Mathematics Degrees

SPECIAL TOPICS IN MATHEMATICS

MATH 702 Initial Value Problems in the Space of Generalized Analytic Functions.(3-0)3
Initial value problems in Banach spaces, scales of Banach spaces, solution of IVP in scales of Banach spaces, the classical Cauchy-Kowalewski theorem, the Holmgren theorem, basic properties of generalized analytic functions, IVP with generalized analytic initial data.
MATH 710 Low Dimensional Topology (3-0)3
Preliminaries: Vector bundles, connections, characteristic classes, Hodge Theory. Spin Geometry of four-manifolds: Spin Structure, Dirac operator, Atiyah-Singer Index Theorem. Seiberg Written Module Space. Compactness of module space. Seiberg-Witten Invariants. Topology of four manifolds: Intersection forms of four manifolds, realizability of unimodular, symmetric bilinear forms as intersection forms.
MATH 711 Impulsive Differential Equations (IDE) (3-0)
General Description of IDE: Description of mathematical model. Systems with impulses at fixed times. Systems with impulses at variable times. Discontinuous dynamical systems. Impulsive oscillator. Linear Systems of IDE: General properties of solutions. Stability of solutions. Adjoint systems, Perron theorem. Linear Hamiltonian systems of IDE. Stability of Solutions of IDE: Stability criterion based on first order approximation. Stability in systems of IDE with variable times of impulsive effect. Direct Lyapunov method. Periodic and Almost Periodic Systems of IDE: Nonhomogeneous linear periodic systems. Nonlinear periodic systems. Almost periodic functions and sequences. Almost periodic IDE. Integral Sets of Systems of IDE: Bounded solutions of nonhomogeneous linear systems. Integral sets of quasilinear systems with hyperbolic linear part and with non-fixed moments of impulse actions.
MATH 712 Large Cardinals and Combinatorial Principles in Set Theory (3-0)3
Filter and ideals in partial orders, trees, Ramsey theory. Generalized Continuum Hypothesis, Martin’s axiom. Closed unbounded sets, stationary sets. Principle, Suslin hypothesis, Kurepa hypothesis. Inaccessible, ineffable, compact and measurable cardinals.
MATH 738 Model Theory (3-0) 3
Propositional and first-order logic. The compactness theorem and consequences. Theories that are: complete, model-complete, quantifier-elliminable, categorical. Structures that are: prime, minimal, universal, saturated, stable.
MATH 738 Coding Theory (3-0) 3
Coding constructions, Bounds on the sizes of codes, sphere packing bound, Plotkin bound, Singleton bound, Griesmer bound, Johnson Bound, self-dual codes, codes over rings, codes and invariant theory, quasi-cyclic codes, finite geometry and coding theory, duality issues in coding theory, duality and product codes, covering radius of some classes of codes, orthogonal arrays and coding theory, decoding of codes, algebraic decoding and list decoding, complexity issues in coding theory, low density codes, turbo codes, frameproof codes, watermarking, sequences in coding theory and cryptology.
MATH 741 Analytic Function Spaces and Their Operators (3-0)3
Operators on Hilbert and Banach spaces, Bergman, Bloch, Besov, and Hardy spaces, functions of bounded mean oscillation, Carleson measures, duality, Berezin transform, Toeplitz, Hankel, and composition operators.
MATH 742 Topics in Partial Differential Equations (3-0)3
Sobolev spaces: Weak Derivatives, Approximation by Smooth functions, Extensions, Traces, Sobolev Inequalities, The Space H^-1. Second-Order Elliptic Equations: Weak Solutions, Lax-Milgram Theorem, Energy Estimates, Fredholm Alternative, Regularity, Maximum Principles, Eigenvalues and Eigenfunctions. Linear Evolution Equations: Second-order Parabolic equations (Weak Solutions, Regularity, Maximum principle), Second-order Hyperbolic Equations (Weak Solutions, Regularity, Propagation of disturbances), Hyperbolic Systems of First-order Equations, Semigroup theory.