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MATH 700. Set Theory and Logic. (3)
An introduction to logic, mathematical proof, and elementary
set theory; elementary logic, the basic constructions
of set theory, relations, partitions, functions, cartesian
products, disjoint untions, orders, and a construction of
the natural numbers; also ordinal and cardinal numbers,
the Axiom of Choice, and transfinite induction.
Special emphasis will be given to proving theorems.
MATH 701. Elementary Topology 1. (3) Introduction
to axiomatic topology including a study of compactness,
connectedness, local properties, separation axioms,
and metrizability. Pr.: MATH 633 .
MATH 702. Elementary Topology 2. (3) Path
connectedness, fundamental groups, covering spaces,
introduction to topological and differentiable manifolds.
Pr.: MATH 701.
MATH 704. Introduction to the Theory of Groups. (3)
Introduction to abstract group theory; to include
permutation groups, homomorphisms, direct products, Abelian
groups, Jordan-Holder and Sylow theorems. Pr.:
MATH 512 .
MATH 706. Theory of Numbers. (3) Divisibility,
congruences, multiplicative functions, number theory from an
algebraic viewpoint, quadratic reciprocity, Diophantine equations,
prime numbers.
Pr.: MATH 221
and either MATH 511
or MATH 512 .
MATH 710. Introduction to Category Theory. (3) Categories,
duality, special morphisms, functors, natural transformations,
limits and colimits, adjoint situations, and applications.
Pr.: MATH 701 and MATH 730.
MATH 711. Category Theory. (3) Set-valued functors
and concrete categories, factorization structures, algebraic
and topological functors, categorical completions, Abelian categories.
Pr.: MATH 710.
MATH 713. Advanced Applied Matrix Theory. (3)
A development of the concepts of eigenvalues by
considering applications in differential equations and quadratic forms.
A discussion of the Jordan canonical form, functions
of matrices, vector and matrix norms, and various related
numerical methods.
Pr.: MATH 551 or MATH 515 .
MATH 721. Analysis I. (3) Metric spaces, limits,
continuity, sequences and series, connectedness,
compactness, Baire category, uniform convergence,
theorems of Stone-Weierstrass and Arzela.
Pr.: MATH 240 or graduate standing.
MATH 722. Analysis II. (3) Lebesgue and Riemann-Stieltjes
integration on the real line, differentiation on the real
line, elementary transcendental functions. Pr.: MATH 721.
MATH 730. Abstract Algebra I. (3) Groups, rings, fields
vector spaces and their homomorphisms. Elementary
Galois theory and decomposition theorems for
linear transformations on a finite dimensional vector space.
Pr.: MATH 512 or consent of instructor.
MATH 731. Abstract Algebra II. (3) Continuation of
MATH 730. Pr.: MATH 730 or consent of instructor.
MATH 740. Calculus of Variations. (3) Necessary
conditions and the
Euler-Lagrange equations. Hamilton-Jacobi theory, Noether's
theorems, direct methods, application to geometry and
physics. Pr.: MATH 722 or equivalent.
MATH 755.Dynamic Modeling Processes. (3) Topics
to include equilibrium and stability, limit circles,
reaction-diffusion, and shock phenomena, Hopf bifurcation and
cusp catastrophes, chaos and strange attractors, bang-bang
principle. Application from physical and biological
sciences and engineering. Pr.:
MATH 240 and MATH 551 .
MATH 772. Elementary Differential Geometry. (3) Curves
and surfaces in Euclidean spaces, differential
forms and exterior differentiation, differential
invariants and frame fields, uniqueness
theorems for curves and surfaces, geodesics, introduction to Riemannian
geometry, some global theorems, minimal surfaces.
Pr.: MATH 240 .
MATH 791. Topics in Mathematics for Secondary School Teachers.
(3)
Topics of importance in the preparation of secondary school
teachers to teach modern mathematics. May be repeated for credit.
MATH 801. Numerical Solution of Differential Equations I. (3)
Single and multistep methods for initial value
problems for ordinary differential equations;
discretization and round-off error; consistency,
convergence, and stability of these methods; stiff
equations and implicit methods; two point boundary value
problems; initial and boundary value problems for partial
differential equations; finite difference methods; marching
schemes for parabolic and hyperbolic problems; consistency,
stability, convergence. and the Lax equivalence theorem;
treatment of boundary conditions; boundary value problems
for elliptic equations; relaxation, alternating direction,
and strongly implicit iterative methods; nonlinear
problems; finite element methods.
Pr.: MATH 655 and
knowledge of a programming language.
MATH 802. Numerical Solution of Differential Equations II. (3)
Continuation of MATH 801. Pr.: MATH 801.
MATH 810. Higher Algebra I.
(3) Theory of groups, theory of rings and ideals,
polynomial domains, theory of fields and their extensions.
Pr.: MATH 731.
MATH 811. Higher Algebra II.
(3) Continuation of MATH 810. Pr.: MATH 810.
MATH 821. Real Analysis I.
(3) Measurability, integration theory, regular Borel
measures, the Riesz representation theorem, and Lebesgue
measure in Euclidean spaces. Pr.: MATH 722.
MATH 822. Real Analysis II.
(3) The Lp-spaces, Banach spaces, and Hilbert spaces,
complex measures and the Radon-Nikodym theorem, the Fubini
theorem on double integration, and differentiation. Pr.:
MATH 821.
MATH 825. Complex Analysis I.
(3) Holomorphic functions, harmonic functions, the
Cauchy integral theorem, normal families and the Reimann
mapping theorem, and the Mittag-Leffler theorem. PR.: MATH
822 or consent of department.
MATH 826. Complex Analysis II.
(3) Analytic continuation, the Picard theorem, Hp-
spaces, elementary theory of Banach algebra, the theory of
Fourier transforms, and the Paley-Wiener theorems. Pr.:
MATH 825.
MATH 852. Functional Analysis I.
(3) Topics to be selected from
linear topological spaces, seminormed linear spaces, Banach
spaces, Hilbert spaces, Banach algebras, spectral theory,
harmonic analysis, and others. May be taken four times for
a total of 12 credit hours. Pr.: MATH 852.
MATH 853. Functional Analysis II.
(3) Continuation of Functional
Analysis I. May be repeated for credit. Pr.: MATH 852.
MATH 855. Methods of Applied Mathematics I.
(3) An introduction to the mathematical techniques of
problem solving in the sciences and engineering.
Construction of mathematical models; problem formulation,
dimensional analysis and scaling; solution methods for
differential equations and difference equations; methods
for obtaining approximate solutions; regular and singular
perturbations methods, asymptotic series, applications to
specific equations and scientific problems.
Pr.: MATH 630 ,
633, and 551.
MATH 856. Methods of Applied Mathematics II.
(3) A continuation of MATH 855. Asymptotic expansion of
integrals; the methods of stationary phase and steepest
descent; summations of series, the Shanks transformation
and the Pade fractions; boundary layer theory; the WKB and
Langer approximations; the method of averaging and the
method of multiple scales. Pr.: MATH 855.
MATH 861. Numerical Analysis I.
(3) Topics covered may include elementary functional
analysis relevant to numerical analysis; numerical solution
of different or integral equations; analysis of stability
and convergence; numeric linear algebra including large-
scale systems; approximation theory.
Pr.: MATH 634 and MATH 655 .
MATH 862. Numerical Analysis II.
(3) Continuation of MATH 861. Pr.: MATH 861.
MATH 864. Theory of Ordinary Differential Equations I.
(3) The modern theory of ordinary differential
equations including general theory and the theory of linear
differential equations. Pr.: MATH 641, 722 and 731.
MATH 865. Theory of Ordinary Differential Equations II.
(3) Continuation of MATH 864 to include nonlinear
equations and differential equations in Banach spaces. Pr.:
MATH 864.
MATH 866. Partial Differential Equations I.
(3) Elliptic, parabolic, and hyperbolic partial
differential equations of the second order. First order
partial differential equations, characteristics. Linear
and nonlinear hyperbolic systems, nonlinear elliptic
equations.
Pr.: MATH 634 and MATH 641 .
MATH 867. Partial Differential Equations II.
(3) Continuation of MATH 866. Pr.: MATH 866.
MATH 871. General Topology II.
(3) Topological spaces and topological invariants;
continuous mappings and their invariants; perfect mappings;
topological constructs (product, quotient, direct and
inverse limit spaces). Pr.: MATH 700 and 701.
MATH 872. General Topology II.
(3) Compact spaces and compactification, uniform and
proximity spaces, metric spaces and metrization, topology
of Euclidean n-space, function spaces, complete spaces, introduction to
homotopy theory. Pr.: MATH 871.
MATH 881. Differentiable Manifolds I.
(3) Differentiable structures, tangent bundles, tensor
bundles, vector fields and differential equations, integral
manifolds, differential forms, Stokes' Theorem, DeRham
cohomology, Riemannian metrics, introduction to Lie groups,
topics in algebraic topology from a differentiable
viewpoint. Pr.: MATH 702.
MATH 882. Differentiable Manifolds II.
(3) Continuation of MATH 881. Pr.: MATH 881.
MATH 896. Topics in Mathematics.
(Var.) Pr.; Background of courses needed for
topic undertaken and consent of instructor.
MATH 897. Seminar in Mathematics Education.
(1-3)
MATH 898. Master's Research.
(Var.) Pr.: Consent of Instructor.
MATH 899. Master's Thesis.
(Var.)
MATH 910. Universal Algebra I.
(3) Topics include congruences, homomorphisms, direct
and subdirect products, varieties, Birkoff's theorem, and
the Mal'cev conditions. In addition, special topics will
be selected from Stone duality, ultra products, Boolean
products, and connections with model theory. Pr.: MATH 811.
MATH 911. Universal Algebra II.
(3) Continuation of MATH 910. Pr.: MATH 910.
MATH 914. Lattice Theory I.
(3) Posets, quantum logics,
orthocomplemented, orthomodular, and Boolean lattices; the
concepts of atomicity, completeness, reducibility,
modularity, M-symmetry, O-symmetry, distributivity,
algebraic coordinatization, and specific realization. Pr.:
Consent of instructor.
MATH 915. Lattice Theory II.
(3) Continuation of MATH 914.
MATH 920. Theory of Groups.
(3) Group representations and group characters,
transfer, signalizer functors, theory of pushing-up, groups
of Lie type, (B,N)-pairs, chamber systems and buildings,
sporadic simple groups, amalgam methods, Bass-Serre theory.
Pr.: MATH 811.
MATH 925. Group Representations and Character Theory I.
(3) The basic topics in representation theory are
covered: Schur's Lemma, irreducibility, class functions,
characters, orthogonality relations, Frobenius-Schur
theorem, induced characters and Frobenius reciprocity,
Mackey's theorem, Clifford's theorem, exceptional
characters and applications to group orders, generalized
characters and Brauer's characterizations of characters.
Pr.: MATH 811.
MATH 926. Group Representations and Character Theory II.
(3)
Depending on the interest of the students, topics may be
chosen from the following: modular representations,
Brauer's theory of blocks, characters of the linear groups,
homologically induced representations, representations of
complex Lie algebras. Pr.: MATH 925.
MATH 971. Algebraic Topology I.
(3) Homotopy groups, covering spaces, fibrations,
homology, general cohomology theory and duality, homotopy
theory. Pr.: MATH 702 and 811
MATH 972. Algebraic Topology II.
(3) Continuation of MATH 971. Pr.: MATH 971.
MATH 973 Low Dimensional Topology I---Geometric Topology.
(3) Manifolds, triangulations, differentiable
structures, wild vs. tame embeddings, the Jordan Curve
theorem, Schonflies Theorems, the classification of compact
surfaces, Dehn's Lemma, the Triangulation Theorem and
Huaptvermutung in dimensions 2 and 3, introduction to knot
theory: knot groups, the Alexander polynomial, and related
topics. Pr.: MATH 872 or 881.
MATH 974. Low Dimensional Topology I---Quantum Topology.
(3) Artin's braid groups, Markov's Theorem, the Jones
Polynomial and its generalizations, state-sum invariants of
knots and manifolds, skein-relations, quantum groups and
categories of tangles, topological quantum field theories.
Pr.: MATH 973 or consent of instructor.
MATH 991. Topics in Algebra.
(3) On sufficient demand. Selected topics in modern
algebra. May be repeated for credit. Pr.: Consent of
Instructor.
MATH 992 . Topics in Analysis.
(3) On sufficient demand. Selected topics in modern
analysis. May be repeated for credit. Pr.: Consent of
instructor.
MATH 993. Topics in Harmonic Analysis.
On sufficient demand. Selected topics in harmonic
analysis. May be repeated for credit. Pr.: Consent of
instructor.
MATH 994. Topics in Applied Mathematics.
(3) On sufficient demand. Selected topics in applied
mathematics analysis. May be repeated for credit. Pr.:
Consent of instructor.
MATH 995. Topics in Geometry.
(3) On sufficient demand. Selected topics in geometry.
May be repeated for credit. Pr.: Consent of instructor.
MATH 996. Topics in Topology.
(3) On sufficient demand. Selected topics in topology.
May be repeated for credit. Pr.: Consent of instructor.
MATH 997. Topics in Number Theory.
(3) On sufficient demand. Selected topics in number
theory. May be repeated for credit. Pr.: Consent of
instructor.
MATH 999. Ph.D. Research.
(Var.) I, II, S. Pr.: Sufficient training to carry on the
line or research undertaken and consent of instructor.
Pr.: Consent of instructor.