Pınar Ongan

Places,valuation rings and discrete valuations of a function field; the rational function field; divisors, Weil different adeles, genus; Riemann-Roch Theorem and its consequences; extensions of function fields, ramification, Hurwitz genus formula; constant field extensions, Galois extensions, Kummer and Artin-Schreier extensions.

Characterization of finite fields, roots of irreducible polynomials, traces, norms, and bases, representation of elements of finite fields. Order of polynomials, irreducible polynomials and their construction. Factorization of polynomials. Linear recurring sequences. Introduction to applications of finite fields; algebraic coding theory and cryptology.

Euclidean and unitary vector spaces, normal forms for endomorphisms of Euclidean and unitary spaces (e.g., orthogonal, self-adjoint, normal endomorphisms), linear optimization, simplex algorithm, introduction to coding theory.

Modules. Fields, extension fields, Galois theory. Categories and functors.

Introduction to group theory. Isomorphism theorems. Permutation groups and Cayley’s theorem. Conjugacy classes. Lagrange’s theorem and the Sylow theorems. principle ideal domains. Polynomial ring.

Differentiable manifolds. Smooth mappings; differential of a map. Implicit and inverse function theorems. Submanifolds. Vector fields. Differential forms. Orientation on manifolds. Integration on manifolds. Stokes’ theorem.

Lebesgue measure and integration on the line. Convergence theorems. General measure and integration. Lp spaces. Decomposition of measures. Radon Nikodym theorem. Product measures and Fubini’s theorem.

Introduction to Algebraic Geometry, Euclidean and non-Euclidean Geometry.

Student presentation on an area not covered in classes under the supervision of an instructor.

Review of vector spaces, normed vector spaces, lP and LP spaces, Banach and Hilbert spaces, duality, bounded linear operators and functionals.