Andy qingyun Zeng

At UPenn

Notes (updating in progress)

At Cambridge

This is a second course in functional analysis, which covers roughly part I&III in Rudin's Functional analysis and Ch 3-7 of Allan's Introduction to Banach spaces and algebras. The topics inlcude
1. Hahn–Banach Theorem on the extension of linear functionals. Locally convex spaces. Duals of the spaces L^p(μ) and C(K).
2.The Radon–Nikodym Theorem and the Riesz Represen- tation Theorem. Weak and weak-* topologies. Theorems of Mazur, Goldstine, Banach–Alaoglu. Reflexivity and local reflexivity. Hahn–Banach Theorem on separation of convex sets. Extreme points and the Krein–Milman theorem. Partial converse and the Banach–Stone Theorem.
3. Banach algebras, elementary spectral theory. Commutative Banach algebras and the Gelfand representation theorem.
4.Holomorphic functional calculus. Hilbert space operators,
5. C∗-algebras. The Gelfand–Naimark theorem. Spectral theorem for commutative C∗-algebras. Spectral theorem and Borel functional calculus for normal operators.
6.Some additional topics. For example, the Fr ́echet–Kolmogorov Theorem, weakly compact subsets of L1(μ), the Eberlein–Sˇmulian and the Krein–Sˇmulian theorems, the Gelfand–Naimark–Segal construction

Lecture Notes (updating in progress)
Reiview of measure theory
Resumè on Hilbert Spaces and Spectral Theory

The purpose of this course is to introduce some techniques and methodologies in the mathematical treat- ment of Partial Differential Equations (PDE). The theory of PDE is nowadays a huge area of active research, and it goes back to the very birth of mathematical analysis in the 18th and 19th century. It is at a crossroad with physics and many areas of pure and applied mathematics.
The course begins with an introduction to four prototype linear equations: Laplace’s equation, the heat equation, the wave equation and Schr ̈odinger’s equation. Emphasis will be given to the modern functional analytic techniques relying on the notion of Cauchy problem and estimates rather than explicit solutions, although the interaction with classical methods (e.g. the fundamental solution, Fourier representations) will be discussed. The following basic unifying concepts will be studied: well-posedness, energy estimates, elliptic regularity, characteristics, propagation of singularities, maximum principle. The course will end with a discussion of some of the open problems in PDE.
1. Introduction: from ODEs to PDEs
2. The Cauchy-Kovalevskaya theorem
3. Elliptic equations of second orders
4. Hyperbolicity: transport equations and wave equations

This course is intended as an introduction to the theory of elliptic partial differential equations. Elliptic equations play an important role in geometric analysis and a strong background in linear elliptic equations provides a foundation for understanding other topics including minimal submanifolds, harmonic maps, and general relativity. We will discuss both classical and weak solutions to elliptic equations, considering when solutions to the Dirichlet problem exist and are unique and considering the regularity of solutions. This involves establishing maximum principles, Schauder estimates, and other estimates on solutions. As time permits, we will discuss other topics including the De Giorgi-Nash theory, which can be used to prove the Harnack inequality and establish Hölder continuity for weak solutions, and quasilinear elliptic equations.
This course essentially covers first 8 chapters in Elliptic Partial Differential Equations of Second Order by David Gilbarg and Neil S. Trudinger.

1. Introduction to elliptic PDEs
2. Weak maximum principle
3. Strong maximum principle
4. A priori estimates
5. Hölder spaces
6. Interior Schauder estimates
7. Global Schauder estimates
8. Dirichlet problems
9. Regularity theory
10. Equations in divergence form, weak solutions
11. Applications of Lax-Milgram
12. DeGiogi-Nash theory
13. Consequence of DeGiogi-Nash

Here is a slightly more elementry notes (involves discussion about Laplace/Poisson equations, harmonic functions, etc.): Elliptic PDEs (Michealmas 2007) given by Prof. Neshan Wickramasekera who is also my Director of Studies at the Churchill College

Another good reference is Elliptic partial differential equations. Courant Lecture Notes, Vol. 1 (2011) by Qing Han and Fanghua Lin.

We discuss the local and global theories for quasilinear wave equations and their applications to physical theories including fluid mechanics and general relativity. The following topics will be covered:
1. Quantitative behaviour of solutions to the linear wave equation in Minkowski spacetime
2. Energy methods and the local theory for quasilinear wave equations
3. Application in general relativity: local well-posedness of the Einstein equations
4. Examples of subcritical nonlinear wave equations
5. The null condition and the small-data global theory for quasilinear wave equations 6. Application in fluid mechanics: formation of shocks in spherical symmetry
7. Application in general relativity: stability of the Minkowski spacetime

Lecture notes and Review on Fourier analysis and Sobolev theory
A good reference is Lecture notes in analysis by S. Klainerman
Part of the course is drawn from the following (recent) research papers:
1. H. Lindblad and I. Rodnianski, Global stability of Minkowski spacetime in harmonic gauge, Annals of Math (2010)
2. G. Holzegel, S. Klainerman, J. Speck and W. Wong, Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations: An Overview

Notes (updating in progress)

The official notes