Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler–Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $Spec\left(\mathbb{Z}\right)$. In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the zeta function $\zeta (\mathcal{X},s)$ at $s=0$ in terms of a perfect complex of abelian groups $R{\Gamma }_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé’s conjecture and to the Tamagawa number conjecture of Bloch–Kato, and deduce its validity in simple cases.

Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion, ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. This provides many examples of surfaces with negative second Segre number and big cotangent bundle.

We give an algebraic construction of standard modules—infinite-dimensional modules categorifying the Poincaré–Birkhoff–Witt basis of the underlying quantized enveloping algebra—for Khovanov–Lauda–Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an “affine quasihereditary algebra.” In simply laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces $(X,\mathsf{d},\mathfrak{m})$ which is stable under measured Gromov– Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm, and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local, and local-to-global properties. In these spaces, which we call $\mathrm{RCD}(K,\infty )$ spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry–Émery estimates and the $L^{\infty }\text{-}\mathrm{Lip}$ Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with $\mathsf{d}$, that the local energy measure has density given by the square of Cheeger’s relaxed slope, and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.