General Elective

MATH 503 / APPLIED MATHEMATICS I**Sınıf:** **Credit:** 3**Precondition:**

Linear algebra: Vector and inner product spaces, linear operators, eigenvalue problems; Vector calculus: Review of differential and integral calculus, divergence and Stokes' theorems. Ordinary differential equations: Linear equations, Sturm-Liouville theory and orthogonal functions, system of linear equations; Methods of mathematics for science and engineering students.

MATH 504 / NUMERICAL METHODS I**Sınıf:** **Credit:** 3**Precondition:**

A graduate level introduction to matrix-based computing. Stable and efficient algorithms for linear equations, least squares and eigenvalue problems. Both direct and iterative methods are considered and MATLAB is used as a computing environment.

MATH 506 / NUMERICAL METHODS II**Sınıf:** **Credit:** 3**Precondition:**

Development and analysis of numerical methods for ODEs, an introduction to numerical optimization methods, and an introduction to random numbers and Monte Carlo simulations. The course starts with a short survey of numerical methods for ODEs. The related topics include stability, consistency, convergence and the issue of stiffness. Then it moves to computational techniques for optimization problems arising in science and engineering. Finally, it discusses random numbers and Monte Carlo simulations. The course combines the theory and applications (such as programming in MATLAB) with the emphasis on algorithms and their mathematical analysis.

MATH 521 / ALGEBRA I**Sınıf:** **Credit:** 4**Precondition:**

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**Sınıf:** **Credit:** 4**Precondition:**

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 / ALGEBRATIC NUMBER THEORY**Sınıf:** **Credit:** 4**Precondition:**

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**Sınıf:** **Credit:** 4**Precondition:**

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**Sınıf:** **Credit:** 4**Precondition:** MATH. 533 or consent of the instructor

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 531 / REAL ANALYSIS I**Sınıf:** **Credit:** 4**Precondition:**

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**Sınıf:** **Credit:** 4**Precondition:** MATH. 531 or consent of the instructor

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

MATH 533 / COMPLEX ANALYSIS I**Sınıf:** **Credit:** 4**Precondition:**

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**Sınıf:** **Credit:** 4**Precondition:** MATH. 533 or consent of the instructor

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

MATH 535 / FUNCTIONAL ANALYSIS**Sınıf:** **Credit:** 4**Precondition:** MATH. 532 or consent of the instructor

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

MATH 536 / APPLIED FUNCTIONAL ANALYSIS I**Sınıf:** **Credit:** 4**Precondition:**

Review of linear operators in Banach spaces and Hilbert spaces; Riesz ·Schauder theory; fixed point theprems of Banach and Schauder; semigroups of linear operators; Sobolev spaces and basic embedding theorems; boundary - value problems for elliptic equations; eigenvalues and eigenvectors of second order elliptic operators; initial boundary-value problems for parabolic and hyperbolic equations.

MATH 537 / APPLIED FUNCTIONAL ANALYSIS II**Sınıf:** **Credit:** 4**Precondition:**

Existence and uniqueness of solutions of abstract evolutionary equations. Global non-existence and blow up theorems. Applications to the study of the solvability and asymptotic behavior of solutions of initial boundary-value problems for reaction diffusion equations, Navier-Stokes equations, nonlinear Klein-Gordon equations and nonlinear Schrödinger equations.

MATH 538 / DIFFERENTIAL GEOMETRY**Sınıf:** **Credit:** 4**Precondition:**

Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature

MATH 541 / PROBABILITY THEORY**Sınıf:** **Credit:** 4**Precondition:**

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

MATH 544 / STOCHASTIC PROCESS AND MARTINGALES**Sınıf:** **Credit:** 4**Precondition:** MATH. 541 or consent of the instructor

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**Sınıf:** **Credit:** 4**Precondition:**

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 differentialequations, 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 550 / SELECTED TOPICS IN MATHEMATICS**Sınıf:** **Credit:** 3**Precondition:**

MATH 551 / SELECTED TOPICS IN ANALYSIS I**Sınıf:** **Credit:** 3**Precondition:**

MATH 552 / SELECTED TOPICS IN ANALYSIS II**Sınıf:** **Credit:** 3**Precondition:**

MATH 563 / ALGEBRIC CODING THEORY**Sınıf:** **Credit:** 4**Precondition:**

Error correcting coding theory. Hamming, Golay, cyclic, 2-error correcting BCH codes, Reed-Solomon, Convolutional, Reed-Muller and Preparata codes. Interaction of codes and combinatorial designs.

MATH 564 / COMBINATORIAL DESIGN THEORY**Sınıf:** **Credit:** 4**Precondition:**

Balanced incomplete block designs, group divisible designs and pairwise balanced designs. Resolvable designs, symmetric designs and designs having cyclic automorphisms. Pairwise orthogonal latin squares. Affine and projective geometries. Embeddings and nestings of designs.