Volume 2017, Issue 726

In the present paper, we prove that if y′′=f⁢(y,y′,t,α,β,…)${y^{\prime\prime}\hskip-0.569055pt=\hskip-0.569055ptf(y,y^{\prime},t,\alpha,% \beta,\ldots)}$ is a “generic” Painlevé equation (that is, an equation in one of the families PI${P_{\rm I}}$–PVI${P_{\rm VI}}$ but with the relevant complex parameters α,β,…${\alpha,\beta,\ldots}$ algebraically independent), and if y1,…,yn${y_{1},\ldots,y_{n}}$ are solutions such that y1,y1′,y2,y2′,…,yn,yn′$y_{1},y_{1}^{\prime},y_{2},y_{2}^{\prime},\ldots,y_{n},y_{n}^{\prime}$ are algebraically dependent over ℂ⁢(t)${{\mathbb{C}}(t)}$, then already for some 1≤i<j≤n${1\leq i<j\leq n}$, yi,yi′,yj,yj′${y_{i},y_{i}^{\prime},y_{j},y_{j}^{\prime}}$ are algebraically dependent over ℂ⁢(t)${{\mathbb{C}}(t)}$. The proof combines results by the Japanese school on “irreducibility” of the Painlevé equations, with the trichotomy theorem for strongly minimal sets in differentially closed fields.

In this note, we will prove that given a sequence of Ricci flows (Mk,gk⁢(t))${(M_{k},g_{k}(t))}$ of dimension n , t∈[-1,0]${t\in[-1,0]}$, with Ricci curvature bounded below uniformly and volume non-collapsing, (Mk,gk⁢(0))${(M_{k},g_{k}(0))}$ converge as a sequence of manifolds to a limit length space (X,d)${(X,d)}$. The regular part of (X,d)${(X,d)}$ must be open and the convergence is of C1,α${C^{1,\alpha}}$ regularity. As a corollary of this result, we prove that in dimension 3, a Ricci flow can always extend unless either its Ricci curvature goes to -∞${-\infty}$ or it shrinks to a point.

We study the L2${L^{2}}$ gradient flow of the Yang–Mills functional on the space of connection 1-forms on a principal G -bundle over the sphere S2${S^{2}}$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space Ω⁢G${\Omega G}$ of based loops in the compact Lie group G . An isomorphism between these two Morse homologies is obtained by coupling a perturbed version of the Yang–Mills gradient flow with the L2${L^{2}}$ gradient flow of the classical energy functional on loops. Our result gives a positive answer to a question due to Atiyah.

Let F be a totally real number field and let p be a prime unramified in F . We prove the existence of Galois pseudo-representations attached to mod⁡pm${\operatorname{mod}p^{m}}$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for F of level prime to p .

In 1994, Brüdern and Fouvry [1] initiated the investigation of Lagrange’s four squares theorem with almost prime variables. In this paper, we prove that every sufficiently large integer, congruent to 4 modulo 24, can be represented as a sum of four squares of integers, each of which has at most four prime factors. Instead of the four-dimensional vector sieve developed by Brüdern and Fouvry [1], we establish this result by combining the three-dimensional sieve and the switching principle.

Let X be a complex projective variety and D a reduced divisor on X . Under mild conditions on the singularities of (X,D)$(X,D)$, which includes the case of smooth X with simple normal crossing D , and by running the minimal model program, we obtain by induction on dimension via adjunction geometric criteria guaranteeing various positivity conditions for KX+D$K_{X}+D$. Our geometric criterion for KX+D$K_{X}+D$ to be numerically effective yields also a geometric version of the cone theorem and a criterion for KX+D$K_{X}+D$ to be pseudo-effective with mild hypothesis on D . We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi–Yau manifolds, an optimal geometric sharpening of the Nakai–Moishezon criterion for the ampleness of a divisor of the form KX+D$K_{X}+D$, a criterion verified under a canonical hyperbolicity assumption on (X,D)$(X,D)$. Without these conjectures, we verify this ampleness criterion with mild assumptions on D , being none in dimension two and D≠0$D\neq 0$ in dimension three.

Nous appelons corps de Siegel une extension algébrique du corps des nombres rationnels sur laquelle un lemme de Siegel vaut. C’est classiquement le cas pour les corps de nombres mais aussi pour le corps des nombres algébriques d’après Roy–Thunder et Zhang. Nous donnons de nouveaux exemples. Nous montrons aussi qu’il existe des corps qui ne sont pas de Siegel, à savoir les corps de degré infini qui satisfont la propriété de Northcott, introduite par Bombieri–Zannier. Notre démarche repose sur l’étude de plusieurs séries de minima successifs associés à un espace adélique. Les différentes propriétés du corps se lisent sur des quantités généralisant la constante d’Hermite. Dans le cas des nombres algébriques, nous calculons leurs valeurs exactes. We define a Siegel field to be a subfield K of the algebraic numbers over which a Siegel lemma holds. Known examples include number fields (classical geometry of numbers) and the field of algebraic numbers itself (Roy–Thunder and Zhang). We investigate the situation for other fields. We provide new examples such as real algebraic numbers or Hilbert class fields towers, as well as counterexamples. We also relate the Siegel condition to other properties of K , for example the Northcott property introduced by Bombieri–Zannier. For this we define several series of successive minima for an adelic vector space over K , which give rise to diverse numerical invariants of K (akin to Hermite’s constant). When K is the field of algebraic numbers, we determine their exact values.

It is known that one can associate a Kontsevich-type formality morphism to every Drinfeld associator. In the present paper, we show that this morphism may be extended to a Kontsevich–Shoikhet formality morphism of cochains and chains, by describing the action of the Grothendieck–Teichmüller group on such objects (up to homotopy).

We construct the quantum curve for the Gromov–Witten theory of the complex projective line.