The HADES seminar on Tuesday, November 3rd, will be at 3:30pm in Room 736.

Speaker: Ethan John Ebbighausen

Abstract: In 1985, Kenneth Falconer introduced a continuous version of the Erdos Distance Problem that would later drive developments in many aspects of fractal geometry and restriction theory. In this survey talk, we approach several key techniques and turning points that took this problem from simple estimates to the bastion OpenAI recently claimed to solve.

The HADES seminar on Tuesday, October 27th, will be at 3:30pm in Room 736.

Speaker: Vitor Borges

Abstract:A central problem in nonlinear dispersive equations is to understand when the spreading of waves controls the cumulative effect of nonlinear interactions. I will describe the progression from linear and bilinear methods to trilinear restriction estimates, which exploit additional geometric information about three interacting waves. As an application, I will present global well-posedness and scattering for a class of three-dimensional quadratic Schrödinger equations with small initial data in the scaling-critical space. This is based on joint work with Ioan Bejenaru and Tiklung Chan.

The HADES seminar on Tuesday, October 20th, will be at 3:30pm in Room 736.

Speaker: Aren Martinian

Abstract: Gluing techniques in general relativity are incredibly versatile, for example in combining multiple gravitational systems together or improving understanding of black hole initial data. Such gluing constructions have been done in a wide variety of settings, including those of constant nonzero curvature. In this talk, we give a new proof of Corvino-Schoen type gluing, also known as gluing up to linear obstructions, for asymptotically hyperbolic spacetimes satisfying improved quantitative smallness conditions in arbitrary dimension over the existing work of Chruściel and Delay, also generalizing this work to the non-time-symmetric case k ≠ 0. In particular, we describe three Bogovskii solution operators, inspired by the framework of Mao-Oh-Tao, with each providing a distinct and important role in the construction.

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.