MATH 505 (3) Readings from Original Sources Mathematics studied through the reading, analysis, and discussion of original papers. May be repeated once for credit with consent of instructor. Prerequisite: To be determined by instructor.
MATH 510 (3) Mathematical Communication Selected topics in advanced mathematics chosen to demonstrate appropriate use of technology and effective organization and presentation of mathematics in oral and written form. Includes three aspects of mathematical writing: writing expository mathematics, writing formal mathematics, and writing as a tool to learn; preparation of mathematical lectures; development software modules/notebooks. Prerequisites: MATH 350 and at least nine (9) other units of upper-division mathematics, or consent of instructor.
MATH 520 (3) Algebra Review and continuation of the study of algebra begun in MATH 470. Covers some of the following: the theory of finite group theory including the Sylow Theorems, polynomial ring, unique factorization, number fields, and finite fields. The latter half of the course will cover field extensions of Galois theory, including the classic theorems on the unsolvability of the general quintic and the impossibility of certain ruler and compass constructions, such as trisecting an angle. Prerequisite: MATH 470 or consent of instructor.
MATH 521 (3) Computational and Applied Algebra Modern advances in computing and the theory of Grobner bases and resultants have created a new branch of computational algebra with many applications. Additionally, other algebraic topics such as semigroups and finite fields play an important role in discrete math and applications to cryptography and coding theory. Covers some of the following: Grobner bases, resultants, and applications to such fields as algebraic geometry, robotics, computer vision, and integer programming; semigroups, finite fields, partially ordered sets, Boolean algebras, applications to finite-state machines, cryptography, and coding theory. Prerequisite: MATH 470 or consent of instructor.
MATH 522 (3) Number Theory Introduction to number theory from the algebraic and/or analytic point of view. Includes some of the following: congruences, finite fields and rings and quadratic reciprocity; quadratic forms and Diophantine equations; elliptic curves; the Gaussian integers, the Eisenstein integers, and unique factorization in these rings; other quadratic and cyclotomic fields and ideal factorization; introduction to analytic number theory, primes in arithmetic progressions, and the prime number theorem. Prerequisite: MATH 470 or consent of instructor.
MATH 523 (3) Cryptography and Computational Number Theory Algorithms for factorization and primality testing: psuedo-primes, quadratic sieve, Lucas test, continued fractions, factorization using elliptic curves, public key cryptosystems such as RSA, which is widely used for secure transfer of data on the Internet. Additional background material (such as the rudiments of elliptic curves) will be introduced as needed. Combines theoretical ideas with computer lab experimentation and implementation. Some familiarity with a computer language is useful but not required. Some familiarity with a computer language is useful but not required. Prerequisite: MATH 350 or 370 or consent of instructor.
MATH 528 (3) Advanced Linear Algebra Vector spaces; dual spaces; linear transformations, bilinear forms and their matrix representations; Jordan and other canonical forms; finite-dimensional spectral theory; connections to other branches of mathematics. Prerequisite: MATH 374 or consent of instructor.
MATH 530 (3) Measure Theory Lebesque measure, measurable functions, the Lebesque integral, Fubini's theorem, Lp-spaces, and differentiation. Prerequisite: MATH 430 or consent of instructor.
MATH 532 (3) Ordinary Differential Equations Theory and applications of ordinary differential equations. Existence and uniqueness of solutions, methods for solving equations, linear differential equations, singularities, qualitative analysis of solutions, systems of equations. Prerequisite: MATH 374 and 430, or consent of instructor.
MATH 534 (3) Partial Differential Equations Theory and applications of partial differential equations. Cauchy problems, boundary problems, the Cauchy-Kovalevsky theorem, Fourier series, harmonic functions, elliptic equations, hyperbolic equations. Prerequisites: MATH 260, 374 and 430, or consent of instructor.
MATH 535 (3) Multivariate Advanced Calculus Analysis in several variables including multivariable derivatives and integrals, inverse function theorem, implicit function theorem, generalizations of the fundamental theorem of calculus (e.g., Stokes theorem). Some of these topics may be presented from the point of view of differential forms. Prerequisites: MATH 260, 374 and 430, or consent of instructor.
MATH 536 (3) Complex Analysis Study of functions of a complex variable, including analytic functions, contour integrals, Cauchy's theorem, poles and residues, Liouville's theorem, Laurent series, the Residue theorem, analytic continuation, and conformal mappings. Prerequisite: MATH 430 or consent of instructor.
MATH 538 (3) Applicable Analysis Foundations of functional analysis; linear and metric spaces; different modes of convergence; Hilbert space; applications. May include topics such as calculus of variations, fixed point theorems, and operator theory. Prerequisites: MATH 374 and 430, or consent of instructor.
MATH 540 (3) Concrete Mathematics A blend of continuous and discrete topics including sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. Prerequisite: MATH 350 or 370 or 470 or 472 or 474 or consent of instructor.
MATH 542 (3) Algorithmic Graph Theory Introduction to graphs; algorithmic complexity; depth-first and breadth-first search; trees; paths and distance; network flows; matchings and factorizations; Eulerian and Hamiltonian graphs; traveling salesman problem; planarity; vertex and edge colorings. Prerequisite: MATH 350 or 370 or 470 or 472 or 474 or consent of instructor.
MATH 544 (3) Applied Combinatorics Counting; Ramsey theory; experimental design; finite projective planes; combinatorial optimization; combinatorial set systems; matroids; axiomatic social choice; scheduling theory; location of facilities on networks. Prerequisite: MATH 350 or 370 or 470 or 472 or 474 or consent of instructor.
MATH 550 (3) Geometry Geometric ideas selected from the following fields: Euclidean geometry, hyperbolic geometry, projective geometry, introductory algebraic geometry, and computational geometry. Combines theoretical ideas with laboratory experience. Prerequisites: MATH 374 and MATH 470, or consent of instructor.
MATH 552 (3) Introduction to Differential Topology and Geometry Introduction to curves, surfaces, and possibly higher dimensional manifolds from the point of view of differential topology and/or differential geometry. Includes some of the following: curves (e.g., Frenet-Serret theorem and its consequences, isoparametric inequality, four-vertex theorem, line integrals, Fenchel's theorem), the topological classification of surfaces, vector fields, curvature on surfaces (leading up to some of the following: geodesics, minimal surfaces, Gauss's Theorema Egregium, and the Gauss-Bonnet theorem), introduction to higher dimensional manifolds, differential forms and integration (possibly including Stoke's theorem and global invariants such as the Euler Characteristic and the De Rham Cohomology). Prerequisites: MATH 260, 374, and 430, or consent of instructor.
MATH 555 (3) General Topology Topological spaces, open and closed sets, metric spaces, continuity, compactness, connectedness. Other subjects may include separation axioms, fundamental groups, classification of surfaces, completion of metric spaces. Prerequisites: MATH 350 or 430 or consent of instructor.
MATH 561 (3) Computational Linear Algebra Provides a thorough background in the formulation and analysis of algorithms for numerical linear algebra. Includes fundamentals of scientific computation, subspaces, rank-revealing matrix factorizations, numerical solutions of linear systems, linear least squares, regularization, perturbation theory, and iterative methods. Combines theoretical ideas with laboratory experience. Knowledge of computer language is required. Prerequisites: MATH 374 or consent of instructor.
MATH 564 (3) Nonlinear Programming Theory and techniques for solving constrained and unconstrained nonlinear programming problems. Techniques include Quasi-NewtonSecant Methods, Broyden's Method, conjugate gradient methods, and line search methods. Theoretical aspects include convexity, Lagrangian Multipliers, optimality conditions, convergence, primal problem, duality, saddle points, and line searches. This course meets for four hours per week. Prerequisites: MATH 374 or MATH/CS 480 or consent of instructor.
MATH 570 (3) Introduction to Stochastic Processes Elements of stochastic processes, discrete-time and continuous-time Markov chains, random walks, branching processes, birth and death processes, and Poisson point processes. Applications to queues and stochastic networks, resource management, biology and physics. May include optimal stopping, hidden Markov models, renewal processes, martingales, Brownian motion and Gaussian processes. Prerequisite: MATH 430 and MATH 440.
MATH 571 (3) Probability and Random Processes Framework for probability theory: probability spaces as measure spaces, random variables, expectation and conditional probability. Major results such a limit theorems for sums of random variables, zero-one laws, and ergodic theorems. Applications may include branching processes, Markov chains, Markov random fields, martingales, percolation, Poisson processes, queuing theory, random walks, and renewal processes. Combines theoretical ideas with laboratory experience using appropriate software packages. Prerequisites: MATH 430 or 440 or consent of instructor.
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