In the fall of 2021 we are offering the following graduate Analysis courses:
Math 721: A First Course in Real Analysis
Real analysis concentrating on measures, integration, and differentiation and including an introduction to Hilbert spaces.
Math 821: Introduction to Fluid Dynamics
The aim of this course is to provide an introduction to the field of Fluid Dynamics i.e on the theory of incompressible fluid equations, focusing particularly on the Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but we will focus on these two equations in this course. We will begin with a derivation of the Euler and Navier-Stokes equations, and consider both their compressible and incompressible forms. Topics to be discussed include:
1) Eulerian and Lagrangian coordinates
2) Local well-posedness
3) Long time solutions/blow up criteria (profile decomposition method, specifically for Navier-Stokes eq)
4) Rough solutions
5) Vortex patches, vortex filaments
6) Free boundary problems
After this, we will move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and a discussion of the Onsager conjecture. If time permits, we will probably continue to more advanced and recent topics as we get closer to the end of the semester. There is no specific book we will follow. The chances are that the class will also present some recent results (most relevant papers in the field) in the field and for completeness I will introduce the mathematical tools needed in this process.
Math 823: Advanced Topics in Complex Analysis
This course is devoted to complex function theory and its applications in harmonic (Fourier) analysis. We will start with the basics of complex function theory in the unit disk and upper half-plane, and discuss Hardy spaces, Nevanlinna and Smirnov classes of analytic functions, inner/outer factorizations and related results. In the theory of entire functions we will review the Paley-Wiener theory, Carthright class, Phragmen-Lindeloef theorems and factorizations of entire functions. At the end of the term we plan to discuss applications of complex analysis to harmonic analysis such as Levinson's and Beurling's gap theorems in Fourier analysis and Beurling-Malliavin theory on completeness of families of complex exponentials in L2 spaces.
Math 827: Fourier Analysis
Topics include: Singular integrals, BMO, spaces of smoothness, interpolation theory (real and complex method), Fourier multipliers, various maximal functions, the role of curvature. If there is time we'll cover some special topic related to recent research.