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.
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.
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.
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.
Matchings, edge colorings and vertex colorings of graphs. Connectivity, spanning trees, and disjoint paths in graphs. Cycles in graphs, embeddings. Planar graphs, directed graphs. Ramsey Theory, matroids, random graphs.
Generalities on modules, categories, and functors. The socle and the Jacobson radical of a module. Semisimple modules. Chain conditions on modules. The Hopkins-Levitzki Theorem. The Wedderburn-Artin Theorem and its applications toM linear representations of finite groups. The ?Hom? functors and exactness. Injective modules. Essential monomorphisms, injective hulls. Projective modules. Superfluous epimorphisms, projective covers. Indecomposable direct sum decompositions of modules. The Krull-Remak-Schmidt-Azumaya Theorem. Krull dimension and Goldie dimension of modules and lattices.
Topological spaces, subspaces, continuous functions, base for a topology, separation axioms, compactness, locally compact spaces, connectedness, path connectedness, finite product spaces, set theory and Zorn?s lemma, infinite product spaces, quotient spaces, homotopic paths, the fundamental group,induced homomorphisms, covering spaces, applications of the index, homotopic maps, maps into the punctured plane, vector fields, the Jordan curve theorem.
Fundamental group, Seifert-van Kampen theorem, CW complexes, covering spaces and deck transformations; simplicial and singular homology, homotopy invariance, exact sequences and excision, cellular homology, Mayer-Vietoris sequences; cohomology, universal coefficient theorem, cup product, Kunneth formula, orientation, Poincare duality.
Literature survey and presentation on a subject determined by the instructor.
Individual term project accompanied with the advisor.