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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We introduce and analyse the almost sure convergence of a new stochastic algorithm for the global minimization of Morse functions on compact Riemannian manifolds. This diffusion process is called fraudulent because it requires the knowledge of minimal value of the function to minimize. Its investigation is nevertheless important, since in particular it appears as the limit behavior of non-fraudulent and time-inhomogeneous swarm mean-field algorithms used in global optimization.

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We solve an asymptotic variant of a smooth version of the $A_n$-realization problem for plane curves. As an application, we determine the cobordism distance between torus links of type $T(d,d)$ and $T(2,N)$ up to an error of at most $3d$. We also discuss the limits of knot theoretic approaches aimed at solving the $A_n$-realization problem.

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We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways:

• We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus.
• We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

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We study a natural class of Fermi–Ulam Models featuring good hyperbolicity properties that we call dispersing Fermi–Ulam models. Using tools inspired by the theory of hyperbolic billiards we prove, under very mild complexity assumption, a Growth Lemma for our systems. This allows us to obtain ergodicity of dispersing Fermi–Ulam Models. It follows that almost every orbit of such systems is oscillatory.

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The aim of this article is to derive discontinuous finite elements vector spaces which can be put in a discrete de Rham complex for which the matching between the continuous and discrete cohomology spaces can be proven for periodic meshes.

First, the triangular case is addressed, for which we prove that this property holds for the classical discontinuous finite element space for vectors.

On Cartesian meshes, this result does not hold for the classical discontinuous finite element space for vectors. We then show how to use the de Rham complex found for triangular meshes for enriching the finite element space on Cartesian meshes in order to recover a de Rham complex, on which the same property is proven.

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We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group. We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.

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Let $S$ be a closed surface of genus $g\ge 2$, furnished with a Borel probability measure $\lambda $ with total support. We show that if $f$ is a $\lambda $-preserving homeomorphism isotopic to the identity such that the rotation vector $\operatorname{rot}_f(\lambda )\in H_1(S,\mathbb{R})$ is a multiple of an element of $H_1(S,\mathbb{Z})$, then $f$ has infinitely many periodic orbits.

Moreover, these periodic orbits can be supposed to have their rotation vectors arbitrarily close to the rotation vector of any fixed ergodic Borel probability measure.

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We begin by presenting a version of the Poincaré–Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex $K$. To that end we construct a cellular sheaf on $K$ whose cohomology with compact support is isomorphic to the homology of the initial cosheaf. Thereafter we use the first result to generalise the tropical version of the Lefschetz Hyperplane Section Theorem to some singular tropical toric varieties and singular tropical hypersurfaces.

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We consider a scattering theory for difference operators on $\mathcal{H}=\ell ^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-range potential $V:\mathbb{Z}^d\rightarrow \mathbb{R}^n$. One of the motivating examples is discrete Schrödinger operators on $\mathbb{Z}^d$-periodic graphs. We construct time-independent modifiers, so-called Isozaki–Kitada modifiers, and we prove that the modified wave operators with the above-mentioned Isozaki–Kitada modifiers exist and that they are complete.

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