# Lecture notes

These are some notes from lectures I have given.

## A crash course in interpolation theory

These notes grew out of lecture notes for a summer school I taught at the Scuola Matematica Interuniversitaria in Cortona, Italy in July 2021. They cover the classical interpolation theory of Lebesgue and Lorentz spaces, the basics of abstract interpolation between pairs of compatible Banach spaces, and applications of these ideas in various areas of analysis. Topics in the classical theory include: Lorentz spaces, the Marcinkiewicz and Riesz-Thorin theorems, and various applications to some useful operators, including the Hardy-Littlewood maximal function, the Fourier transform, and certain integral operators. The material on abstract interpolation focuses solely on the real method, mostly using the K-function. Topics include: compatibility, sums and intersections, upper and lower closures, interpolation spaces and their mapping properties (linear and nonlinear), some equivalent norms, reiteration theory, and concrete examples of interpolating between Lebesgue, Sobolev, and Hölder spaces.

## Linear constant coefficient ordinary differential systems

These are some notes on the general theory of linear ordinary differential systems with constant complex coefficients. Here general roughly means some combination of: without assuming the principal symbol of the operator is invertible (i.e. the non-elliptic case), with completely general initial conditions, and with extra decay conditions imposed at infinity. They grew out of some notes that I took while trying to better understand some components of the famous second paper by Agmon, Douglis, and Nirenberg on boundary value problems for elliptic systems of PDEs. One can think of these notes as a primer on the ODE analysis needed to understand ADN2, though there is more here than strictly needed for that purpose.

## A crash course in complex analysis

These notes grew out of a set of notes delivered during the last week of the honors course Mathematical Studies: Analysis II at Carnegie Mellon in the Spring of 2020. They are meant as an amuse bouche preceding a more serious course in complex analysis. They cover the absolute essentials, but within the context of complex Banach-valued holomorphic functions. Topics include: holomorphic functions; paths, loops, and homotopies; Cauchy-Goursat in various forms (culminating in the loop chain homology version); analyticity and the Cauchy integral formula; various estimates of holomorphic functions and other forms of holomorphic rigidity; Laurent series, poles, meromorphic functions, and the residue theorem.

## Monstrous functions

This is a lecture I gave to the CMU math club in the Fall of 2018. I was playing off of the proximity to Halloween and talking about monstrous functions such as Weierstrass' monster, a uniformly continuous but nowhere differentiable function on the reals. I was teaching myself TikZ at the time and was pretty pleased with the graphics I managed to put together here.

## Derivation of the fluid equations

These lectures comprised the first two weeks of a special topics course on the analysis of fluid equations I gave at CMU in the Spring of 2016. They are a refinement of lectures I gave at the Peking University Summer School in Applied Mathematics in the Summer of 2015. They cover the fundamental laws of continuum mechanics and derive the equations of compressible and incompressible fluid mechanics in the spirit of the Coleman-Noll procedure.

## Quasilinear symmetric hyperbolic systems

These lectures were given as part of the graduate course PDE II at CMU in 2015. They cover some existence theory for quasilinear symmetric hyperbolic systems and some nice applications. The existence results are largely based on this excellent paper of Kato.

## From Stokes flow to Darcy's law

These lectures were given at a CNA Working Group on homogenization in the Spring of 2014 and at Xiamen University in the Summer of 2018. They cover Tartar's rigorous derivation of Darcy's law via homogenization of the Stokes equations, which was originally published in the appendix of this book.

## Gamma-convergence of the Ginzburg-Landau energy

These were given at the CNA Summer School in 2013. They cover background material on the harmonic mapping problem, the relaxation approach, vortices, and Gamma-convergence results.

## Fluid-solid interaction

These were given at a CNA Working Group on fluids in the Fall of 2012. I was experimenting with handwriting applications on a tablet at the time: hence the weird format. The lectures cover the basics of rigid body mechanics, the equations of motion for fluids with rigid bodies included, and then discuss this paper of Feireisl.