Semester 2 Syllabus
Back to Int PhD course structure
1. Advanced Calculus & Geometry ( MTH-105.4 )
Review of multivariable differentiation from real analysis course, implicit function theorem and its applications, integral calculus, line/surface Integrals, Green/divergence/Stokes theorem.
Curves (in plane and space) - local properties, Curves - global properties.
Surfaces, first/second fundamental form.
2. Functional Analysis ( MTH-106.4 )
Normed Linear Spaces, Banach spaces, continuous linear functional, dual spaces.
Hahn-Banach theorem, applications of Baire Category Theorem: open mapping, closed graph and uniform boundedness theorems.
Weak, weak * topologies, Banach- Alaoglu Theorem, reflexivity.
Hilbert spaces, Reisz representation theorem, Adjoint, Hermitian, Normal, unitary operators, Compact operators. Spectral theorem for Compact Hermitian operators
3. Measure Theory ( MTH-107.4 )
Construction of Lebesgue measure on ℝ using inner measure and outer measure, σ - algebras, abstract measures, measurable functions.
Lebesgue integration, Fatou, Monotone and dominated convergence theorems, product measures, Fubini theorem.
Lebesgue decomposition, Radon-Nikodym theorem, Lp spaces, duality, Lebesgue differentiation theorem, absolutely continuous functions, monotone functions.
Convolution and Fourier transforms.
4. Numerical Analysis ( MTH-108.4 )
Round off Errors and Computer Arithmetic.
Interpolation: Lagrange Interpolation, divided differences, Hermite interpolation, splines.
Numerical differentiation, Richardson extrapolation.
Numerical integration: Trapezoidal, Simpson’s, Newton-Cotes, Gauss quadrature, Romberg integration, multiple integrals.
Solutions of Linear Algebraic equations: Direct Methods, Gauss elimination, Pivoting, matrix factorizations.
Iterative Methods: Matrix norms, Jacobi and Gauss-Seidel methods, relaxation Methods.
Computation of Eigenvalues and Eigenvectors: Power Method, Householder's method, QR algorithm.
Numerical solutions of nonlinear algebraic equations: bisection, secant and Newton's method.
Zeros of polynomials, Horner and Muller methods, equations in higher dimensions.
Ordinary differential equations, initial value problems: Euler method, higher order methods of the Runge-Kutta type. Multi-step methods, Adams-Bashforth, Adams- Moulton methods, Systems of ODEs.
Ordinary differential equations, boundary value problems, shooting methods, finite differences, Rayleigh-Ritz methods.
Fast Fourier transforms.
5. Non-credit course on Python programming ( MTH-109.4 )
(Will appear in transcript with S/U grade)