Lecture recordings - email Semyon if you have problems with access!

This is a collection of 6 two-week minicourses aimed at beginning graduate students interested in Analysis and PDE. It is organized by Semyon Dyatlov and Gigliola Staffilani. Here is the schedule:

Sep 15–Sep 26: Aleksandr Logunov, Elementary Questions in Real Algebraic Geometry and Analysis

Sep 29–Oct 10: Semyon Dyatlov, Quantum chaos
Oct 15–Oct 24: Larry Guth, Kakeya conjecture part 1
- Alex Cohen and I will give a sequence of two mini-courses about the Kakeya conjecture, especially the recent proof in $LaTeX: \mathbb R^3$ by Hong Wang and Joshua Zahl. I will start by introducing the problem, giving history and background, and discussing some ideas that appear in the proof. In particular, we will discuss the connection between Kakeya and Fourier analysis, multilinear Kakeya and induction on scales, obstacles to Kakeya coming from similar statements that are false, the sticky case of Kakeya, and the connection to sum/product theory. This will set the stage for part 2...
Oct 27–Nov 7: Alex Cohen, Kakeya conjecture part 2
- Picking up where Larry left off, I will discuss how Wang and Zahl prove the Kakeya conjecture by reducing to the special case where the tube set is self-similar (also called sticky). Wang and Zahl's key estimate says that the union of a well-spaced family of tubes has large volume. We prove two different multiscale decompositions that allow us split the key estimate into sub-problems of the same form. Using the first decomposition along with the sticky case, we prove a high-density lemma: If there is an excess of tubes, the estimate gets better. Using the second decomposition along with the high density lemma, we prove the key estimate.
Nov 12–Nov 21: Christoph Kehle, The moduli space of dynamical spherically symmetric black hole spacetimes
- The existence and dynamical formation of black holes are among the most spectacular predictions of general relativity. The Schwarzschild family, together with its charged analog, the Reissner–Nordström family, provides explicit solutions of the Einstein–Maxwell equations that describe eternal, spherically symmetric black holes.
  The ultimate goal of this minicourse is to develop the tools needed to study the infinite-dimensional moduli space of dynamical solutions in the spherically symmetric Einstein–Maxwell–scalar field model. This framework already captures the richness of genuine dynamical phenomena, including the event horizon red-shift and its degeneracy, the Aretakis instability, and the extremal black hole formation threshold with its universal scaling laws. Some nodding familiarity with the basics of (pseudo-)Riemannian geometry and hyperbolic PDEs will be helpful.
Dec 1–Dec 10: Sky Cao, Singular stochastic partial differential equations (SPDEs)
- SPDEs arise in various areas of physics, much like their deterministic counterparts. (Examples include QFT and statistical mechanics.) However, a key distinction is that many SPDEs of interest are singular: their solutions are only expected to lie in negative-regularity distribution spaces. Because of this singularity, even the local theory of nonlinear singular SPDEs remained poorly understood until roughly twelve years ago. This changed with the breakthrough frameworks of regularity structures (Hairer) and paracontrolled calculus (Gubinelli–Imkeller–Perkowski), which for the first time enabled local existence results for a broad class of singular SPDEs. In this mini-course, I will focus on the paracontrolled approach, which combines insights from harmonic analysis (via Littlewood–Paley theory and Bony's paraproduct) and probability theory. Time permitting, I will also touch on aspects of the global theory for singular SPDEs, which is currently one of the central directions in the field.

Information:

The course meets most Mondays, Wednesdays, and Fridays 11AM–noon in room 4-265.
There is no class the first 2 weeks of class and Thanksgiving week. Some other classes might be cancelled depending on the pacing of the individual minicourses.
Each minicourse will have problem set(s), which will be graded for students enrolled for credit. At the end of the semester Semyon and Gigliola will collect all the problem set scores and combine them into a letter grade. We will use the 5 best scores (i.e. you can skip psets for one course).
Undergraduate students are not allowed to take the course for credit, but they are welcome to attend as Listeners.