The APDE seminar on Monday, 9/28, will be given by Federico Pasqualotto (UC San Diego) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Recent results on quasiconvexity

Abstract: In this talk, I will present three interconnected results centered on quasiconvexity: the quasiconvexity range for the Dacorogna—Marcellini functional, new examples of integrands on R^{2×4} that are rank-one convex but not quasiconvex, and the resolution of a conjecture of T. Iwaniec on the Beurling—Ahlfors transform. The first two results are based on joint work with Andrea Agazzi, Giuseppe Bruno, and André Guerra. The last result is based on a wider community project aimed at incorporating AI into mathematical research. We will end with a brief discussion on our approach to AI-assisted collaboration.

The APDE seminar on Monday, 9/21, will be given by Xiao Ma (University of Michigan) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Recent Advances on Hilbert’s Sixth Problem

Abstract:  In this talk, we present our recent progress on Hilbert’s sixth problem—deriving fluid equations from microscopic dynamics. We will provide the necessary physical background and introduce the key techniques developed in our work, including the Feynman diagram representation and the cutting algorithm.

The APDE seminar on Monday, 9/14, will be given by Ovidiu-Neculai Avadanei (Caltech) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

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 $\alpha>\frac{1}{3}$.

The APDE seminar on Monday, 8/24, will be given by Ugur G. Abdulla (Okinawa Institute of Science and Technology) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Kolmogorov Problem and Wiener-type Criteria in Potential Theory

Abstract: The central problem in the Analysis of PDEs is understanding the nature of singular-
ities that reflect natural phenomena. This talk will present a full characterization of the fundamental boundary singularity, and equivalently, the unique solvability of the singular
Dirichlet problem for the elliptic and parabolic PDEs. The results are threefold. We prove
a new Wiener-type criterion for the ”geometric” characterization of the removability of the
fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of
the complementary set near the singularity point. In the special case when the surface
of revolution forms the boundary of the open set near the singularity point, we establish
a Kolmogorov-Petrovsky-type test to characterize the removability of the singularity and
uniqueness. Finally, in the special case when a continuous graph locally represents the boundary of the open set, the minimal thinness criterion for the removability of the singularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point. From the probabilistic point of view, the criteria present an asymptotic
law for conditional Brownian motion. In the topological context, the criteria present a full characterization of the neighborhood base of the boundary singularity point in the minimal-
fine topology. In the more general framework, the talk will outline a program for the full characterization of the singularities formed by the elliptic and parabolic PDEs.

The APDE seminar on Monday, 5/4, will be given by Claude Warnick (University of Cambridge) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Small data global well posedness of the vacuum Einstein equations in centred Newman-Unti gauge

Abstract:The global stability of Minkowski spacetime is a foundational question in mathematical relativity, and the proof of this result by Christodoulou and Klainerman in their `94 monograph is a landmark achievement in the field. The result has been re-proved in several ways, most notably by Lindblad-Rodnianski `04. In this talk I shall present a new proof with Jonathan Luk and Sun-Jin Oh based on a single coordinate system constructed from outgoing lightcones emanating from a singe central geodesic. In this coordinate system the Einstein equations have a remarkably elegant structure, as noted by Newman-Unti in `62, which can be exploited to give an efficient proof of a wide range of results in the literature.

The APDE seminar on Monday, 4/27, will be given by Juhi Jang (USC) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Stable Larson-Penston collapse

Abstract: I will discuss recent progress on mathematical construction of self-similar solutions to the Euler-Poisson system describing gravitational collapse and nonlinear stability of the Larson-Penston collapse against radially symmetric perturbations. At the heart of the latter stability result is the ground state character of the Larson-Penston solution featuring important monotonicity properties. The talk is based on joint works with Yan Guo, Mahir Hadzic and Matthew Schrecker.

The APDE seminar on Monday, 4/20, will be given by Josef Greilhuber (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Non-density of nodal lines in the clamped plate problem

Abstract: It is well known that the nodal set (i.e., zero set) of an eigenfunction of the Laplacian – modelling a fundamental mode of vibration of an elastic membrane – is dense at the scale of its characteristic wave-length.

In contrast, we show that the nodal set of high energy eigenfunctions of the clamped plate problem – a fourth order PDE modeling a vibrating metal plate – is not necessarily dense and can in fact exhibit macroscopic “nodal voids”.

Specifically, we construct small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius ~0.44.

Remarkably, this radius is sharp, simultaneously providing the asymptotic upper bound for the size of such circular nodal voids among small perturbations of the disk.

The APDE seminar on Monday, 4/13, will be given by Ioan Bejenaru (UCSD) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: An effective resolution space for the Schroedinger equation

Abstract: In the analysis of nonlinear dispersive PDEs it is important to design resolution spaces which replicate key estimates available for free solutions. While most resolution spaces transfer the linear estimates, also known as Strichartz estimates, this is not always the case for bilinear/multilinear restriction estimates. In this talk we propose a new structure which transfers the classical bilinear L^2 estimate without loss, among other desirable properties. This is done in the context of the Schroedinger equation.

The APDE seminar on Monday, 4/6, will be given by our own Daniel Tataru in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global solutions for 3D gravity water waves

Abstract: The aim of the talk is to describe work in progress on the problem of local and global well-posedness in the small data, low regularity regime for gravity waves in a fluid of infinite depth, and infinite width, in spatial dimension n ≥ 3 and higher. This is joint work with Mihaela Ifrim and Ben Pineau.