GRADUATE COURSES
214-204. Graduate Tutorial. 3 crs.
214-205. Graduate Tutorial. 3 crs.
214-208. Introduction to Modern Algebra
I. 3 crs. Groups, subgroups, cyclic groups, quotient groups,
Lagranges Theorem, permutation groups, homomorphism and isomorphism
theorems, Cayley's theorem, rings, subrings, ideals, fields,
homomorphism and isomorphism theorems.
214-209. Introduction to Modern Algebra
II. 3 crs. Sylow's theorems for finite groups, p-groups,
abelian groups, group action on sets, domains, prime and maximal
ideals, unique factorization domain. Prereq.: 214-208
214-210. Modern Algebra I. 3 crs. Groups,
group actions on sets, structure of finitely generated abelian
groups, category theory, exact sequences, rings, P.l.D's, modules,
projective, injective and free modules.
214-211. Modern Algebra ll. 3 crs. Structure
of finitely generated modules over P.l.D's, fields, Galois theory,
vector spaces and classical groups G(n.R), algebras over a field.
214-214. Number Theory I. 3 crs. Congruences;
primitive roots and indices; quadratic residues; number-theoretic
functions; primes; sums of squares; Pell's theorem; and rational
approximations.
214-215. Number Theory II. 3crs. Continuation
of 214-214, including binary quadratic forms; algebraic numbers;
rational number theory, irrationality and transcendence; Dirichlet's
theorem; and the prime number theorem. Prereq: 214-214.
214-218. Mathematical Logic I. 3 crs.
Axiomatic and formal mathematics; consistency and completeness;
recursive functions; undecidability and intuitionism. Prereq:
Graduate status.
214-219. Mathematical Logic ll. 3 crs.
Continuation of 214-218, including model theory and first-order
set theory. Prereq.: 214-218.
214-220. Introduction to Analysis I. 3
crs. Logical connectives, qualifiers, mathematical proof, basic
set operations, relations, functions, cardinality, axioms of
set theory, natural number and induction, ordered fields. The
completeness axiom, topology of the reals, Heine-Borel theorem,
convergence Bolzano-Weierstrass theorem, limit theorems, monotone
sequence and Cauchy sequence, subsequences, infinite series and
convergence criterion, convergence tests, power series.
214-221. Introduction to Analysis II. 3
crs. Limits of functions, continuity, uniform continuity, differentiation,
the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's
theorem, Riemann Integral, properties of the Riemann Integral,
the fundamental theorem of calculus, pointwise and uniform convergence,
applications of uniform convergence. Prereq.: 214-220.
214-222. Real Analysis I. 3 crs. Topology
of n-dimension Euclidean space, functions of bounded variation,
absolute continuity, differentiation, Riemann-Stieltjes integration.
Lebesgue measure and integration theory; Lp spaces, separability,
completeness, duality, L spaces and the Riesz- Fischer theorem.
214-223. Real Analysis II. 3 crs. Continuation
of 214-222. Abstract measures, mappings of measure spaces, integration
sets and product spaces, the Fubini, Tonelli and Radon- Nikodyn
theorems, the Riesz representation theorem, Haar measures on
locally compact groups.
214-224. Applications of Analysis. 3
crs. Operators defined by convolution, maximal functions, Fourier
transform in classical spaces of functions, distributions; harmonic
and subharmonic functions; applications to P.D.E and probability
theory, Bochner theorem and central limit theorem. Prereq.: 214-223.
214-229. Complex Analysis I. 3crs. Linear
fractional transformations, conformal mapping, holomorphic functions,
Cauchy's theorem (including the homotopic version), properties
of holomorphic functions, the argument principle, residues, power
series, Laurent series, meromorphic functions.
214-230. Complex Analysis II. 3 crs.
Continuation of 214-229. Montel's theorem, normal families, Riemann
Mapping Theorem Picard's theorem, Mittag-Leffler's theorem, Weierstrass'
theorem, simply connected domains, Riemann surfaces, meromorphic
functions on compact Riemann surfaces.
214-231. Functional Analysis I. 3 crs.
Banach spaces; the dual topology and weak topology; the Hahn-Banach,
Krein- Milman and Alaoglu theorems; the Baire category theorem;
the closed graph theorem; the open mapping theorem; the Banach-Steinhaus
theorem; elementary spectral theory; and differential equations.
Prereq: Graduate status.
214-232. Functional Analysis II. 3 crs.
Continuation of 214- 231, including topological vector spaces;
bounded operators; Banach algebras; spectra and symbolic calculus;
Gelfand and Fourier transforms; and distributions. Prereq: 214-231.
214-234. Advanced Ordinary Differential
Equatlons I. 3 crs. Existence, uniqueness, and representation
of solutions of ordinary differential equations; periodic solutions,
singular points, oscillation theorems, and boundary value problems.
Prereq.: Graduate status.
214-235. Advanced Ordinary Differential
Equations II. 3 crs. Continuation of 214-234. including
qualitative theory stability and Liapunov functions; focal,
nodal, and saddle points; limit sets: and the Poincare-Bendixson
theorem. Prereq.: 214-234.
214-236. Partial Differential Equations
I. 3 crs. First-order partial differential equations, method
of characteristics; Cauchy-Kovalevskaya theorem; second-order
equations, classification existence, and uniqueness results;
formulation of some of the classical problems of mathematical
physics. Prereq.: Graduate status.
214-237. Partial Differential Equations
II. 3 crs. Continuation of 214-236, showing applications
of functional analysis to differential equations including
distributions, generalized functions, semigroups of operators,
the variational method, and the Riesz-Schauder theorem. Prereq:
214-236.
214-239. Fourier Series and Boundary Value
Problems. 3 crs. Fourier analysis, Bessel's inequality,
Parseval's relation, Hilbert spaces, compact operators, eigenfunction
expansions, and Sturm-Liouville problems. Prereq.: Graduate
status.
214-240. Mathematics Statistlcs I. 3
crs. Probability; random variables; distributions; moment generating
functions: limit theorems; parametric families of distributions;
sam- pling distributions; sufficiency; and likelihood functions.
Prereq.: Graduate status.
214-241. Mathematical Statistics II. 3
crs. Continuation of 214-240 including point and interval estimations;
hypotheses testing; decision functions; regression; non-parametric
inferences; and analysis of categorical data.
214-242. Stochastic Processes. 3 crs.
Continuation of 214- 241 including conditional probability, conditional
expectation, normal processes, convariance, stationary processes,
renewal equations, and Markov chains. Prereq.: 214-241.
214-243. Dynamical System I. 3 crs.
Systems of differential equations existence, uniqueness and continuity
of solutions, linear systems, including constant coefficients,
asymptotic behaviour, periodic coefficients; stability of linear
and almost linear systems, the Poincare-Bendix theorem; global
stability (Lyapunov method); differential equations and dynamical
systems - including closed orbits structural stability and 2-dimensional
flow. Prereq.: Graduate status.
214-244. Dynamical Systems II. 3 crs.
Introduction to Chaos; local bifurcations: - center manifolds,
normal forms, equilibria: and periodic orbits; averaging and
perturbation: - Poincare maps, Hamiltonian In systems and Melnikov's
methods; hyperbolic sets, symbolic dynamics and strange attractors;
Smale Horseshoe, invariant sets, Markov partitions and statistical
properties; global bifurcations; - Lorentz and Hopf bifurcations;
Chaos in discrete dynamical system. Prereq.: 214-243.
214-245. Methods of Applied Mathematics
I. Principles and techniques of modern applied mathematics
with case studies involving deterministic problems, random
problems, and Fourier analysis. Prereq.: Graduate status.
214-246. Methods of Applied Mathematics
II.. 3 crs. Asymptotic sequences and series, special functions,
asymptotic expansions of integrals and solutions of ordinary
differential equations, and singular perturbations. Prereq.:
214-245.
214-247. Numerical Analysis I. 3 crs.
Numerical solutions of ordinary and partial differential equations
including convergence stability, and consistence of schemes.
Prereq.: Graduate status.
214-248. Numerical Analysis II. 3 crs.
Continuation of 214- 247 including numerical methods for partial
differential equations using functional analysis techniques;
the Lax equivalence theorem; Courant-Friedrich Levy condition;
Kreiss matrix theorem; and finite element methods. Prereq.:214-247.
214-250. Topology I. 3 crs. Topological
basis, continuous, open closed topological maps, product spaces,
connectedness, compactness, local connectedness, local compactness;
identitication and weak topologies, separation axioms, metrizable
spaces, covering spaces, homotopy, fundamental groups.
214-251. Topology II. 3 crs. Compactifications,
Baire spaces, function spaces, topological vector spaces.
214-252. Algebraic Topology I. 3 crs.
Homotopy, covering spaces, fibrations, polyhedra, simplicial
complexes, simplicial and singular homology, and Eilenberg-Steenrod
axioms. Prereq.: 214-251.
214-253. Algebraic Topology II. 3 crs.
Continuation of 214- 252 including products; cohomology; homotopy,
CW spaces, obstructions; sheaf theory; and spectral sequences.
Prereq.: 214-252.
214-259. Differential Geometry I. 3
crs. Differential manifolds, tensors, affine connections, and
Riemannian manifolds. Prereq.: Graduate status.
214-260. Differential Geometry II. 3
crs. Continuation of 214-259 inclucing Riemannian geometry; submanifolds;
variations of the length integral; the Morse index theorem; complex
manifolds; Hermitian vector bundles; and characteristic classes.
Prereq.: 214-259.
214-270. Several Complex Variables I. 3
crs. Basic facts about holomorphic functions; zero sets of holomorphic
functions, analytic sets and Weierstrass Preperation theorem;
domains of holomorphy, convexity w.r.t holomorphic curves plurisubharmonic
functions, pseudoconvexity Levi problem; holomorphic convexity,
Stein domains and complete Reinhardt domains; differential forms;-
complex manifolds, complex structure on TpM, almost complex structures,
exterior derivatives forms of the (p,q)-type, cohomology. Prereq.:
214-229, 214-230.
214-271. Several Complex Variables II..
3 crs. Holomorphic convexity, Stein domains and complete Reinhardt
domains; differential forms; complex manifolds, complex manifolds,
complex structure on TpM, almost complex structures, exterior
derivative forms of the (p,q)-type, cohomology.
214-280. Topics in History of Mathematics.
3 crs. Topic to be selected by the instructor. Prereq.: Graduate
status.
214-290. Reading in Mathematics. 3 crs.
Topic to be selected by the instructor. Prereq.: Graduate status.
214-300. Graduate Seminar. 3 crs. Topic
to be selected by the instructor. Prereq.: Graduate status.
214-350. M.S. Thesis. 6 crs. Topic to
be selected by mutual consent of the student and the instructor.
Prereq.: Consent of graduate chairperson.
214-410,419. Topics in Algebra. 3 crs.
ea. Further topics in algebra to be selected by the instructor.
Prereq.: Consent of instructor.
214-430,439. Topics in Analysis. 3 crs.
ea. Further topics in real and complex analysis to be selected
by the instructor. Prereq.: Consent of instructor.
214-450, 459. Topics in Applied Mathematics.
3 crs. ea. Further topics in applied mathematics to be selected
by the instructor. Prereq.: Consent of instructor.
214-470,479. Topics in Topology and Geometry.
3 crs. ea. Further topics in geometry and topology to be selected
by the instructor. Prereq.: Consent of instructor.
214-500, 501. Graduate Seminar. 3 crs.
ea. Topics to be selected by the instructor. Prereq.: Consent
of instructor.
214-550. Ph.D. Dissertation. 12 crs.
Prereq.: Consent of Ph.D. adviser.
