Courses
ENGL 500
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 506
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 525
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 531
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.
MATH 534
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.
MATH 504
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 522
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 528
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 533
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 536
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 503
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 521
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 527
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 532
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 535
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.
