Part
V Graduate Courses
800
Level Courses
| 810 |
Error-Correcting
Codes Spring
Prerequisite:
Math 411 or MTH 414 or MTH 415
Block codes, maximum likelihood decoding,
Shannon's theorem. Generalized Reed-Solomon
codes, modification of codes, subfield codes.
Alternant and Goppa codes, cyclic codes and
BCH codes. |
| 818 |
Algebra
I (Core Course) Fall
Prerequisite:
MTH 411 or equivalent
Group theory: Sylow theory, permutation groups,
Jordon-Holder theory, Abelian groups, free
groups. Ring theory: algebra of ideals, unique
factorization, polynomial rings, finitely
generated modules over PID's. |
| 819 |
Algebra
II (Core Course) Spring
Prerequisite:
MTH 818
Modules and vector spaces, projective modules,
tensor algebra. Fields and Galois groups,
algebraic and transcendental numbers, non-commutative
rings. The Jacobson radical, the structure
of semisimple rings with the descending chain
condition. |
| 822 |
Calculus
on Manifolds Fall
Prerequisites:
MTH 421
Recommended
Background: MTH 414 or equivalent
A modern treatment of differential and integral calculus on manifolds in Euclidean space. Differential forms, generalized Stokes’ Theorem. Interaction among linear algebra, topology, and analysis |
| 828 |
Real
Analysis I (Core Course) Fall
Prerequisites:
MTH 421, MTH 461 or equivalent
Lebesgue
measure on real line, general measure theory.
Convergence theorems, Lusin's theorem, Egorov's
theorem, Lp-spaces, Fubini's theorem. Functions
of bounded variation, absolutely continuous
functions, Lebesgue differentiation theorem. |
| 829 |
Complex
Analysis I (Core Course) Spring
Prerequisites:
MTH 421, MTH 425 or equivalent
Cauchy theorem, identity principle, Liouville's
theorem, maximum modulus theorem. Cauchy formula,
residue theorem, Rouche's theorem. Casorati-Weierstrass
theorem, Arzela-Ascoli theorem. Conformal
mapping, Schwarz lemma, Riemann mapping theorem. |
| 840 |
Chaos
and Dynamical Systems Spring
Prerequisites:
MTH 320 and MTH 414
Recommended
Background: MTH 441, some experience with
mathematical software such as Mathematica
or Matlab
Chaotic or random motions in differential
and difference equations. |
| 841 |
Boundary
Value Problems I Fall
Prerequisites:
MTH 414, MTH 421 or equivalent
Boundary value problems for ordinary and partial
differential equations. Sturm-Liouville theory.
Fourier series and generalized Fourier series.
Eigenfunction expansions. Variational methods.
Fredholm integral equations. Potential theory.
Green's functions. Special functions. Integral
transform methods. |
| 842 |
Boundary
Value Problems II Spring
Prerequisite:
MTH 841
Continuation of MTH 841. |
| 843 |
A
Survey of Industrial Mathematics
Fall
Prerequisites:
MTH 414, MTH 421 or equivalent
The course has three objectives: to survey
mathematics of particular importance to industry,
to gain experience in team project report
generation, and to gain experience in oral
presentation of technical reports. |
| 844 |
Projects
in Industrial Mathematics Spring
Prerequisite: Approval of Department
Representatives from Industry or Government
will come to campus to pose problems of interest
to
their unit. Students will divide into teams
of 2 or 3 to tackle one of the posed problems.
By the end of term each student team will present
both a written and oral report of their findings
to the industrial or governmental unit that
posed the problem. Teams will be advised by
a faculty member plus a liaison from the unit
that posed the problem. |
| 848 |
Ordinary
Differential Equations (Core Course)
Fall
Prerequisites: MTH 414, MTH 421 or equivalent
Existence and uniqueness theorems. Theory of
linear differential equations. Floquet theory.
Stability theory and Poincare-Bendixson theory.
Green's functions and boundary value problems. |
| 849 |
Partial
Differential Equations (Core Course)
Spring
Prerequisites: MTH 414, MTH 421 or equivalent
Cauchy-Kowalewski theorem. Characteristics.
Initial-boundary value problems for parabolic
and hyperbolic equations. Energy methods,
boundary value problems for elliptic equations,
potential theory. Green's function, maximum
principles, Schauder's method. |
| 850 |
Numerical
Analysis I (Core Course) Fall
Prerequisites: MTH 414, MTH 421 or equivalent
Convergence and error analysis of methods
in computational linear algebra. Direct methods
for linear systems, matrix decompositions,
iterative methods. Eigenvalue programs: the
power, QR, Lanczos and other methods. Singular
and generalized eigenvalues. |
| 851 |
Numerical
Analysis II (Core Course) Spring
Prerequisite: MTH 850
Interpolation theory and approximation of
functions. Numerical solutions of nonlinear
equations. Numerical integration methods. |
| 852 |
Numerical
Methods for Ordinary Differential Equations
(Core Course) Fall
Prerequisite: MTH 414, 421
Linear multi-step methods and single step
nonlinear methods for initial value problems.
Consistency, stability and convergence. Finite
difference, finite element, shooting methods
for boundary value problems. |
| 864 |
Geometric
Topology Spring
Prerequisites: MTH 421
Topology of surfaces and higher dimensional
manifolds, studied from combinatorial, algebraic
or differential viewpoints. |
| 868 |
Geometry
and Topology I (Core Course) Fall
Prerequisite: MTH 411, MTH 421 or equivalent
Fundamental group and covering spaces, vanKampen's
theorem. Homology theory, differentiable manifolds,
vector bundles, transversality, calculus on
manifolds. Differential forms, tensor bundles,
deRham theorem, Forbenius theorem. |
| 869 |
Geometry
and Topology II (Core Course) Spring
Prerequisite: MTH 868
Continuation of MTH 868. |
| 870 |
Set
Theory and Foundations of Mathematics
Spring
Prerequisite: MTH 411 or MTH 421 or equivalent
Zermelo-Fraenkel axioms. Cardinals and ordinals
and their arithmetics. Axiom of choice and
maximal principles. Transfinite induction
and recursion, consistency and independence. |
| 880 |
Combinatorics
Fall
Prerequisite: MTH 411 or MTH 482 or equivalent
Enumerative combinatorics, recurrence relations,
generating functions, asymptotics, applications
to graphs, partially ordered sets, generalized
Möebius inversions, combinatorial algorithms. |
| 881 |
Graph
Theory Spring
Prerequisite: MTH 880
Graph theory, connectivity, algebraic and
topological methods. Networks, graph algorithms,
Hamiltonian and Eulerian graphs, extremal
graph theory, random graphs. |
| 890 |
Reading
in Mathematics Every Semester
Prerequisite: Approval of Department
Individualized study for Master’s level
students. |
900
Level Courses
The
descriptions of the 900 level course given below are
a list of topics covered in the past.
| 910 |
Commutative
Algebra I Fall of odd numbered years
Prerequisite: MTH 819
Noetherian rings and modules, localization
and tensor products, primary decomposition,
Krull dimension, graded rings and modules,
Hilbert's Nullstellensatz, integral extension,
discrete valuation rings, Dedekind domains. |
| 911 |
Commutative
Algebra II Spring of even numbered
years
Prerequisite: MTH 910
Ext and Tor, regular sequences, Cohen-Macauley
rings, regular rings, Gorenstein rings, completion,
modules of differentials, Cohen's structure
theorems. |
| 912 |
Group
Theory I Fall of even numbered years
Prerequisite: MTH 819
Permutation groups, solvable and nilpotent
groups, simple groups. Representation and
character theory. Extension theory and cohomology
groups. |
| 913 |
Group
Theory II Spring of odd numbered
years
Prerequisite: MTH 912
Groups of Lie type, linear groups, locally
finite groups, free groups and free products,
the subgroup theorems. |
| 914 |
Lie
Groups and Algebras I Fall of odd
years
Prerequisite: MTH 819
Nilpotent and semisimple algebras, the adjoint
representation, root spaces, Weyl groups,
Dynkin diagrams, classification of simple
algebras. |
| 915 |
Lie
Groups and Algebras II Spring of
even numbered years
Prerequisite: MTH 914
Weights, symmetric spaces, groups of Lie type,
finite groups of Lie type, Lang's theorem. |
| 916 |
Algebraic Geometry I
Prerequisite: MTH 818 and MTH 819
Affine and projective algebraic varieties and their properties. Morphism and singularieties. Schemes and coherent sheaves. Sheaf cohomology and other related topics.
|
| 917 |
Algebraic Geometry II
Prerequisite: MTH 916
Continuation of MTH 916. |
| 920 |
Functional
Analysis I Spring
Prerequisite: MTH 828
Hilbert spaces: Riesz representation theorem,
Parseval's identity, Riesz-Fisher theorem,
Fourier series operators. Banach spaces: Hahn-Banach
theorem, open mapping and closed graph theorems,
Banach-Steinhaus theorem. |
| 921 |
Functional
Analysis II Fall of even-numbered
years
Prerequisites: MTH 829 and MTH 920
Topological vector spaces, convexity, Krein-Milman
theorem, Banach algebras, operators on Banach
spaces, spectral theorem, C*-algebras. |
| 922 |
Harmonic
Analysis Fall of odd-numbered years
Prerequisites: MTH 829 and MTH 920
Fourier series, mean and pointwise convergence,
conjugate functions, Fourier transform, Plancherel
theorem, Paley-Wiener theorem, interpolation
of operators, Hausdorff-Young theorem. |
| 928 |
Real
Analysis II Fall
Prerequisites: MTH 828
Positive Borel measure, complex measure, Riesz
representation theorem, Radon-Nikodym theorem,
Lebesgue decomposition theorem. Differentiable
transformations and change of variables, differentiation
of measures, maximal functions. |
| 929 |
Complex
Analysis II Spring
Prerequisites: MTH 828, MTH 829
Phragmen-Lindeloff method. Hadamard's theorem,
Runge's theorem, Weierstrass factorization
theorem, Mittag-Leffler theorem, and Picard's
theorem. Poisson integrals, Harnack's inequality,
Dirichlet problem. Hp-spaces, Blaschke products. |
| 930 |
Riemannian
Geometry I Fall
Prerequisite: MTH 869
Riemannian metrics, connections, curvature,
geodesics. First and second variation, Jacobi
fields, conjugate points. Rauch comparison
theorems, Hodge theorem, Bochner technique,
spinors. Further topics on curvature or submanifold
theory. |
| 931 |
Riemannian
Geometry II Spring
Prerequisite: MTH 930
Continuation of MTH 930. |
| 935 |
Complex
Manifolds I Fall of odd numbered
years
Prerequisites: MTH 829, MTH 869
Riemann surfaces, Serre duality, Riemann-Roch
theorem. Weierstrass points, Abel's theorem,
Plucker formulas. Hermitian metrics, connections,
curvature, Hodge theorem. Kaehler metrics,
Kodaira vanishing theorem, Chern classes. |
| 936 |
Complex
Manifolds II Spring of even numbered
years
Prerequisite: MTH 935
Continuation of MTH 935. |
| 940 |
Applied
Analysis I Fall
Prerequisite: MTH 828
Sobolev spaces, trace theorem, imbedding theorems,
sectorial forms. Linear elliptic boundary
and eigenvalue problems. |
| 941 |
Applied
Analysis II Spring
Prerequisite: MTH 940
Fixed point theorems. Variational methods.
Applications to nonlinear integral and elliptic
differential equations. Semigroup theory. |
| 942 |
Foundations
of Applied Mathematics I Fall
Prerequisites: MTH 841, MTH 842 or equivalent
Modeling in classical applied mathematics.
Newtonian and continuum mechanics. Special
mathematical techniques. |
| 943 |
Foundations
of Applied Mathematics II Spring
Prerequisite: MTH 848
Asymptotic series. Fundamental techniques.
Perturbation of integrals. Stationary phase.
Steepest descent. Perturbation of eigenvalue
problems. Periodic and almost periodic solutions.
Averaging. Multiple scales. Singular perturbation.
Boundary layer theory. Matched asymptotic
expansions. Applications. |
| 950 |
Numerical
Methods for Partial Differential Equations
I Spring odd-numbered years
Prerequisite: MTH 852 strongly recommended
Analysis of finite difference methods for
ordinary and partial differential equations.
Evolutionary problems, stability of initial
value problems, ADI methods, wave equations
and first order hyperbolic systems. Elliptic
problems, approximation of boundary conditions,
multigrid methods. Applications to structural
analysis. |
| 951 |
Numerical
Methods of Partial Differential Equations
II Spring even-numbered years
Prerequisite: 950
Finite element methods, variational formulations,
Sobolev spaces, finite element spaces, a priori
estimates, collocation methods, applications
to computational fluid dynamics. |
| 960 |
Algebraic
Topology I Fall
Prerequisite: MTH 869
Cohomology, products, duality, basic homotopy
theory, bundles, obstruction theory, spectral
sequences, characteristics classes, and other
related topics. |
| 961 |
Algebraic
Topology II Spring
Prerequisite: MTH 960
Continuation of MTH 960. |
| 990 |
Reading
in Mathematics Every Semester
Prerequisite: Approval of Department
Individualized study for doctoral level students. |
Others
MTH 991: Special Topics in Algebra
MTH 992: Special Topics in Analysis
MTH 993: Special Topics in Geometry
MTH 994: Special Topics in Applied Mathematics
MTH 995: Special Topics in Numerical Analysis and
Operations Research
MTH 996: Special Topics in Topology
MTH 998: Special Topics in Combinatorics and Graph
Theory
MTH
999: Doctoral Dissertation Research
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