The HADES seminar on Tuesday, September 29th, will be at 3:30pm in Room 736.
Speaker: Maciej Zworski
Abstract: The purpose of the talk is to provide an introduction to basic aspects of second quantisation and its relevance to many body physics. It will be done from a perspective of a non-expert with the principal source being the appendix by Kevin Stubbs to the Lecture Notes with Zhongkai Tao https://math.berkeley.edu/
The HADES seminar on Tuesday, September 22nd, will be at 3:30pm in Room 736.
Speaker: Ryan Unger
Abstract: Extremal black holes are special solutions of Einstein’s equations with maximal spin or charge for their mass. In this talk, I will describe a series of conjectures explaining the role of extremal black holes in gravitational collapse. I will focus on upcoming progress on these conjectures in the context of the spherically symmetric Einstein—Maxwell-charged scalar field model. This is joint work with Yannis Angelopoulos (University of Crete) and Christoph Kehle (University of Zürich).
The HADES seminar on Tuesday, September 15th, will be at 3:30pm in Room 736.
Speaker: Ovidiu Avadanei
Abstract: We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,\alpha}\cap L^2$, where $\alpha\in\left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold. Motivated by this, we define Lagrangian classes of coherent initial data and conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $\alpha\geq\frac{1}{3}$. When $\alpha>\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $\alpha=\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under cusp-tail, Dini coherence, near-field compatibility, and bounded transverse-distortion hypotheses, this singular value cannot collapse in finite time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3\alpha}$, which rules out Shkoller-type clock collapse for $\alpha\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller’s subcritical blow-up mechanism in the case $\alpha>\frac{1}{3}$.
This talk is an extended version of yesterday’s Analysis and PDE Seminar, in the sense that it will contain additional details.