MATH 501 Analysis (3 0 3)
General measure and integration theory. General convergence theorems. Decomposition of measures. Radon-Nikodym theorems. Outer measure. Caratheodory extension theorem. Product measures. Fubini’s theorem. Riesz representation theorem.

MATH 503 Algebra I (3 0 3)
Groups: generalities, groups acting on a set, Sylow theorems, free group, direct product and sums. Rings: generalities, commutative rings, principle ideal domains, unique factorization domains, Euclidean domains. Noetherian rings. Hilbert’s theorem. Field of fractions. Localization.

MATH 504 Algebra II (3 0 3)
Galois theory. Categories and functions: generalities, additive, abelian categories, Yoneda’s dilemma. Module categories: definitions, projective, injective modules, semi-simple rings, modules over Noetherian rings and principal ideal domains, Morita theory. Homological methods: the functors Ext, Tor, (co-) homology, derived categories, stable categories, applications to cohomology of groups, schemes.

MATH 505 Differentiable Manifolds (3 0 3)
Differentiable manifolds. Smooth mappings. Tangent, cotangent bundles. Differential of a map. Submanifolds. Immersions. Imbeddings. Vector fields, tensor fields. Differential forms. Orientation on manifolds. Integration on manifolds. Stoke’s theorem.

MATH 513 Representation Theory of Finite Groups (3 0 3)
Ring theoretic preliminaries. Group representations and their characters. Characters, integrality and application to the structure theory of finite groups. Product of characters. Induced characters. Reduction and extension of characters. Brauer’s theorem on characterization of characters.

MATH 515 Commutative Algebra (3 0 3)
Rings and ideals. Modules. Rings and modules of fractions. Preliminary decomposition. Integral dependence.

MATH 535 Topology (3 0 3)
Topological spaces. Neighbourhoods. Basis. Subspace topology, product and quotient topologies. Compactness. Tychonoff’s Theorem. Heine-Borel theorem. Separation properties. Urysohn’s Lemma and Tietze Extension theorem. Stone-Cech compactification. Alexandroff one point compactification. Convergence of sequences and nets. Connectedness. Metrizability. Complete metric spaces. Baire’s theorem.

MATH 537 Algebraic Topology I (3 0 3)
Fundamental group, covering spaces. Singular homology: homotopy invariance, homology long exact sequence, Mayer-Vietoris sequence, excision; applications of homology. Homology and attaching cells, CW-complexes. Definition of simplicial homology and its relation to singular homology. Cohomology groups.

MATH 538 Algebraic Topology II (3 0 3)
Homology and cohomology with coefficients. Universal coefficient theorems for homology and cohomology. Künneth formula. Cup product and cross product. Cohomology algebra. Homotopy groups.

MATH 545 Riemannian Geometry (3 0 3)
Review of differentiable manifolds and tensor fields. Riemannian metrics, the Levi-civita connectıons. Geodesics and exponential map. Curvature tensor, sectional curvature. Ricci tensor, scalar curvature. Riemannian submanifolds. Gauss and Codazzi equations.

MATH 555 Theory of Functions of a Complex Variable (3 0 3)
Analytic functions. Singular points and zeros. The argument principle. Conformal mappings. Riemann mapping theorem. Mittag-Lefler theorem. Infinite products. Canonical products. Analytical continuation. Elementary Riemann surfaces.

MATH 558 Introduction to Functions of Several Complex Variables (3 0 3)
Holomorphic functions. Comparison of one and several variables. Domains of holomorphy, subharmonicity, pseudoconvexity; invariant metrics, holomorphic maps, Stein and CR-manifolds, integral formulas.

MATH 566 Banach Spaces (3 0 3)
Isomorphic theory of Banach spaces. Isometric theory of Banach spaces. Structure theory and basic properties of classical Banach spaces. Banach lattices. Positive operators.

MATH 569 Functional Analysis I (3 0 3)
Notion of topological vector space, normed linear spaces. Baire category theorem and its consequences: open mapping, closed graph theorems, uniform boundedness prınciple. Convexity and separation. Hahn-Banach and Krein-Milman theorems. Duality theory: weak topologies, Bipolar and Alaoglu-Bourbaki theorems.

MATH 570 Functional Analysis II (3 0 3)
Compact operators. Fredholm operators, normal operators. Spectral theory of normal operators. Elementary theory of commutatiive Banach algebras. Commutative C*-algebras and Gelfand representation theorem.

MATH 581 Numerical Analysis I (3 0 3)
Gaussian elimination and its variants. Sensitivity of linear systems. Orthogonal matrices and the least squares problem. Eigenvalues and eigenvectors. The singular value decomposition.

MATH 582 Numerical Analysis II (3 0 3)
Interpolation. Approximation of functions. Numerical differentiation and integrations. Root finding methods.

MATH 583 Partial Differential Equations (3 0 3)
Cauchy-Kowalevski theorem. Linear and quasilinear first order equations. Existence and uniqueness theorems for second order elliptic, parabolic and hyperbolic equations. Correctly posed problems. Green’s function.

MATH 584 Hilbert Space Techniques in Partial Differential Equations (3 0 3)
Equations without solutions. Some notions from Hilbert spaces, weak solutions, a necessary condition, Friedrich mollifier, elliptic operators. Fourier transforms. Hypoelleptic operators. Proof of regularity for constant and variable coefficient cases. Weak solutions for Cauchy problem. Properties of hyperbolıc operators. Existence of solutions.

MATH 587 Ordinary Differential Equations (3 0 3)
Basic theory: initial value problems. Linear systems: linear homogenous and non homogeneous systems. Linear systems with constant and periodic coefficients. Oscillation theory. Stability: definitions of stability and its boundedness. Lyapunov functions. Lyapunov stability and instability. Domain of attraction. Perturbation of linear systems. Stability of an equilibrium point. The stable manifold. Stability of periodic solutions. Asymptotic equivalence.

MATH 589 Numerical Solutions of Partial Differential Equations (3 0 3)
Finite difference method, stability, convergence and error analysis. Initial and boundary conditions, irregular boundaries. Parabolic equations. Explicit and implicit methods, stability analysis, error reduction, variable coefficients, derivative boundary conditions, solution of tridiagonal systems. Elliptic equations, iterative methods, rate of convergence. Hyperbolic equations. The Lax-Wendroff method, variable coefficients, systems of conservation laws, stability. Finite volume method.

MATH 590 Graduate Seminar (Non-credit)
Presentation and discussion of current issues and works by graduate students in their relevant fields.

MATH 591 Special Studies (0 4 0)
This course is required for students who are en rolled in “Thesis” course.

MATH 599 Thesis (Non-credit)
Directed independent research on a specific topic approved by the student’s adviser.