Weekly Bulletin

tba

Given an arbitrary reversible Finsler metric on $S^2$, the question of how many closed geodesics such a metric admits has a long history. In this talk, we present a result showing that the number of prime closed geodesics grows at least quadratically with respect to length. The proof relies on an improvement of Franks’ theorem concerning the periodic points of area-preserving annulus maps, a model contact system, and the theory of cylindrical contact homology in the complement of a link.

Whenever you see a string of quadrature nodes, you can consider it as a branch cut defined by the poles of a rational approximation to the Cauchy transform of a weight function. The aim of this talk is to explain this statement and show how it opens the way to calculation of targeted quadrature formulas for all kinds of applications. Gauss quadrature is an example, but it is just the starting point, and many more examples will be shown. I hope this talk will change your understanding of quadrature formulas. This is joint work with Andrew Horning.

<div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">Suppose \(f\) is a <u>general</u> infinitely differentiable topologically mixing surface diffeomorphism with positive topological entropy.</span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">We show that  the measure of maximal entropy is Bernoulli, and that all smooth observables exhibit exponential decay of correlations, the strong  almost sure  invariance principle,  and (consequently) the central limit theorem and the law of the iterated logarithm.</span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">The proof, joint with Crovisier and Buzzi,  is based on a new dynamical property called "strong positive recurrence," which guarantees the existence of a symbolic model with spectral gap for the associated transfer operator. </span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">(Joint with S. Crovisier & J. Buzzi)</span></span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"> </div>

Let X be a centred random vector in \R^d and consider the random matrix \Gamma=\sum_{i=1}^N <X_i,->e_i, whose rows are independent copies of X. Given T \subset S^{d-1}, how much of T's structure is captured by \Gamma T? Our focus here is on fine notions of similarity - specifically, whether the (empirical) distributions of (<X_i,\theta>)_{i=1}^N are close in an appropriate sense to the distributions of <X,\theta>, uniformly in \theta \in T. The answer to this question was not known even when X=G, the standard gaussian. In this talk I will present the current state-of-the-art - which holds under minimal assumptions, and in particular leads to an (almost) optimal answer in the gaussian case. If time permits, I will present some of the questions that are still open. Joint work with D. Bartl.

In 1981, Margulis and Soifer established a remarkable property of finitely generated linear groups: such a group contains uncountably many maximal subgroups if and only if it is not virtually solvable. In particular, this yields the existence of maximal subgroups of infinite index in SL_n(Z). We will show how their methods can be adapted to establish an analogous result for groups of birational transformations of the plane. This is based on joint work with Serge Cantat, Julie Déserti and Seung Uk Jang.

The talk first presents some numerical experiments with time-dependent tree tensor network algorithms for the approximation of quantum spin system dynamics. It continues with the basics in the design of time integration methods that are robust to the typical presence of small singular values, that have good structure-preserving properties (norm, energy conservation or dissipation), and that allow for rank (= bond dimension) adaptivity and for parallelism. This discussion of basic concepts forms the main part of the talk and will be done for the smallest possible type of tensor network differential equations, namely low-rank matrix differential equations, which are of interest in their own right. Once this nontrivial technically simplest case is understood, there is a systematic path to the extension of the integrators and their favourable properties to general tree tensor networks. This talk is based on joint work with many colleagues and former and present students, among which I wish to single out Othmar Koch for the first mathematical work on dynamical low-rank approximation (DLRA) in 2007, Ivan Oseledets for jointly discovering the first robust DLRA integrator in 2014 (the projector-splitting integrator), Gianluca Ceruti and Jonas Kusch for jointly developing the Basis Update & Galerkin (BUG) integrators since 2022, and Hanna Walach, Gianluca Ceruti, Dominik Sulz and Charlotte Verhoeven for the recent systematic extension from low-rank matrices to general tree tensor networks both in theory and in increasingly efficient implementations.

A classical consequence of the Prékopa–Leindler inequality is that log-concavity is preserved along the heat flow. Beyond the classical heat semigroup, however, this phenomenon typically breaks down. In this talk, I will introduce a weaker notion of log-concavity that remains stable under generalized heat semigroups. I will show how this framework yields new log-semiconcavity estimates for ground states of Schrödinger operators with non-convex potentials, as well as propagation results for functional inequalities along generalized heat flows.I will also discuss how these weak log-concavity properties behave under conditioning and marginalization, extending classical ideas of Brascamp and Lieb beyond the log-concave setting. The main ingredient is a stochastic control perspective on quadratic Hamilton–Jacobi–Bellman equations, combined with a second-order analysis of reflection couplings along HJB characteristics.

In many situations, when a global symmetry is restricted to a subspace, we lose the group action, but a residual symmetry remains visible. In this talk, I will illustrate how groupoids address this issue and explain why they are the most suitable mathematical objects for describing symmetries. I will then define linear representations of groupoids and prove a theorem establishing a one-to-one correspondence between representations of groupoids and representations of their isotropy groups. Joint work with prof. Alberto Ibort and prof. Juan Manuel Perez-Pardo.

<p>The Quantum Hydrodynamics (QHD) system is a fundamental model for quantum fluids, such as Bose-Einstein condensates and superfluid Helium. Mathematically, QHD is described by a compressible Euler system augmented by a dispersive quantum stress tensor. While the formal connection to nonlinear Schrödinger (NLS) equations via the Madelung transformation is well-established, the rigorous, general analytical framework relies on the polar factorization method.<br>In the first part of this talk, I will review existing results for the Cauchy problem. Focusing on the one-dimensional case, I will discuss a wave-function lifting method that, under physically consistent assumptions, reconstructs a wave function directly from hydrodynamic data. This allows for a "genuine" hydrodynamic treatment of the theory without a priori reliance on the Schrödinger picture. Furthermore, I will introduce a novel entropy related to the system’s chemical potential, which provides the necessary a priori estimates to determine a compactness class for 1D weak solutions.<br>Finally, I will address the initial-boundary value problem (IBVP). I will motivate our choice of boundary conditions and their correspondence to the IBVP for the 1D NLS equation, then present a global existence result for finite-energy weak solutions. I will conclude by discussing potential applications and future directions for this research.</p>

In this talk, we address the problem of modeling loss frequency using regression when the counts have a non-standard distribution. We propose a novel approach based on mixture-of-experts specifications on discrete phase-type distributions. Compared to continuous phase-type counter parts, this approach offers fast estimation via expectation-maximization algorithms, making it more feasible for use in real-life scenarios. Our model is both robust and interpretable in terms of risk classes and can be naturally extended to the multivariate case through two different constructions. Using simulated and real-world data, we showcase how our approach provides an effective solution for modeling loss frequency in non-standard situations.

In recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera has established a min-max construction of minimal hypersurfaces in closed Riemannian manifolds, based on the analysis of singular limits of sequences of solutions of the Allen—Cahn equation, a semi-linear elliptic equation arising in the theory of phase transitions. In this talk, I will describe some recent boundary regularity results for such limit-interfaces, which provide the first steps towards an Allen—Cahn min-max construction of free boundary minimal hypersurfaces in Riemannian manifolds with boundary. This is based on joint works with Akashdeep Dey, Wenkui Du, Martin Li, and Davide Parise.