First order and first degree equations. The homogeneous differential equations with constant coefficients. The methods of undetermined coefficients, reduction of order, and variation of parameters. The Cauchy-Euler equation. Series solutions. Systems of linear differential equations. Applications.

**Pre-Requisites:**
MATH102

Review of free, projective, and injective modules, direct limits. Watt's theorems. Flat modules. Localization. Noetherian, semisimple, Von Neumann regular, hereditary, and semi-hereditary rings. Homology, homology functors, derived functors. Ext. and Tor. homological dimensions, Hilbert syzygy theorem.

**Pre-Requisites:**
MATH551

Basics of rings and ideals. Rings of fractions, integral dependence, valuation rings, discrete valuation rings, Dedekind domains, fractional ideals. Topologies and completions, filtrations, graded rings and modules. Dimension theory.

**Pre-Requisites:**
MATH551

Basic principles of option pricing, binomial model, the Black-Scholes model, arbitrage, complete and incomplete markets, trading strategies, European options, American options. Topics include Risk-neutral Valuation, options on stock Indices, currencies, futures, the Greek letters, Interest Rate Derivatives, Black-Scholes PDE and formula

**Pre-Requisites:**
MATH564

Advanced topics not covered in regular courses.

None

Prerequisite: Graduate Standing

Prerequisite: Graduate Standing

Variable Content. Advanced mathematical topics not covered in regular courses.

Variable Content. Advanced mathematical topics not covered in regular courses.

Preparation and defense of the MS thesis. This is an NP/NF/IP course.

**Pre-Requisites:**
MATH599*

**Co-Requisites:**
MATH 599

Selected topics from: Groups, rings, modules, and general algebraic systems.

Review of the Fredholm and Hilbert-Schmidt theories for Fredholm integral equations of the second kind. Kernels with weak and logarithmic singularities. Singular integral equations of the first and second kind (Abel, Carleman, and Wiener- Hopf equations). Nonlinear integral equations (Volterra and Hammerstein equations). Application of the Schauder fixed point theorem. Nonlinear eigenvalue problems and integral equation methods for nonlinear boundary-value problems. Nonlinear singular integral equations. Applications to engineering and physics (the nonlinear oscillator, the airfoil equation, nonlinear integral equations arising the radiation transfer, hydrodynamics, water waves, heat conduction, elasticity, and communication theory).

**Pre-Requisites:**
MATH535 Or MATH535

None

None

Prerequisite: Admission to Ph.D. Program

Prerequisite: Admission to Ph.D. Program

Variable Content. Advanced mathematical topics not covered in regular courses.

Variable Content. Advanced mathematical topics not covered in regular courses.

None