Courses
Real Analysis - AMCS 235
This is a basic introductory graduate level course on Real Analysis. It starts with a review of notions connected with the real analysis of continuous functions including the Ascoli-Arzela Theorem on compactness of sequences of continuous functions, and the Stone-Weierstrass Theorem for the approximation of continuous functions. The main body of the course will be the theory of measure and integration. The last part of the course deals with various topics in Real Analysis, like Differentiation, approximation of $L^p$ functions, and Fourier Series. The course intends to prepare the students with the main background needed in modern Advanced Mathematics related to the fields of Real Analysis and approximations of functions.
Topics
- Review of continuous functions, Metric spaces, Sequences of functions, uniform convergence, the Weierstrass approximation theorem, Compactness in metric spaces, the Ascoli-Arzela theorem.
- Lebesgue integral: $\sigma$-algebras, measurable functions, measure, integrable functions, $L^p$ spaces, modes of convergence, decomposition of measures (Radon-Nikodym), Generation of measures (Lebesgue, Lebesgue-Stieljes),
- Further topics in Real Analysis: Product measures (Tonelli, Fubini), Differentiation, functions of bounded variation, Approximation via convolutions.
Functional Analysis - AMCS 338
This course covers topics in Real Analysis and Functional Analysis and their applications. It starts with a review of the theory of metric spaces, the Lp spaces, and the approximation of real functions. It proceeds to the theory of Hilbert spaces, Banach spaces and the main theorems of functional analysis, linear operators in Banach and Hilbert spaces, the spectral theory of compact, self-adjoint operators and its application to the theory of boundary value problems and linear elliptic partial differential equation. It concludes with approximation methods in Banach spaces.
Topics
- Banach Spaces - Function Spaces ($L^p$-space, $L^\infty$, $BC(\Omega)$, Hölder, Sobolev spaces), Approximation in $L^p$ spaces
- Sobolev spaces, Embedding and Compactness theorems
- Dual Spaces (dual of continuous functions, dual of $L^p$), Functionals on other spaces. Hahn-Banach, Dual of $W^{1,p}$, The weak and weak-$\star$ topologies.
- Baire category, Uniform Boundedness, Open Mapping, Closed Graph
- Bounded operators, Integral operators, Inverses and Neumann Series, Adjoint, Kernel, Compact Operators, Green's operator
- Hilbert Spaces, Examples, Riesz Representation, Orthonormal bases, Projections, Spectral theory for compact self-adjoint operators
- Fourier Series (Dirichlet kernel, Fejer kernel), Fourier Integral.
Nonlinear Partial Differential Equations in Continuum Physics - AMCS 394
The objective of this course is to introduce students to the modern theory of Nonlinear Partial Differential Equations in Continuum Mechanics and Mathematical Physics. We present topics of current interest in these subjects, their interconnections, and also how various mathematical questions reflect or are motivated from ideas in continuum physics or mathematical physics where these models originate.
Topics
- Mathematical derivation of continuum mechanics, the second law of thermodynamics and stability
- Entropy solutions for conservation laws, existence theory of classical solutions, existence theory of entropy solutions for scalar conservation laws
- Variational derivation of the equations of mechanics, existence theory of polyconvex elastodynamics
- The Monge-Kantorovich problem, Wasserstein distance, the method of minimizing movements, the JKO-scheme for gradient flows.
- Navier-Stokes equations, Viscoelasticity, Microscopic models for suspensions