Weekly Bulletin

I will report on recent results on geometric flows associated to harmonic mappings with free boundary. Those maps are instrumental in several geometric problems, such as extremal metrics for the Steklov spectrum for instance and one can formulate several possible parabolic equations whose stationnary solutions are such maps. I will describe these formulations, each of which offering interesting applications and analytic problems. In various cases, one can derive partial regularity results for weak solutions and describe the structure of the singular set. I will try to give an overview of such results. However, a formulation, related to the Plateau flow, poses more challenging issues and I will formulate some conjectures about its singularity formation. The construction of solutions blowing up in finite or infinite time uses a new gluing technique, which has been successfully used recently to investigate singularity formations in other flows, such as Fast Diffusion equations or Yang-Mills heat flow.

<p><span style="color: black; font-size: 12pt; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif;" data-olk-copy-source="MessageBody">Given a Lagrangian inside a symplectic manifold, one can define a metric, called the Hofer distance, on the space of Hamiltonian isotopies of this Lagrangian. It is known to be bounded in the case of a circle inside the plane, while it is unbounded for a diameter inside the disc (Khanevsky, 2009), or the standard Lagrangian inside the Euclidean ball of even dimension (Seyfaddini, 2013). The question remains open in most cases, such as the equator inside the sphere or a circle inside the disc. In this talk, I will show the unboundedness of this distance for a disjoint union of circles inside the disc. This result relies on a theorem of Morabito, together with a standard argument of Khanevsky.</span></p>

<div dir="ltr">Let X be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice \Gamma. </div> <div dir="auto">For instance, X could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group.</div> <div dir="ltr">Let H be a discrete subgroup of isometries of X with critical exponent (exponential growth rate) strictly less than half of the growth rate of \Gamma. </div> <div dir="ltr"> We show that the injectivity radius of X/H grows linearly along almost every geodesic in X (with respect to the Patterson-Sullivan measure on the Gromov boundary of X).</div> <div dir="ltr"> <div>The proof will involve an elementary analysis of a novel concept called the "sublinearly horosherical limit set" of H which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups.</div> <div> </div> <div> </div> <div>This talk is based on joint work with Inhyeok Choi and Keivan Mallahi-Kerai.</div> </div>

We consider testing the goodness of fit of semiparametric regression models, such as generalised additive models, partially linear models, or quantile additive regression models. We propose an approach that involves first splitting the data in two parts. On one part, we "hunt" for any signal that may be present in the score-type residuals following a fit of the model. On the remaining data, we test for the presence of the potential signal thus found. For the first hunting stage of the procedure, our framework allows for carrying this out based on a user-chosen flexible regression method, such as a random forest. The method is thus able to leverage the power of modern machine learning methods to detect complex alternatives. A challenge with the testing step is that any first-order bias in the residuals may lead to rejection under the null. We therefore employ a debiasing step which we show amounts to performing a particular weighted least squares regression. We show that the type I error may be controlled under relatively mild conditions, and that we have power under alternatives where with high probability the hunted signal is correlated with the true signal present in the score residuals.

Over the last two years, subsampled natural gradient descent (SNGD) has demonstrated breakthrough performance for parametric optimization tasks in scientific machine learning, including neural network wavefunctions and physics-informed neural networks. In this talk, I will first present an accelerated algorithm called SPRING which improves the convergence of SNGD at no extra cost. I will then consider the theoretical aspects of these algorithms and explain why previous analytical approaches based on stochastic optimization theory fail to explain the empirical observations. Finally, I will show how a different perspective rooted in randomized linear algebra overcomes these limitations and provides an accurate and detailed understanding of the convergence properties of both SNGD and SPRING. Beyond explaining existing algorithms, the new analytical framework also suggests new pathways towards faster and more robust training algorithms for scientific machine learning

<p>We will discuss recent advances on the mathematical kinetic theory of plasma. First, we will present a new Landau damping result for the (screened) Vlasov-Poisson equation in the presence of an ion, justifying the stopping power law in plasma physics. Since the equation is non-dissipative and time-reversible, the key challenge lies in a detailed study of dispersion/phase-mixing.<br />This is in contrast to the collisional kinetic equations for plasma, which are parabolic PDEs formally satisfying an entropy dissipation theorem. We present new results on the regularity and well-posedness of the Landau- and Balescu-Lenard equations, featuring a new blow-down mechanism for singular data. Based on joint works with M. Duerinckx, M.P. Gualdani and R. Höfer.</p>

I will discuss recent work with Skorobogatov, and work in progress with Lyczak, establishing the Hasse principle for classes of degree 4 del Pezzo surfaces, conditional on finiteness of certain Tate--Shafarevich groups. Key to the method is the study of an auxiliary family of Kummer surfaces.