**MATH-204. Graduate Tutorial**. 3 crs.
**MATH-205.** **Graduate Tutorial.** 3 crs.
**MATH-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.
**MATH-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.: MATH-208
**MATH-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.
**MATH-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.
**MATH-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.
**MATH-215. Number Theory II. **3crs. Continuation of MATH-214, including binary quadratic forms; algebraic numbers; rational number theory, irrationality and transcendence; Dirichlet's theorem; and the prime number theorem. Prereq: MATH-214.
**MATH-218. Mathematical Logic I.** 3 crs. Axiomatic and formal mathematics; consistency and completeness; recursive functions; undecidability and intuitionism. Prereq: Graduate status.
**MATH-219.** **Mathematical Logic ll.** 3 crs. Continuation of MATH-218, including model theory and first-order set theory. Prereq.: MATH-218.
**MATH-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.
**MATH-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.: MATH-220.
**MATH-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.
**MATH-223.** **Real Analysis II.** 3 crs. Continuation of MATH-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.
**MATH-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.: MATH-223.
**MATH-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.
**MATH-230. Complex Analysis II.** 3 crs. Continuation of MATH-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.
**MATH-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.
**MATH-232. Functional Analysis II.** 3 crs. Continuation of MATH- 231, including topological vector spaces; bounded operators; Banach algebras; spectra and symbolic calculus; Gelfand and Fourier transforms; and distributions. Prereq: MATH-231.
**MATH-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.
**MATH-235. Advanced Ordinary Differential Equations II.** 3 crs. Continuation of MATH-234. including qualitative theory stability and Liapunov functions; focal, nodal, and saddle points; limit sets: and the Poincare-Bendixson theorem. Prereq.: MATH-234.
**MATH-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.
**MATH-237. Partial Differential Equations II.** 3 crs. Continuation of MATH-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: MATH-236.
**MATH-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.
**MATH-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.
**MATH-241.** **Mathematical Statistics II.** 3 crs. Continuation of MATH-240 including point and interval estimations; hypotheses testing; decision functions; regression; non-parametric inferences; and analysis of categorical data.
**MATH-242. Stochastic Processes.** 3 crs. Continuation of MATH- 241 including conditional probability, conditional expectation, normal processes, convariance, stationary processes, renewal equations, and Markov chains. Prereq.: MATH-241.
**MATH-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.
**MATH-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.: MATH-243.
**MATH-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.
**MATH-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.: MATH-245.
**MATH-247. Numerical Analysis I.** 3 crs. Numerical solutions of ordinary and partial differential equations including convergence stability, and consistence of schemes. Prereq.: Graduate status.
**MATH-248. Numerical Analysis II.** 3 crs. Continuation of MATH- 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.:MATH-247.
**MATH-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.
**MATH-251. Topology II.** 3 crs. Compactifications, Baire spaces, function spaces, topological vector spaces.
**MATH-252. Algebraic Topology I.** 3 crs. Homotopy, covering spaces, fibrations, polyhedra, simplicial complexes, simplicial and singular homology, and Eilenberg-Steenrod axioms. Prereq.: MATH-251.
**MATH-253. Algebraic Topology II.** 3 crs. Continuation of MATH- 252 including products; cohomology; homotopy, CW spaces, obstructions; sheaf theory; and spectral sequences. Prereq.: MATH-252.
**MATH-259.** **Differential Geometry I.** 3 crs. Differential manifolds, tensors, affine connections, and Riemannian manifolds. Prereq.: Graduate status.
**MATH-260.** **Differential Geometry II.** 3 crs. Continuation of MATH-259 inclucing Riemannian geometry; submanifolds; variations of the length integral; the Morse index theorem; complex manifolds; Hermitian vector bundles; and characteristic classes. Prereq.: MATH-259.
**MATH-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.: MATH-229, MATH-230.
**MATH-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.
**MATH-280. Topics in History of Mathematics**. 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.
**MATH-290. Reading in Mathematics**. 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.
**MATH-300.** **Graduate Seminar.** 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.
**MATH-350. M.S. Thesis.** 6 crs. Topic to be selected by mutual consent of the student and the instructor. Prereq.: Consent of graduate chairperson.
**MATH-410,419. Topics in Algebra. **3 crs. ea. Further topics in algebra to be selected by the instructor. Prereq.: Consent of instructor.
**MATH-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.
**MATH-450, 459. Topics in Applied Mathematics**. 3 crs. ea. Further topics in applied mathematics to be selected by the instructor. Prereq.: Consent of instructor.
**MATH-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.
**MATH-500, 501. Graduate Seminar**. 3 crs. ea. Topics to be selected by the instructor. Prereq.: Consent of instructor.
**MATH-550. Ph.D. Dissertation. **12 crs. Prereq.: Consent of Ph.D. adviser. |