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.
The HADES seminar on Wednesday, May 6th, will be at 3:30pm in Room 736.
Speaker: Yuda Chen
Abstract: The decoupling inequality, introduced by Bourgain and Demeter in their celebrated proof of the ℓ² decoupling theorem for the paraboloid, has become a fundamental tool in modern harmonic analysis.
This talk will survey its key applications. We first discuss its pivotal role in the Vinogradov Mean Value Theorem. We then focus on more recent applications to proving discrete Strichartz estimates. Finally, we conclude by proposing several conjectures concerning potential extensions of these estimates.
The HADES seminar on Tuesday, April 21st, will be at 3:30pm in Room 740.
Speaker: Dongxiao Yu
Abstract: We study the 3D non-relativistic compressible Euler equations. We assume that the flow is irrotational and isentropic. We treat all equations of state except that of the Chaplygin gas, for which shocks are not expected to form. For an open set of initial data with tails at infinity, we provide a complete and precise description of the maximal globally hyperbolic development (MGHD). The boundary of this MGHD consists of an initial singularity known as the crease, a singular boundary where gradient blow-up occurs, and a Cauchy horizon emanating from the crease. The analysis involves delicate competition between dispersion and resonant nonlinear terms. Moreover, we prove that this MGHD is unique by applying a uniqueness theorem of Eperon-Reall-Sbierski. This is joint work with Leonardo Abbrescia and Jared Speck.
The HADES seminar on Tuesday, March 17th, will be at 3:30pm in Room 740.
Speaker: Warren Li
Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct solutions that (locally) blow-up on an arbitrary spacelike hypersurface, unique up to the choice of a function we call auxiliary scattering data, and that such blow-up hypersurfaces and auxiliary scattering data is stable to perturbations away from the singularity. For instance, we show smooth perturbations of the ODE blow-up solution yields a smooth spacelike blow-up hypersurface. This is based on joint work with Isti Kadar (ETH).
The HADES seminar on Tuesday, February 17th, will be at 3:30pm in Room 740.
Speaker: Liyang Shao
Abstract:
Badly approximable vectors are one of the central topics in Diophantine approximation. Though being null in Lebesgue measure, these vectors are known to have ‘thick’ structure, e.g. full Hausdorff dimension, or even stronger, the winning property that was first proven by Schmidt in the unweighted setup in the 1960s, and in the weighted setup recently by Beresnevich-Nesharim-Yang.
In this talk, we will first briefly introduce how the study of such vectors can be rooted in counting rational points, which is connected to subjects including harmonic analysis and homogeneous dynamics. Then we will describe how non-divergence estimates from homogeneous dynamics can give a winning property of inhomogeneous weighted badly approximable vectors. The second part is joint work with Shreyasi Datta.
The HADES seminar on Tuesday, December 16th, will be at 3:30pm in Room 762.
Speaker: Shi-Zhuo Looi
Abstract: The classical wave equation is a basic model for the propagation of waves. In even space dimensions, solutions are known to develop long-lived polynomially decaying tails inside the region where the wave has passed, in contrast with the sharp finite propagation of disturbances in odd dimensions.
In this talk, I will discuss how such even-dimensional tails behave in the presence of forcing and nonlinear effects, as well as on non-stationary spacetime backgrounds.
The HADES seminar on Tuesday, December 2nd, will be at 3:30pm in Room 740.
Speaker: Xueying Yu
Abstract: Strichartz estimates are fundamental tools for understanding the dispersive behavior of solutions to Schrödinger equations. In particular, bilinear Strichartz estimates provide sharper information by capturing interactions between two waves with different frequencies, which play a key role in many nonlinear problems. In this talk, we will first review several classical bilinear Strichartz estimates for the Schr\”odinger equation, with an emphasis on the strategies used in their proofs. We then present a new bilinear estimate in the setting of Schr\”odinger equations on negatively curved spaces.
The HADES seminar on Wednesday, November 19th, will be at 4:00pm in Room 732.
Speaker: Serban Cicortas
Abstract: We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small scattering data in the distant past or the distant future. We will also explain why the case of even spatial dimension $n$ poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.