In a previous article, Sarkar and Wang [15] gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version

We study the pseudospectrum of a class of nonselfadjoint differential operators. Our work consists of a microlocal study of the properties that rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this article a simple, necessary, and sufficient condition on the Weyl symbol of these operators which ensures the stability of their spectra. When this condition is violated, we prove that strong spectral instabilities occur for the high energies of these operators in some regions that can be far away from their spectra

We give an explicit formula for certain global trilinear forms that appear in Jacquet's conjecture in terms of local trilinear forms and the central values of triple product $L$-functions

We construct Golod-Shafarevich groups with property $(T)$ and thus provide counterexamples to a conjecture stated in a recent article of Zelmanov [Z2]. Explicit examples of such groups are given by lattices in certain topological Kac-Moody groups over finite fields. We provide several applications of this result, including examples of residually finite torsion nonamenable groups

The main result of this article is a (practically optimal) criterion for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg [40], [41] and F. Campana [6]. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon and J. McKernan [16] and S. Takayama [29], [30] concerning the extension of twisted pluricanonical forms. More applications will be offered in [4], the sequel to this article

We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where “sufficiently general” means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard number equal to one