Mersen (France)

In the first part of this article we establish, for a compact pseudomanifold and a given perversity in the sense of Goresky and MacPherson a Morse theoretical cochain complex, which computes the intersection cohomology of the space. In the second part we use this cochain complex as well as the model Witten Laplacian to define the Bismut–Zhang torsion of a pseudomanifold. Conjecturally the Bismut–Zhang torsion will serve as the “topological” side in a Cheeger–Müller theorem for spaces with iterated conical singularities.

A Mall bundle on a Hopf manifold H=ℂ n ∖0 ℤ is a holomorphic vector bundle whose pullback to ℂ n ∖0 is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincaré–Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of ℂ n is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-resonant Hopf manifolds are linear; previously, this result was obtained by Kodaira using the Poincaré–Dulac theorem.

We study the spectrum of the semiclassical Witten Laplacian Δ f associated to a smooth function f on ℝ d . We assume that f is a confining Morse–Bott function. Under this assumption we show that Δ f admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring–Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.

In this paper, we generalize the tools that were introduced in [13] in order to study the Andreadakis problem for subgroups of IA n . In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IA n . We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group F n-1 , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.

We construct new invariant measures supported on mass level sets for the cubic defocusing nonlinear Schr\"odinger equation in dimensions $1$ and $2$.

This note is a companion article to the recent paper [Löcherbach, et al. Electronic J. Probab. (2025) in press]. We consider mean field systems of interacting particles. Each particle jumps with a jump rate depending on its position. When jumping, a macroscopic quantity is added to its own position. Moreover, simultaneously, all other particles of the system receive a small positive random kick which is distributed according to a one-sided α−stable law and scaled in N−1/α, where 0 < α < 1. In between successive jumps of the system, the particles follow a deterministic flow with drift depending on their position and on the empirical measure of the total system. In a more general framework where jump amplitudes and state space do not need to be non-negative, we have shown in [Löcherbach, et al. Electronic J. Probab. (2025) in press] that the mean field limit of this system is a McKean-Vlasov type process which is solution of a non-linear SDE, driven by an α−stable process. Moreover we have obtained in [Löcherbach, et al. Electronic J. Probab. (2025) in press] an upper bound for the strong rate of convergence with respect to some specific distance disregarding big jumps of the limit stable process. In the present note we consider the specific situation where all jump amplitudes are positive, and particles take values in ℝ+. We show that in this case it is possible to improve upon the error bounds obtained in [Löcherbach, et al. Electronic J. Probab. (2025) in press] by using an adhoc distance obtained after applying a concave space transform to the trajectories. The distance we propose here takes into account the big jumps of the limit α−stable subordinator.

In this paper, we study the value distribution of the Sudler product ∏ n=1 N |2sin(πnα)| and the Diophantine product ∏ n=1 N (2e∥nα∥) for various irrational α, as N ranges in a long interval of integers. At badly approximable irrationals these products exhibit strong concentration around N 1/2 , and at certain quadratic irrationals they even satisfy a central limit theorem. In contrast, at almost every α we observe an interesting anticoncentration phenomenon when the typical and the extreme values are of the same order of magnitude. Our methods are equally suited for the value distribution of Birkhoff sums ∑ n=1 N f(nα) for circle rotations. Using Diophantine approximation and Fourier analysis, we find the first and second moment for an arbitrary periodic f of bounded variation, and (almost) prove a conjecture of Bromberg and Ulcigrai on the appropriate scaling factor in a so-called temporal limit theorem. Birkhoff sums also satisfy a central limit theorem at certain quadratic irrationals.

We study metric measure spaces that admit “thick” families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z×ℝ k for some k≥1. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.