# Course Descriptions

MATH 521

Algebra I

Credits4

Free groups, group actions, group with operators, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss's lemma, PID and UFD, localization and local rings,chain conditions, Jacobson radical.

MATH 522

Algebra II

Credits4

Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions. Modules, direct sums, free modules, sums and products, exact sequences, morphisms, Hom and tensor functors, duality, projective, injective and flat modules, simplicity and semisimplicity, density theorem, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.

MATH 525

Algebraic Number Theory

Credits4

Valuations of a field, local fields, ramification index and degree, places of global fields, theory of divisors, ideal theory, adeles and ideles, Minkowski's theory, extensions of global fields, the Artin symbol.

MATH 527

Number Theory

Credits4

Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, p-adic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and L-functions, ABC-conjecture, heights, class numbers for quadratic fields, a sketch of Wiles' proof.

MATH 528

Analytic Number Theory

Credits4

Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet L-functions, the prime number theorem, Polya- Vinogradov inequality, the large sieve, average results on the distribution of primes.

MATH 529

Analytic Number Theory II

Credits3

The prime number theorem for arithmetic progressions. Sums over primes, exponential sums. The large sieve, Bombieri-Vinogradov theorem,Selberg’s sieve. Results on the distribution of primes.

MATH 531

Real Analysis I

Credits4

Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.

MATH 532

Real Analysis II

Credits4

Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, category and uniform boundedness theorem, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.

MATH 533

Complex Analysis I

Credits4

Review of the complex number system and the topology of C, elementary properties and examples of analytic functions, complex integration, singularities, maximum modulus theorem, compactness and convergence in the space of analytic functions.

MATH 534

Complex Analysis II

Credits4

Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.

MATH 535

Functional Analysis

Credits4

Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Smulian theorem, Eberlein-Smulian theorem, linear operators on Banach spaces.

MATH 541

Probability Theory

Credits4

An introduction to measure theory, Kolmogorov axioms, independence, random variables, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, product measures and joint probability, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems.

MATH 544

Stochastic Processes and Martingales

Credits4

Stochastic processes, stopping times, Doob-Meyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, Radon-Nikodym theorem, Brownian motion, reflection principle, law of iterated logarithms.

MATH 545

Mathematics of Finance

Credits4

From random walk to Brownian motion, quadratic variation and volatility, stochastic integrals, martingale property, Ito formula, geometric Brownian motion, solution of Black-Scholes equation, stochastic differential equations, Feynman-Kac theorem, Cox-Ingersoll-Ross and Vasicek term structure models, Girsanov's theorem and risk neutral measures, Heath-Jarrow-Morton term structure model, exchange-rate instruments.

MATH 551

Partial Differential Equations I

Credits4

Existence and uniqueness theorems for ordinary differential equations, continuous dependence on data. Basic linear partial differential equations : transport equation, Laplace's equation, diffusion equation, wave equation. Method of characteristics for non-linear first-order PDE's, conservation laws, special solutions of PDE's, Cauchy-Kowalevskaya theorem.

MATH 552

Partial Differential Equations II

Credits4

Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for second-order elliptic equations, maximum principles, second-order linear parabolic and hyperbolic equations, methods for non-linear PDE's, variational methods, fixed point theorems of Banach and Schauder.

MATH 571

Topology

Credits4

Fundamental concepts, subbasis, neighborhoods, continuous functions, subspaces, product spaces and quotient spaces, weak topologies and embedding theorem, convergence by nets and filters, separation and countability, compactness, local compactness and compactifications, paracompactness, metrization, complete metric spaces and Baire category theorem, connectedness.

MATH 572

Algebraic Topology

Credits4

Basic notions on categories and functors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and their homology.

MATH 575

Differentiable Manifolds

Credits3

Differentiable manifolds, smooth maps, submanifolds, vectors and vector fields, Lie brackets, Lie Groups, Lie group actions, integral curves and flows, Lie algebras, Lie derivative, Killing fields, differential forms, Integration.

MATH 576

Riemannian Geometry

Credits3

Differentiable manifolds, vectors and tensors, riemannian metrics, connections, geodesics, curvature, jacobi fields, riemannian submanifolds, spaces of constant curvature.

MATH 577

Complex Manifolds

Credits3

Complex Manifolds, Kahler and Calabi-Yau Manifolds, Homology and Cohomology, Fiber Bundles, Connections on Fiber Bundles, Characteristic Classes, Index Theorems.

MATH 579

Graduate Seminar

Credits0

Presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.

MATH 581

Selected Topics in Analysis I

Credits3

MATH 582

Selected Topics in Analysis II

Credits3

MATH 583

Selected Topics in Foundations of Mathematics

Credits3

MATH 584

Selected Topics in Algebra and Topology

Credits3

MATH 585

Selected Topics in Probability and Statistics

Credits3

MATH 586

Selected Topics in Differential Geometry

Credits3

MATH 587

Selected Topics in Differential Equations

Credits3

MATH 588

Selected Topics in Applied Mathematics I

Credits3

MATH 589

Selected Topics in Combinatorics

Credits3

MATH 590

Readings in Mathematics

Credits1

Literature survey and presentation on a subject to be determined by the instructor.

MATH 601

Measure Theory

Credits4

Fundamentals of measure and integration theory, Radon-Nikodym Theorem, Lp spaces, modes of convergence, product measures and integration over locally compact topological spaces.

MATH 611

Differential Geometry I

Credits4

Survey of differentiable manifolds, Lie groups and fibre bundles, theory of connections, holonomy groups, extension and reduction theorems, applications to linear and affine connections, curvature, torsion, geodesics, applications to Riemannian connections, metric normal coordinates, completeness, De Rham decomposition theorem, sectional curvature, spaces of constant curvature, equivalence problem for affine and Riemannian connection.

MATH 612

Differential Geometry II

Credits4

Submanifolds, fundamental theorem for hypersurfaces, variations of the length integral, Jacobi fields, comparison theorem, Morse index theorem, almost complex and complex manifolds, Hermitian and Kaehlerian metrics, homogeneous spaces, symmetric spaces and symmetric Lie algebra, characteristic classes.

MATH 623

Integral Transforms

Credits4

Fourier transforms, exponential, cosine and sine, Fourier transform in many variables, application of Fourier transform to solve boundary value problems, Laplace transform, use of residue theorem and contour integration for the inverse of Laplace transform, application of Laplace transform to solve differential and integral equations, Fourier-Bessel and Hankel transforms for circular regions, Abel transform for dual integral equations.

MATH 624

Numerical Solutions of Partial Differential and Integral Equations

Credits4

Parabolic differential equations, explicit and implicit formulas, elliptic equations, hyperbolic systems, finite elements characteristics, Volterra and Fredholm integral equations.

MATH 627

Optimization Theory I

Credits4

Fundamentals of linear and nonlinear optimization theory. Unconstrained optimization, constrained optimization, saddlepoint conditions, Kuhn-Tucker conditions, post-optimality, duality, convexity, quadratic programming, multistage optimization.

MATH 628

Optimization Theory II

Credits4

Design and analysis of algorithms for linear and non-linear optimization. The revised simplex method, algorithms for network problems, dynamic programming, quadratic programming techniques, methods for constrained nonlinear problems.

MATH 631

Algebraic Topology I

Credits4

Basic notions on categories and functions, the fundamental groups, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and homology of simplicial complexes.

MATH 632

Algebraic Topology II

Credits4

Singular homology, exact sequences, the Mayer-Vietoris exact sequence, the Lefschetz fixed-point theorem, cohomology, cup and cap products, duality theorems, the Hurewicz theorem, higher homotopy groups.

MATH 635

An Introduction to Nonlinear Analysis

Credits3

Calculus in Banach spaces. Implicit function theorems. Degree theories. Fixed Point Theorems. Bifurcation theory. Morse Lemma. Variational methods. Critical points of functionals. Palais-Smale condition. Mountain Pass Theorem.

MATH 643

Stochastic Processes I

Credits4

Survey of measure and integration theory, measurable functions and random variables, expectation of random variables, convergence concepts, conditional expectation, stochastic processes with emphasis on Wiener processes, Markov processes and martingales, spectral representation of second-order processes, linear prediction and filtering, Ito and Saratonovich integrals, Ito calculus, stochastic differential equations, diffusion processes, Gaussian measures, recursive estimation.

MATH 644

Stochastic Processes II

Credits4

Tightness, Prohorov's theorem, existence of Brownian motion, Martingale characterization of Brownian motion, Girsanov's theorem, Feynmann-Kac formulas, Martingale problem of Stroock and Varadhan, applications to mathematics of finance.

MATH 645

Mathematical Statistics

Credits4

Review of essentials of probability theory, subjective probability and utility theory, statistical decision problems, a comparison game theory and decision theory, main theorems of decision theory with emphasis on Bayes and minimax decision rules, distribution and sufficient statistics, invariant statistical decision problem, testing hypotheses, the Neyman-Pearson lemma, sequential decision problem.

MATH 660

Advanced Number Theory

Credits4

Basic algebraic number theory; number fields, ramification theory, class groups, Dirichlet unit theorem; zeta and L-functions; Riemann, Dedekind zeta functions, Dirichlet, Hecke L-functions, primes in arithmetic progressions, prime number theorem; cyclotomic fields, reciprocity laws, class field theory, ideles and adeles, modular functions and modular forms.

MATH 680

Seminar in Pure Mathematics I

Credits4

Recent developments in pure mathematics.

MATH 681

Seminar in Pure Mathematics II

Credits4

Recent developments in pure mathematics.

MATH 682

Seminar in Applied Mathematics I

Credits4

Recent developments in applied mathematics.

MATH 683

Seminar in Applied Mathematics II

Credits4

Recent developments in applied mathematics.

MATH 690

M.S. Thesis

MATH 699

Guided Research

Credits4

Research in the field of Mathematics, by arrangement with members of the faculty; guidance of doctoral students towards the preparation and presentation of a research proposal.

MATH 790

Ph.D. Thesis