2003-2005 - Graduate School - The University of Alabama

Mathematics171MATH 554 Mathematical Statistics I (equivalent to ST 554). Three hours.Prerequisites: MATH 237 and MATH 382.Distributions of random variables, moments of random variables, probability dis tri bu tions,joint distributions, and change of variable techniques.MATH 555 Mathematical Statistics II (equivalent to ST 555). Three hours.Prerequisite: MATH 554.Order statistics, asymptotic distributions, point estimation, interval estimation, andhypothesis testing.MATH 556 Mathematical Statistics III (equivalent to ST 610). Three hours.Prerequisite: MATH 555.Generalized inverse matrices; distribution of quadratic forms; regression analysis whenthe model is of full rank; regression using dummy variables and analysis of variancemodels; and regression analysis when the model is not of full rank.MATH 557 Stochastic Processes with Applications I. Three hours.Prerequisite: MATH 554 or ST 554.Introduction to the basic concepts and applications of stochastic processes. Markovchains, continuous-time Markov processes, Poisson and renewal processes, andBrownian motion. Applications of stochastic processes including queueing theory andprobabilistic analysis of computational algorithms.MATH 559 Stochastic Processes with Applications II. Three hours.Prerequisite: MATH 355 or permission of the department.Continuation of MATH 557. Advanced topics of stochastic processes including Mar tin -gales, Brownian motion and diffusion processes, advanced queueing theory, stochasticsim u la tion, and probabilistic search algorithms (simulated annealing).MATH 560 Introduction to Differential Geometry. Three hours.Prerequisites: MATH 380 or MATH 382, and permission of the department.Introduction to basic classical notions in differential geometry: curvature, torsion, geodesic curves, geodesic parallelism, differential manifold, tangent space, vector fi eld, Liederivative, Lie algebra, Lie group, exponential map, and representation of a Lie group.MATH 565 Introduction to General Topology. Three hours.Prerequisite: MATH 380.Basic notions in topology that can be used in other disciplines in mathematics. Topicsinclude topological spaces, open sets, closed sets, basis for a topology, continuousfunctions, separation axioms, compactness, connectedness, product spaces, quotientspaces, and metric spaces.MATH 566 Introduction to Algebraic Topology. Three hours.Prerequisites: MATH 565 and a course in abstract algebra.Homotopy, fundamental groups, covering spaces, covering maps, and basic homologytheory, including the Eilenberg Steenrod axioms.MATH 570 Principles of Modern Algebra I. Three hours.Prerequisite: MATH 257.A fi rst course in abstract algebra. Topics include groups, permutations groups, Cayley’stheorem, fi nite Abelian groups, isomorphism theorems, rings, poly no mi al rings, ideals,integral domains, and unique factorization domains.MATH 571 Principles of Modern Algebra II. Three hours.Prerequisite: MATH 470 or equivalent.The basic principles of Galois theory are introduced in this course. Topics covered arerings, polynomial rings, fi elds, algebraic extensions, normal extensions, and the fun da -men tal theorem of Galois theory.MATH 573 Abstract Algebra I. Three hours.Prerequisite: MATH 470 or equivalent.Fundamental aspects of group theory are covered. Topics include Sylow theorems,semi-direct products, free groups, composition series, nilpotent and solvable groups,and infi nite groups.MATH 574 Cryptography. Three hours.Prerequisite: MATH 307, MATH 470/570, or permission of department.Introduction to a rapidly growing area of cryptography, an application of algebra, especiallynumber theory.