### General electives can be selected from all the courses offered at Koç University.

MATH 544 / STOCHASTIC PROCESS AND MARTINGALES
Sınıf: Credit: 4Precondition: 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: 4Precondition:

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: 3Precondition:

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

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

MATH 553 / SELECTED TOPICS IN FOUNDATIONS OF MATHEMATICS
Sınıf: Credit: 3Precondition:

MATH 554 / SELECTED TOPICS IN ALGEBRA AND TOPOLOGY
Sınıf: Credit: 3Precondition:

MATH 555 / SELECTED TOPICS IN PROBABILITY AND STATISTICS
Sınıf: Credit: 3Precondition:

MATH 556 / SELECTED TOPICS IN DIFFERENTIAL GEOMETRY
Sınıf: Credit: 3Precondition:

MATH 557 / SELECTED TOPICS IN DIFFERENTIAL EQUATIONS
Sınıf: Credit: 3Precondition:

MATH 558 / SELECTED TOPICS IN APPLIED MATHEMATICS
Sınıf: Credit: 3Precondition:

MATH 559 / SELECTED TOPICS IN COMBINATORICS
Sınıf: Credit: 3Precondition:

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

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: 4Precondition:

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.

MATH 565 / GRAPH THEORY
Sınıf: Credit: 4Precondition:

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.

MATH 566 / TOPICS IN MODULE AND RING THEORY
Sınıf: Credit: 3Precondition:

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.

MATH 571 / TOPOLOGY
Sınıf: Credit: 4Precondition:

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.

MATH 572 / ALGEBRAIC TOPOLOGY
Sınıf: Credit: 4Precondition:

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.

MATH 579 / READINGS IN MATHEMATICS
Sınıf: Credit: 1Precondition:

Literature survey and presentation on a subject determined by the instructor.

MATH 584 / SELECTED TOPICS IN ANALYSIS&ALGEBRATIC NUBERS
Sınıf: Credit: 3Precondition: