General Elective

MASE 595 / MS THESIS**Sınıf:** **Credit:** 0**Precondition:**

Independent research

MECH 541 / MANUFACTURING OF COMPOSITE MATERIALS**Sınıf:** **Credit:** 3**Precondition:**

Introduction to composites; composite manufacturing processes; transport equations for composite processing; process modeling; review of numerical methods and programming in MATLAB; advanced thermoplastic- and thermoset-matrix fiber-reinforced composites; liquid composite molding processes; designing, modeling, simulation and hands-on manufacturing of composite parts using resin transfer molding (RTM) and vacuum infusion (VI) processes; on-line control and data acquisition.

MECH 543 / COMPUTER INTEGRATED MANUFACTURING AND AUTOMATION**Sınıf:** **Credit:** 3**Precondition:**

Product realization systems from Computer Aided Design (CAD) to Computer Aided Manufacturing (CAM). Manufacturing Automation. Modern sensors in manufacturing. Computer control of manufacturing systems. Computer Numerical Control (CNC) machine tools. Machining processes. Rapid prototyping. Fundementals of industrial robotics.

MECH 562 / MICRO AND NANOFABRICATION**Sınıf:** **Credit:** 3**Precondition:** MECH. 202 or consent of the instructor

Fabrication and characterization techniques for micro and nano electro mechanical systems, MEMS & NEMS (including: microlithography; wet & dry etching techniques; physical & chemical vapor deposition processes; electroplating; bonding; focused ion beams; top-down approaches - electron-beam lithography, SPM, soft lithography - ; bottom-up techniques based on self-assembly). Semiconductor nanotechnology. Nanotubes & nanowires. Biological systems. Molecular electronics.

TEAC 500 / TEACHING EXPERIENCE**Sınıf:** **Credit:** 0**Precondition:**

Provides hands-on teaching experience to graduate students in undergraduate courses. Reinforces students' understanding of basic concepts and allows them to communicate and apply their knowledge of the subject matter.

ENGL 500 / ACADEMIC WRITING**Sınıf:** **Credit:** 0**Precondition:**

The following objectives will be met through extensive reading, writing and discussion both in and out of class.Build a solid background in academic discourse, both written and spoken. Improve intensive and extensive critical reading skills. Foster critical and creative thinking. Build fundamental academic writing skills including summary, paraphrase, analysis, synthesis. Master cohesiveness as well as proper academic citation when incorporating the work of others.

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 / ALGEBRAIC 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.