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)