# Student Analysis

Organizers: Romain Speciel, Josef Greilhuber

## Past Events

I'll discuss some recent work of Michael Christ which establishes smoothing for the integral of a four-fold product. In particular, I will outline some key ideas involving a reduction to the trilinear case using **spicy** Cauchy-Schwarz, a study of sublevel set estimates for a…

Unlike for pseudodifferential operators, showing even just L²-boundedness for a general Fourier Integral Operators is nontrivial. This is especially true if the corresponding canonical relation cannot be written as a graph over the cotangent bundle of the source manifold. We will have a look at…

We will discuss applications of the clean composition calculus.

We will discuss the clean composition calculus of Guillemin and Duistermaat.

We will move from the local to the global theory of FIOs, providing invariant definitions of relevant notions such as operator symbols. The necessary tools from symplectic geometry will be introduced. If time permits, we'll begin considering some applications.

In this talk, we follow the book ‘Fourier Integral Operators’ by Duistermaat. Fourier integral operators (FIO) are a class of operators that generalise pseudodifferential operators. While pseudodifferential operators include solution operators to elliptic problems, FIO include solution operators…

We will derive the transport equation as the second term in our approach to solving hyperbolic PDEs, describe the meaning of this equation from our symplectic perspective, and, if time permits, outline a solution strategy.

In this talk, I will continue to describe aspects of geometric optics, one of the main themes introduced earlier. I also hope to describe a bit more about how this relates to some interesting properties of hyperbolic PDE. In particular, I hope to motivate a bit more about why you may…

Continuing from last week, we cover Chapter 1 of ‘Semiclassical Analysis’ by Guillemin and Sternberg. We are interested in solving a hyperbolic linear partial differential equation involving a time variable. We reduce it to an ‘eikonal equation’, which we can solve locally by finding a…