This paper explains the theory of spectral sequences via décalage and the Beilinson $t$-structure.

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We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann–Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.

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We give a dynamical, relatively elementary proof of an “absorption theorem” which is closely related to a well-known result due to Matui. The construction is in the spirit of an earlier joint work of the author and S. Robert. In an appendix we explain how to use this result to correct the dynamical proof given by Melleray–Robert of a classification theorem for orbit equivalence of minimal ample groups due to Giordano, Putnam and Skau (the original argument had a gap).

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In this work, we introduce multi-Novikov algebras, a generalisation of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.

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In this article we reconsider the proof of subelliptic estimates for Geometric Kramers–Fokker–Planck operators, a class which includes Bismut’s hypoelliptic Laplacian, when the base manifold is closed (no boundary). The method is significantly different from the ones proposed by Bismut–Lebeau in [BL08] and Lebeau in [Leb05,Leb07]. As a new result we are able to prove maximal subelliptic estimates with some control of the constants in the two asymptotic regimes of high ($b\rightarrow 0$) and low ($b\rightarrow +\infty $) friction. After a dyadic partition in the momentum variable, the analysis is essentially local in the position variable, contrary to the microlocal reduction techniques of the previous works. In particular this method will be easier to adapt on manifolds with boundaries. A byproduct of our analysis is the introduction of a very convenient double exponent Sobolev scale associated with globally defined differential operators. Applications of this convenient parameter dependent functional analysis to accurate spectral problems, in particular for Bismut’s hypoelliptic Laplacian with all its specific geometry, has been developed in [NSW26].

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We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.

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We build a general theory of microlocal (homogeneous) Fourier integral operators in real-analytic regularity, following the general construction in the smooth case by Hörmander and Duistermaat. In particular, we prove that the Boutet–Sjöstrand parametrix for the Szegő projector at the boundary of a strongly pseudo-convex real-analytic domain can be realised by an analytic Fourier integral operator. We then study some applications, such as FBI-type transforms on compact, real-analytic Riemannian manifolds and propagators of one-homogeneous (pseudo)differential operators.

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We study approaches for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs). In this context, the empirical measure is contained within a natural convex set and can be approximated using convex optimization methods. Such an approximation gives rise to a coreset of data points. A key quantity that controls how large such a coreset has to be is the size of the largest ball around the empirical measure that is contained within the empirical convex set. The bulk of our work is concerned with deriving high probability lower bounds on the size of such a ball under various conditions and in various settings: we show how conditions on the density of the data and the kernel function can be used to infer such lower bounds; we further develop an approach that uses a lower bound on the smallest eigenvalue of a covariance operator to provide lower bounds on the size of such a ball; we extend the approach to approximate covariance operators and we show how it can be used in the context of kernel ridge regression. We also derive compression guarantees when standard algorithms like the conditional gradient method are used and we discuss variations of such algorithms to improve the runtime of these standard algorithms. We conclude with a construction of an infinite dimensional RKHS for which the compression is poor, highlighting some of the difficulties one faces when trying to move to infinite dimensional RKHSs.

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We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé–Galton–Watson trees in varying environment.

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At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to obtain an overlap greater than $a$, for some $a>0$, in the whole subcritical phase of inverse temperatures $\beta \in [0,\beta _c)$. Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally. Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.

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We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance (i.e., a strong effect of disorder on the free energy behavior) and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained by D. S. Fisher and coauthors by applying the renormalization group method.

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In a recent paper [CEG23], Chen, Erchenko and Gogolev have proven that if a Riemannian manifold with boundary has hyperbolic geodesic trapped set, then it can be embedded into a compact manifold whose geodesic flow is Anosov. They have to introduce some assumptions that we discuss here. We explain how some can be removed, obtaining in particular a result applicable to all reasonable 3 dimensional examples.

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We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in $\mathbb{R}^{n}\times \mathbb{R}$ (space $\times $ time), i.e., polynomial solutions of the heat equation satisfying $\partial _t p\lnot \equiv 0$ and

When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d/2\rceil $. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\lnot \equiv 0\pmod {4}$ and is 3 if $d\equiv 0\pmod {4}$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, we show that the maximum number of nodal domains is $\Theta (d^n)$ as $d\rightarrow \infty $ and lies between $\lfloor \frac{d}{n}\rfloor ^n$ and $\binom{n+d}{n}$ for all $n$ and $d$. As an application and motivation for counting nodal domains, we confirm existence of the singular strata in Mourgoglou and Puliatti’s two-phase free boundary regularity theorem for caloric measure.

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We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal{F}(\mathbb{R}^3)$. Generalizations for regularity properties of boundary maps between locally conformal representations are also discussed. An ingredient for the proofs is the concept of hyperplane conicality that we introduce for a $\theta $-Anosov representation into a reductive real-algebraic Lie group $\mathsf {G}$. In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson–Sullivan measure.

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The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law.

In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal.

In the present paper we establish links between the dynamical properties of the $\mathbb{Z}^d$-measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the $\mathbb{Z}^d$-action is of product type is a crucial step in this approach.

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