This note is based on a series of lectures delivered at Kyoto University in 2015. This note surveys the homogeneous Besov space ${\dot{B}}_{pq}^s$ on ${\mathbb{R}}^n$ with $1\le p$, $q\le \infty$ and $s\in \mathbb{R}$ in a rather self-contained manner. Among other results, we show that $\mathcal{S}{\text{'}}_{\infty }$ and $\mathcal{S}\text{'}/\mathcal{P}$ are isomorphic, and we also discuss the realizations in ${\dot{B}}_{pq}^s$. The fact that $\mathcal{S}{\text{'}}_{\infty }$ and $\mathcal{S}\text{'}/\mathcal{P}$ are isomorphic can be found in textbooks. The realization of ${\dot{B}}_{pq}^s$ can be found in works by Bahouri, Chemin, and Danchin and by Bourdaud for example. Here, we prove these facts using fundamental results in functional analysis such as the Hahn–Banach extension theorem.

We characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.

According to Auslander’s formula, one way of studying an abelian category $\mathcal{C}$ is to study $\mathrm{mod-}\mathcal{C}$, which has nicer homological properties than $\mathcal{C}$, and then translate the results back to $\mathcal{C}$. Recently, Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander’s formula, including the versions for all modules and for unbounded derived categories. We also include some results concerning recollements of triangulated categories.

After 2000, interest in the Hausdorff operators grew, first in the sense of the variety of spaces on which these operators were considered. Here we give conditions ensuring the boundedness of such operators on Morrey-type spaces. The sharpness of the obtained results is studied, and classes of the Hausdorff operators are described for which the necessary and sufficient conditions coincide.

We consider the derived category of an Artin–Mumford quartic double solid blown up at $10$ ordinary double points. We show that it has a semiorthogonal decomposition containing the derived category of the Enriques surface of a Reye congruence. This answers affirmatively a conjecture by Ingalls and Kuznetsov.

In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $\mathrm{U}(n,n)\left({\mathbf{A}}_F\right)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_{\varphi }$ (of an elliptic modular form $\varphi$) on $\mathrm{U}(n,n)\left({\mathbf{A}}_{\mathbf{Q}}\right)$, proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence prime for $I_{\varphi }$ is controlled by the $\ell$-divisibility of a product of special values of the symmetric square $L$-function of $\varphi$. We close the paper by computing an example when our main theorem applies.

We construct the minimal regular model of the Fermat curve of odd squarefree composite exponent $N$ over the $N$th cyclotomic integers. As an application, we compute upper and lower bounds for the arithmetic self-intersection of the dualizing sheaf of this model.

We introduce a variation of the well-known Newton–Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parameterized by a real number $\epsilon \in {\mathbb{R}}_{\ge 0}$. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way that does not depend on the choice of parameter.

We extend the notion of a fundamental negatively $\mathbb{Z}$-graded Lie algebra ${\mathfrak{m}}_x={\oplus }_{p\le -1}{\mathfrak{m}}_x^p$ associated to any point of a Levi nondegenerate Cauchy-Riemann (CR) manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal{D},\mathcal{J})$ for all $k\ge 2$ and call this invariant the core at $x\in M$. It consists of a $\mathbb{Z}$-graded vector space ${\mathfrak{m}}_x={\oplus }_{p\le k-2}{\mathfrak{m}}_x^p$ of height $k-2$ endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of $(M,\mathcal{D},\mathcal{J})$ and the Levi forms of higher order. In the case of CR manifolds of hypersurface type, we propose a definition of a homogeneous model of type $\mathfrak{m}$, that is, a homogeneous $k$-nondegenerate CR manifold $M=G/G_o$ with core $\mathfrak{m}$ associated with an appropriate $\mathbb{Z}$-graded Lie algebra $\mathrm{Lie}\left(G\right)=\mathfrak{g}=\oplus {\mathfrak{g}}^p$ and subalgebra $\mathrm{Lie}\left(G_o\right)={\mathfrak{g}}_o=\oplus {\mathfrak{g}}_o^p$ of the nonnegative part ${\oplus }_{p\ge 0}{\mathfrak{g}}^p$. It generalizes the classical notion of Tanaka of homogeneous models for Levi nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous $5$-dimensional $2$-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the $7$-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of $7$-dimensional $2$-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the $3$-nondegenerate class.

Let $G$ be a finite subgroup of ${Aut}_k\left(K\right(x_1,\dots ,x_n\left)\right)$, where $K/k$ is a finite field extension and $K(x_1,\dots ,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1,\dots ,x_n)$ is called quasimonomial if it satisfies the following three conditions: (i) $\sigma \left(K\right)\subset K$ for any $\sigma \in G$; (ii) $K^G=k$, where $K^G$ is the fixed field under the action of $G$; (iii) for any $\sigma \in G$ and $1\le j\le n$, $\sigma \left(x_j\right)=c_j\left(\sigma \right){\prod }_{i=1}^n{x}_i^{a_{ij}}$, where $c_j\left(\sigma \right)\in K^{\times }$ and $[a_{i,j}{]}_{1\le i,j\le n}\in {GL}_n(\mathbb{Z})$. A quasimonomial action is called purely quasimonomial if $c_j\left(\sigma \right)=1$ for any $\sigma \in G$ and any $1\le j\le n$. When $k=K$, a quasimonomial action is called monomial. The main question is: Under what situations is $K(x_1,\dots ,x_n{)}^G$ rational (i.e., $=$ purely transcendental) over $k$? For $n=1$, the rationality problem was solved by Hoshi, Kang, and Kitayama. For $n=2$, the problem was solved by Hajja when the action is monomial, by Voskresenskii when the action is faithful on $K$ and purely quasimonomial, which is equivalent to the rationality problem of $n$-dimensional algebraic $k$-tori which split over $K$, and by Hoshi, Kang, and Kitayama when the action is purely quasimonomial. For $n=3$, the problem was solved by Hajja, Kang, Hoshi, and Rikuna when the action is purely monomial, by Hoshi, Kitayama, and Yamasaki when the action is monomial except for one case, and by Kunyavskii when the action is faithful on $K$ and purely quasimonomial. In this paper, we determine the rationality when $n=3$ and the action is purely quasimonomial except for a few cases using a conjugacy classes move technique. As an application, we will show the rationality of some $5$-dimensional purely monomial actions which are decomposable.

We previously showed that the fundamental groupoid of a topological space can be defined by the Seifert–van Kampen theorem. This allowed us to give the first axiomatization of the topological fundamental groupoid. We will prove in this paper that the analogue holds for the étale fundamental groupoid of a Noetherian scheme $X$ as well.