It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. The reverse implication is in general false. It is shown that under a (possibly negative) lower bound condition on a natural notion of curvature associated to a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. The results are essentially the best possible (up to constants) and significantly extend all previously known results, which could deduce only dimension-dependent bounds or could not deduce anything stronger than a linear isoperimetric inequality in the restrictive nonnegative curvature setting. As a corollary, all of these previous results are recovered and extended by generalizing an isoperimetric inequality of Bobkov. Further applications will be described in subsequent works. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.

We prove the Morrison-Kawamata cone conjecture for Kawamata log terminal Calabi-Yau pairs in dimension $2$. For a large class of rational surfaces as well as for K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain.

Let $C$ be the category of finite-dimensional representations of a quantum affine algebra $U_q(\stackrel{\wedge }{\mathfrak{g}})$ of simply laced type. We introduce certain monoidal subcategories $C_{\ell }(\ell \in \mathbb{N})$ of $C$, and we study their Grothendieck rings using cluster algebras.

We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface in order to verify the numerical criterion.