optimization. Prerequisites: 7700, 7300 or equivalent, and 7140.
8480 [440] Advanced Probability (3) (same as STAT 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: 7320 or 8220, or instructor’s consent.
8502 [460] Topics in Geometry (cr. arr.). Prerequisite: instructor’s consent.
8587 [486] Topology Seminar (cr. arr.).
8602 Topics in Financial Mathematics (3).
8615 [435] 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: 8410.
8616 [436] Algebraic Geometry II (3). Continuation of 8615. Prerequisite: 8615.
8618 [470] Introduction to Algebraic Topology (3). Development of singular homology theory with reference to other homology and cohomology theories. Introduction to homological algebra. Prerequisite: 8655.
8628 [409] Functional Analysis I (3). Linear topological spaces, Banach spaces and Hilbert spaces. Operator theory, including the Hahn-Banach, uniform boundedness and closed graph theorems. Prerequisites: 7900 and instructor’s consent, or 8420.
8629 [410] Functional Analysis II (3). Topological vector spaces, duality theory and Banach algebras. Prerequisite: 8628.
8630 [415] Harmonic Analysis I (3). An introduction to Fourier analysis in one and higher dimensions. Topics include Fourier series, conjugate functions, convolutions, Fourier transforms, distributions, interpolation, and maximal functions. Prerequisite: 8420.
8631 [416] 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: 8630.
8642 [418] Nonlinear Differential Equations (3). Existence theorems, criteria for periodic solutions; boundedness of solutions; perturbation theory. Emphasizes second order equations. Prerequisites: 7100 and 7110 or 7700.
8644 [420] Topological Dynamics (3). Periodicity and its generalizations in dynamical systems. Prerequisite: 8420.
8648 [445] Advanced Mathematics for the Physical Sciences (3). Study of selection of topics in quantum mechanics and statistical mechanics, Schrodinger operators and their self-adjointness, semi-classical methods and their application to estimation of eigenvalues, partition functions in many body problems and methods of estimation. Prerequisites: instructor’s consent, 7110, 7700 or Physics 8660 recommended.
8650 [456] Differentiable Manifolds and Riemannian Geometry (3). Tensor product spaces and tensor fields on manifolds, differentiation and integration of differential forms, and Riemannian geometry and applications. Prerequisites: 7700 or 7400.
8655 [468] 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, and metrizable spaces. Prerequisite: 7900 or 7400 or instructor’s consent.
8656 [469] General Topology II (3). Continuation of 8655. Product spaces and Tychonoff’s theorem, introduction to homotopy theory, and fixed point theorems. Prerequisite: 8655.
8670 [424] 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 sparse matrices. Prerequisites: 7700, 7300 or 7310, and 7140, or instructor’s consent.
8675 [422] Numerical Solution of Partial Differential Equations (3). Study of finite difference and finite element methods for solving partial differential equations. Prerequisites: 7140, 7110, or 7700; or instructor’s consent.
8680 [441] Stochastic Processes (3) (same as Statistics 9820). Markov processes, martingales, orthogonal sequences, processes with independent and orthogonal increments, stationarity and linear prediction. Prerequisite: 8480.
8702 [449] 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 and numerical analysis.
8787 [487] Numerical Mathematics Seminar (cr. arr.).
9090 [490] Research (cr. arr.).
9187 [482] Algebra Seminar (cr. arr.).
9287 Functional Analysis Seminar (3).
9387 Harmonic Analysis Seminar (3).
9487 Mathematical Physics Seminar (3).
9502 [479] Topics in Topology (cr. arr.). Advanced topics in topology or topological algebra.
9587 [484] Geometry Seminar (cr. arr.).
9687 Financial Mathematics Seminar (3).
9702 [448] Topics in Numerical Mathematics (cr. arr.). Prerequisite: instructor’s consent.
9787 [488] Applied Mathematics Seminar (cr. arr.).
9887 [480] Analysis Seminar (cr arr.).