We show that the tritronquée solution $y_t$ of the Painlevé equation $P_I$ that behaves algebraically for large $z$ with $argz=\pi /5$ is analytic in a region containing the sector $\{z\ne 0,argz\in [-3\pi /5,\pi \left]\right\}$ and the disk $\{z:|z|<37/20\}$. This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. As a by-product, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than $1/100$ rigorous error bounds.

We prove that every positive-$(p,p)$ integral cycle in an arbitrary almost-complex manifold possesses at every point a unique tangent cone. The argument relies on an algebraic blowup perturbed in order to face the analysis issues of this problem in the almost-complex setting.

We define lines of minima in the thick part of outer space for the free group $F_n$ with $n\ge 3$ generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of $F_n$ defines such a line of minima. Now let $\Gamma$ be a subgroup of the outer automorphism group of $F_n$ which is not virtually abelian. We obtain that if $\Gamma$ contains at least one fully irreducible element, then for every $p\in (1,\infty )$ the second bounded cohomology group $H_b^2(\Gamma ,{\ell }^p(\Gamma \left)\right)$ is infinite-dimensional.

We prove Getzler’s claims about the cohomology of the moduli space of stable curves of genus one, that is, that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) relation and Getzler’s relation. In particular, the even cohomology ring is isomorphic to the tautological ring.

Soit $\mathrm{F}$ un corps localement compact non archimédien de caractéristique résiduelle $p$, soit $\mathrm{D}$ une algèbre à division centrale de dimension finie sur $\mathrm{F}$ et soit $\mathrm{R}$ un corps algébriquement clos de caractéristique différente de $p$. Nous classons les représentations lisses irréductibles de ${GL}_m\left(\mathrm{D}\right)$, $m⩾1$, à coefficients dans $\mathrm{R}$ en termes de multisegments, ce qui généralise des travaux de Zelevinski, Tadić et Vignéras. Nous prouvons que toute $\mathrm{R}$-représentation irréductible de ${GL}_m\left(\mathrm{D}\right)$ a un unique support supercuspidal, et nous obtenons ainsi deux classifications: l’une par les multisegments supercuspidaux, qui classe les représentations en fonction de leur support supercuspidal; l’autre par les multisegments apériodiques, qui classe les représentations en fonction de leur support cuspidal. Ces constructions sont effectuées de façon purement locale, et font un usage substantiel de la théorie des types.

Let $\mathrm{F}$ be a non-Archimedean locally compact field of residue characteristic $p$, let $\mathrm{D}$ be a finite-dimensional central division $\mathrm{F}$-algebra, and let $\mathrm{R}$ be an algebraically closed field of characteristic different from $p$. We classify all smooth irreducible representations of ${GL}_m\left(\mathrm{D}\right)$ for $m⩾1$, with coefficients in $\mathrm{R}$, in terms of multisegments, generalizing works by Zelevinski, Tadić, and Vignéras. We prove that any irreducible $\mathrm{R}$-representation of ${GL}_m\left(\mathrm{D}\right)$ has a unique supercuspidal support and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.