One main purpose of the present paper is to reorganize, in terms of the notion of a \textit{Schwarz system}, a proof, by means of \textit{Schwarzian derivatives}, of the existence of complex projective structures on compact hyperbolic Riemann surfaces and a proof, by means of \textit{Sugiyama-Yasuda locally exact differentials}, of the existence of Frobenius-projective structures of level two on projective smooth curves in characteristic two. Moreover, we also construct quasi-Schwarz systems for certain Frobenius-affine and Frobenius-projective structures on projective smooth curves in positive characteristic.

We show that the Liouville heat kernel decays fast at large distances. In particular, the Liouville semigroup $T_t$ is $C_0$-Feller, where $C_0$ is the space of real-valued continuous functions on ${\mathbb C}$ vanishing at infinity. This is a problem mentioned in the paper [2].

In this paper, we are going to estimate the lower bounds for the first eigenvalues of the buckling problem and clamped plate problem by considering a positive lower bound for the weighted Ricci curvature. Also, we extended the results for the $p$-biharmonic operator and we prove a Lichnerowicz-Obata-Cheng type estimate for the biharmonic operators.

We show that given any two minimal models of a generalized lc pair, there exist small birational models which are connected by a sequence of symmetric flops. We also present some applications of this result.

Let $k$ be a field. In this article, we identify the component of weight 2 of the natural $\mathbf G_{m,k}$-graduation on the $k$-algebra of the arc scheme attached to an affine algebraic variety $X$ with the module of the 2-nd order derivations on $X$. We in particular deduce, from this property, characterizations of the geometry of hypersurfaces (in affine spaces) in terms of the nilpotency on arc scheme.

In this paper, we shall focus on the Wasserstein distance between two jump processes determined by stochastic differential equations in $\mathbb{R}^d$ or the Riemannian manifold $M$. As an application, the study on the Wasserstein distance implies that the law of the subordinated Brownian motion on $M$ is different from the one of the canonical projected process of the Marcus-type equation with jumps valued in the bundle of orthonormal frames $O(M)$.

We compute the rational Betti numbers of the real toric varieties associated to Weyl chambers of types $E_7$ and $E_8$, completing the computations for all types of root systems.

In this paper, we construct a Fukaya category of any infinite type surface whose objects are \textit{gradient sectorial Lagrangians}. This class of Lagrangian submanifolds is introduced by one of the authors in [20] which can serve as an object of a Fukaya category of any Liouville manifold that admits an exhausting proper Morse function, in particular on the Riemann surface of infinite type. We describe a generating set of the Fukaya category in terms of the end structure of the surface when the surface has countably many limit points in its ideal boundary, the latter of which can be described in terms of a subset of the Cantor set. We also show that our Fukaya category is not quasi-equivalent tothe limit of the Fukaya category of surfaces of finite type appearing in the literature.