The period-index problem for fields of transcendence degree $2$
Using geometric methods we prove the standard period-index conjecture for the Brauer group of a field of transcendence degree $2$ over $\mathbf{F}_p$.
View ArticleAlmost contact 5-manifolds are contact
The existence of a contact structure is proved in any homotopy class of almost contact structures on a closed $5$-dimensional manifold.
View ArticleA polynomial upper bound on Reidemeister moves
We prove that any diagram of the unknot with $c$ crossings may be reduced to the trivial diagram using at most $(236 \,c)^{11}$ Reidemeister moves.
View ArticleRationality of $W$-algebras: principal nilpotent cases
We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A...
View ArticleA proof of Demailly’s strong openness conjecture
In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.
View ArticleThe circle method and bounds for $L$-functions – IV: Subconvexity for twists...
Let $\pi$ be an $\mathrm{SL}(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $\chi$ be a primitive Dirichlet character modulo $M$,...
View ArticleIsolation, equidistribution, and orbit closures for the...
We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent...
View ArticleTightness is preserved by Legendrian surgery
This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is...
View ArticleOn Schmidt and Summerer parametric geometry of numbers
Recently, W. M. Schmidt and L. Summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in...
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