We establish the uniqueness of tangent maps for general weakly holomorphic and locally approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a by-product of the approach and the techniques developed, we also obtain the unique tangent cone property for a special class of nonrectifiable positive pseudoholomorphic cycles. This approach also gives a new proof of the main result by Bellettini on the uniqueness of tangent cones for positive integral $(p,p)$-cycles in arbitrary almost complex manifolds.

Let Γ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if Γ is either nonuniform or is uniform of orthogonal type and dimension at least 9, then Γ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first-order theory of Γ is undecidable, that all finitely generated subgroups of Γ are definable, and that Γ is characterized by a single first-order sentence among all finitely generated groups.

We prove the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak–Keller–Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this long-standing problem, we take advantage of a new modulated free energy and prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the proceedings of the Séminaire Laurent Schwartz, we provide the full proof of results announced by the authors in 2019.