Student Analysis
Organizers: Romain Speciel, Josef Greilhuber
Upcoming Events
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.
Past Events
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…
We will kick off our reading of Guillemin and Sternberg's monumental set of lecture notes about "Semiclassical Analysis", with a discussion of the textbook's introduction, and some additional motivating examples. Among these will be another "proof" of the Weyl law, as well as a "Weyl law" for…
TBD.
We will begin our discussion of the proof of the Willmore Conjecture by Marques and Neves by exploring definitions and results from geometric measure theory. Topics include Hausdorff Measure, rectifiable sets and varifolds, and currents. Time permitting, we will introduce Almgren-Pitts minmax…
We will discuss the density of closed geodesics on hyperbolic manifolds and the trace of the resolvents as applications of Selberg's trace formula. Time permitting, we may also define Selberg's zeta function and/or prove the prime geodesic theorem.