We introduce a multi-scale generalization of the notion of scalar product on real and complex vector spaces and study the corresponding Voronoi tilings. This framework allows to analyze limiting metric structures involving several levels of convergence. In particular, we describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings.

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Given a probability measure $\mu $ on a finitely generated group $\Gamma $, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu $. Endowing $\Gamma $ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu $ on $\Gamma $. Our main result exhibits a relatively hyperbolic group with convergent Poincaré series generated by $H_r(n)$.

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We introduce the notion of a flat extension of a connection $\theta $ on a principal bundle. Roughly speaking, $\theta $ admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented $3$-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with $\theta $. As an application, we recover the obstruction of Chern–Simons for the existence of a conformal immersion of a Riemannian $3$-manifold into Euclidean $4$-space. In addition, we obtain corresponding statements for a Lorentzian $3$-manifold, as well as a global obstruction for the existence of an equiaffine immersion into $\mathbb{R}^4$ of a $3$-manifold that is equipped with a torsion-free connection preserving a volume form.

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In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains $0$ arises in this way, as well as a class of compactly supported distributions with density.

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The systole of a contact form $\alpha $ is defined as the shortest period of closed Reeb orbits of $\alpha $. Given a non-trivial $\mathbb{S}^1$-principal bundle over $\mathbb{S}^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact forms on $M$ invariant under the $\mathbb{S}^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb{S}^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.

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We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

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Caffarelli’s contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: the trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim–Milman transport map.

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We study the inviscid Burgers equation on the circle $\mathbb{T}:= \mathbb{R}/\mathbb{Z}$ forced by the spatial derivative of a Poisson point process on $ \mathbb{R}\times \mathbb{T}$. We construct global solutions with mean $\theta $ simultaneously for all $\theta \in \mathbb{R}$, and in addition construct their associated global shocks (which are unique except on a countable set of $\theta $). We then show that as $\theta $ changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This can be seen as an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in $\theta $ to the density of a particle diffusing in the Burgers flow.

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Using the global Kuranishi charts constructed by Hirschi–Swaminathan, we define gravitational descendants and equivariant Gromov–Witten invariants for general symplectic manifolds. We prove that these invariants satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it to derive an explicit formula for the equivariant Gromov–Witten invariants of Hamiltonian GKM manifolds. In particular, the symplectic Gromov–Witten invariants of smooth toric varieties agree with their algebro-geometric counterpart. In the semipositive case, the invariants studied here recover those of Ruan and Tian.

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This article constructs Von Neumann invariants for constructible complexes and coherent $\mathcal{D}$-modules on compact complex manifolds, generalizing the work of the author on coherent $L_2$-cohomology. We formulate a conjectural generalization of Dingoyan’s $L_2$-Mixed Hodge structures in terms of Saito’s Mixed Hodge Modules and give partial results in this direction.

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In the search of a projective analog of Kunz’s theorem and a Frobenius-theoretic analog of Mori–Hartshorne’s theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such a kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank $1$.

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We attack the question of $E_2$-formality of differential graded algebras over $\mathbb{F}_p$ via obstruction theory. We are able to prove that $E_2$-algebras whose cohomology ring is a polynomial algebra on even degree classes are formal. As a consequence we prove $E_2$-formality of the classifying space of some compact Lie groups or of Davis–Januszkiewicz spaces.

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We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model’s energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.

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The infinite-bin model is a one-dimensional particle system on $\mathbb{Z}$ introduced by Foss and Konstantopoulos in relation with last passage percolation on complete directed acyclic graphs. In this model, at each integer time, a particle is selected at random according to its rank and produces a child at the location immediately to its right. In this article, we consider the limiting distribution of particles after an infinite number of branching events have occurred. Under mild assumptions, we prove that the event (called freezing) that a location contains only a finite number of balls satisfies a $0-1$ law and we provide various criteria to determine whether freezing occurs.

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For every integer $g \ge 2$ we show the existence of a compact Riemann surface $\Sigma $ of genus $g$ such that the rank two trivial holomorphic vector bundle $\mathcal{O}^{\oplus 2}_{\Sigma }$ admits holomorphic connections with $\operatorname{SL}(2,\mathbb{R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of Calsamiglia, Deroin, Heu and Loray. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma $, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

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