Notes

Convex Analysis Reading Course

I am organizing a reading course about Convex Analysis in Spring 2019. This is a graduate-student led reading course with guidance from Braxton Osting. We are currently reading the book "Convex Analysis" by R. T. Rockafellar. We meet Fridays from 1-3PM in JWB 308. The lectures are intended to be approximately 1.5 hours, follow by 30 minutes for questions and discussion.

Section 1, 2, 3, 4, 5: Chee Han [Notes]
Section 6, 7: Rebecca
Section 10, 11, 12: Nathan [Notes (Sec10)] [Notes (Sec11)] [Notes (Sec12)]
Section 13: Justin
Section 23, 24, 25, 26: RK
Section 27:

Sobolev Space Reading Course

Together with Nathan Willis, I am co-organising a reading course about Sobolev space in Fall 2018. This is a graduate-student led reading course with guidance from Fernando Guevara Vasquez, Christel Hohenegger, and Akil Narayan. The textbook is L. C. Evans "Partial Differential Equations". The plan is to read Chapter 5 (Sobolev spaces) and if time permits, Chapter 6 (second-order elliptic PDEs). We meet Thursdays from 1-3PM in JWB 108. The lectures are intended to be approximately 1.5 hours a week, follow by 30 minutes for questions and discussion.

August 30: Chee Han, Section 5.2 [Notes]

Motivation, weak derivatives, definition of Sobolev spaces ($W^{k,p}(\Omega)$, $H^{k}(\Omega)$, $W_0^{k,p}(\Omega)$), examples of Sobolev functions, properties of weak derivatives, completeness of Sobolev spaces

September 6 - Chee Han, Section 5.3, Appendix C.4 [Notes]

Mollifiers, mollification of $L_\textrm{loc}^1(\Omega)$ functions and its properties, local (interior) approximation of $W^{k,p}(\Omega)$ functions by mollifiers, global approximation of $W^{k,p}(\Omega)$ functions by $C^{\infty}(\Omega)$ functions

Read ourselves: Global approximation of $W^{k,p}(\Omega)$ functions by $C^{\infty}(\overline\Omega)$ functions

September 13 - Huy, Section 5.4, Appendix C.1 [Notes]

$C^k$ boundary, extension of $W^{k,p}(\Omega)$ to $W^{k,p}(\mathbb{R}^n)$

September 20 - Zexin, Section 5.5

Trace operators, characterisation of $W_0^{k,p}(\Omega)$ functions

September 27 - Dihan, Section 5.6.1 [Notes]

Sobolev conjugate, Gagliardo-Nirenberg-Sobolev inequality

Sobolev embedding of $W^{k,p}(\Omega)$ for $p\in[1,n)$

Poincaré inequality for $W_0^{1,p}(\Omega)$ functions

October 4 - China, Section 5.6.2

Hölder space, Morrey's inequality, Sobolev embedding for $W^{k,p}(\Omega)$ functions with $k\gt 1$

October 18 - Rebecca, Section 5.7

Rellich-Kondrachov theorem

October 25 - Nathan, Section 5.8.1 and 5.8.2 [Notes]

Poincaré inequality for $W^{1,p}(\Omega)$ functions

Difference quotients

November 1 - Nathan, Section 5.8.3 and 5.8.4 [Notes]

Characterisation of $W^{1,\infty}(\Omega)$ functions

Differentiability almost everywhere

Characterisation of $H^k(\mathbb{R}^n)$ by Fourier transform

November 8 - Huy, Section 5.9

Definition of $H^{-1}(\Omega)$ and its characterisation

November 15 - Zexin, Section 6.2.2

Divergence form elliptic operator and its bilinear form

Energy estimates for bilinear form

Existence of weak solutions for linearly perturbed elliptic operator

Definition of adjoint operator

November 29 - China, Section 6.2.3

Fredholm alternative

December 6 - Rebecca, Section 6.5.1

Eigenvalues of symmetric elliptic operators

Math 6640: Introduction to Optimisation

This course is currently taught by Braxton Osting in Fall 2018. The textbook is A. Beck "IntroductIon to Nonlinear Optimization:Theory, Algorithms, and Applications with MATLAB". The lecture notes would be written jointly with Rebecca Hardenbrook and Nathan Willis.

Notes (ongoing)

Math 6880: Fluid Dynamics

This course was taught by Aaron Fogelson and Christel Hohenegger in Fall 2017 and Spring 2018. The course’s webpage for the first half can be found here.

Notes (incomplete)

Math 6730: Asymptotic and Perturbation Methods.

This course was taught by Paul Bressloff in Fall 2017. The textbook is M. H. Holmes "Introduction to Perturbation Methods". These notes were written jointly with Hyunjoong Kim.

Notes