We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. And we prove that a related characterization is equivalent to an asymptotic lower bound on the growth of the Wasserstein distance between heat flows. For weighted Riemannian manifolds, the crucial result will be a precise uniform two-sided bound for \begin{eqnarray*}\frac{d}{dt}\Big|_{t=0}W_2\big(\hat P_t\delta_x,\hat P_t\delta_y\big)\end{eqnarray*}in terms of the mean value of the Bakry-Émery Ricci tensor ${\mathrm{Ric}}+{\mathrm{Hess}} f$ along the minimizing geodesic from $x$ to $y$ and an explicit correction term depending on the bound for the curvature along this curve.

In [Ar68], M. Armstrong proved a beautiful result describing fundamental groups of quotient spaces. In this paper we prove an analogue of Armstrong's theorem in the setting of $F$-divided [dS07] and essentially finite [No76] fundamental group schemes.

We establish a link between the derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. This provides a geometric reformulation of the non-abelian Reidemeister torsion corresponding to the adjoint of the monodromy representation of the hyperbolic structure. These results are then applied to the study of the `1-loop Conjecture' of Dimofte--Garoufalidis, which we generalize to arbitrary 1-cusped hyperbolic 3-manifolds. We verify the generalized conjecture in the case of the sister manifold of the figure-eight knot complement.

In this article, we show that a hypersurface of the nearly Kähler ${\mathbb{C}\mathrm{P}}^3$ or ${\mathbb{F}_{1,2}}$ cannot have its shape operator and induced almost contact structure commute together. This settles the question for six-dimensional homogeneous nearly Kähler manifolds, as the cases of ${\mathbb{S}}^6$ and ${\mathbb{S}}^3 \times {\mathbb{S}}^3$ were previously solved, and provides a counterpart to the more classical question for the complex space forms ${\mathbb{C}\mathrm{P}}^n$ and ${\mathbb{C}\mathrm{H}}^n$. The proof relies heavily on the construction of ${\mathbb{C}\mathrm{P}}^3$ and ${\mathbb{F}_{1,2}}$ as twistor spaces of ${\mathbb{S}^{4}}$ and ${\mathbb{C}\mathrm{P}}^2$.