We provide a proof of the Baum–Connes conjecture (with no coefficients) for word hyperbolic groups, based on the Green metric, a metric adapted to random walks. Résumé Nous proposons une démonstration de la conjecture de Baum–Connes (sans coefficients) pour les groupes hyperboliques en utilisant la distance de Green, une distance adaptée à l’étude des marches aléatoires.
We study the dynamics of complex algebraic families of maps on \mathbb{P}^{N} , and the geometry of their preperiodic points. The goal of this article is to formulate a conjectural characterization of the subvarieties of S \times\mathbb{P}^{N} containing a Zariski-dense set of preperiodic points, where the parameter space S is a quasi-projective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the relative Manin–Mumford conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin–Mumford conjecture (theorem of Raynaud) and the dynamical Manin–Mumford conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that many recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.
We simplify and improve the main fundamental theorems of positive characteristic generic vanishing theory. As a quick corollary of the theory, we prove that a normal variety X of maximal Albanese dimension satisfies H^0(X, ω_X) ≠ 0 and that if Alb(X) is ordinary, then S^0(X, ω_X) ≠ 0.
A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the cubic line complex G associated with the desmic tetrahedra introduced by G. Humbert. We prove that G is a rational Fano threefold with 34 nodes. The number 34 is the maximum number of nodes on a Fano threefold of degree 6 in P-5, and the group of projective automorphisms is isomorphic to S-4 (sic) 2 = (S(4 )x S-4)(sic) 2.
We study the dynamics of complex algebraic families of maps on P-N, and the geometry of their preperiodic points. The goal of this article is to formulate a conjectural characterization of the subvarieties of SxP(N) containing a Zariski-dense set of preperiodic points, where the parameter space S is a quasi-projective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the relative Manin-Mumford conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin-Mumford conjecture (theorem of Raynaud) and the dynamical Manin-Mumford conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that many recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.
We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable real vector bundle. The obstructions are fractional parts of integral Stiefel-Whitney classes, and a fourth of an appropriate combination of Pontryagin, Chern, and Euler classes.
The classical theorem that topological surfaces can be triangulated is proved using the torus trick of Kirby plus a few basic facts about smooth or PL surfaces.
In this paper we study billiards in regular n-sided polygons and monodromy over Teichm & uuml;ller curves from an arithmetic perspective. Our main results show that the combinatorics of billiards in a periodic direction s is controlled by the projection of s to a finite projective line.This arithmetic control coexists, experimentally, with chaotic behavior that first emerges when n=12.
In this expository note, we return to the classification of exceptional actions of PSL2(Z/pZ) on a set with p elements, announced by Galois in his last letter to Chevalier. Mathematics Subject Classification 2020: 20G40 (primary); 05E18 (secondary).
Soit G un groupe de Lie réel compact, et soit f un caractère irréductible de G de degré >1 . Nous montrons qu’il existe un élément g de G , d’ordre fini, tel que f(g)=0 , et nous complétons ce résultat de diverses manières, qui sont détaillées dans l’Introduction.
The Martin boundary of a diffusion process on a manifold consists of positive harmonic functions. In a previous publication, we erroneously assumed that the same holds for discretizations of diffusions. In this article, we correct this mistake and extend some of our previous results.
This text has previously been published in EMS Magazine [Eur. Math. Soc. Mag. 130 (2023), 56–59] and is reproduced here with the authorization of the authors and editors.
In 2017, for the pi-day, Micka & euml;l Launay described a remarkable property of the Fibonacci sequence, which he called "le myst & egrave;re de la farfalle". The name "farfalle" refers to a shape of pasta that is famous in France, a butterfly-shaped one, in connection with a likewise shape appearing in the numerical experiments. In these experiments, the Fibonacci numbers are put on the circle modulo N, and the segments joining consecutive elements of the sequence are drawn. Launay proposed a prize of 3:14 Euros to whom will explain the phenomenon he had discovered. We provide such an explanation, which illustrates recent works by Kurlberg and Rudnick and by Bourgain and Glibichuk.
We present a new, category theoretic point of view on finite Ramsey theory. Our aims are as follows: -- to define the category theoretic notions needed for the development of finite Ramsey Theory, -- to state, in terms of these notions, the general fundamental Ramsey results (of which various concrete Ramsey results are special cases), and -- to give self-contained proofs within the category theoretic framework of these general results. We also provide some concrete illustrations of the general method.
We describe the geometry of conjugation within any split subgroup H of the full isometry group G of n-dimensional Euclidean space. We prove that, for any h is an element of H, the conjugacy class [h](H) of h is described geometrically by the move-set of its linearization, while the set of elements conjugating h to a given h' is an element of [h](H) is described by the fix-set of the linearization of h'. Examples include all affine Coxeter groups, certain crystallographic groups, and the group G itself.
We construct a point set in the Euclidean plane that elucidates the relationship between the fine-scale statistics of the fractional parts of \sqrt{n} and directional statistics for a shifted lattice. We show that the randomly rotated, and then stretched, point set converges in distribution to a lattice-like random point process. This follows closely the arguments in Elkies and McMullen’s original analysis for the gap statistics of \sqrt{n}\bmod 1 in terms of random affine lattices [Duke Math. J. 123 (2004), 95–139]. There is, however, a curious subtlety: the limit process emerging in our construction is not invariant under the standard \operatorname{SL}(2,\mathbb{R}) -action on {\mathbb{R}}^{2} .
Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for A(infinity)-algebras, we give an explicit proof that the universal enveloping algebra UL of a differential graded Lie algebra L is Koszul, via an explicit contracting homotopy from the cobar construction Omega CL of the Chevalley-Eilenberg chain coalgebra CL of L to UL. This may be viewed as an extension of the Poincar & eacute;-Birkhoff-Witt theorem to L-infinity-algebras.
We aim at giving a pedagogical introduction to the non-abelian Hodge correspondence, a bridge between algebra, geometric structures and complex geometry. The correspondence links representations of a fundamental group, the character variety, to the theory of holomorphic bundles. We focus on motivations, key ideas, links between the concepts and applications. Among others we discuss the Riemann–Hilbert correspondence, Goldman's symplectic structure via the Atiyah–Bott reduction, the Narasimhan–Seshadri theorem, Higgs bundles, harmonic bundles and hyperkähler manifolds.
