Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:Math 533

Credits 3

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.

Prerequisite:Math 528

Credits 4

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

Prerequisite:None

Credits 4

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.

Prerequisite:Math 531

Credits 4

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.

Prerequisite:None

Credits 4

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

Prerequisite:MATH 533

Credits 4

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.

Prerequisite:MATH 531 and MATH 532

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:MATH 541

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:MATH 551

Credits 4

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.

Prerequisite:None

Credits 4

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

Prerequisite:MATH 571

Credits 3

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.

Prerequisite:None

Credits 3

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

Prerequisite:None

Credits 3

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

Prerequisite:MATH 576 or consent of the instructor

Credits 0

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

Prerequisite:None

Credits 3

Credits 3

Credits 3

Credits 3

Credits 3

Credits 3

Credits 3

Credits 3

Credits 3

Credits 1

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

Prerequisite:None

Credits 4

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

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:MATH 611

Credits 4

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.

Prerequisite:None

Credits 4

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

Prerequisite:None

Credits 4

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

Prerequisite:None

Credits 4

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.

Prerequisite:MATH 627

Credits 4

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.

Prerequisite:None

Credits 4

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.

Prerequisite:None

Credits 3

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.

Prerequisite:MATH 535 or equivalent

Credits 4

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.

Prerequisite:MATH 552 or consent of instructor.

Credits 4

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.

Prerequisite:MATH 643

Credits 4

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.

Prerequisite:MATH 552 or consent of instructor.

Credits 4

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.

Prerequisite:None

Credits 4

Recent developments in pure mathematics.

Prerequisite:None

Credits 4

Recent developments in pure mathematics.

Prerequisite:None

Credits 4

Recent developments in applied mathematics.

Prerequisite:None

Credits 4

Recent developments in applied mathematics.

Prerequisite:None

Credits 4

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.

Prerequisite:None