In this paper, we show that “$L$-complete null hypersurfaces” (i.e. ruled null hypersurfaces foliated by entirety of light-like lines) as wave fronts in $(n+1)$-dimensional Lorentz–Minkowski space are canonically induced by hypersurfaces in $n$-dimensional Euclidean space. As an application, we show that most of null wave fronts can be realized as restrictions of certain $L$-complete null wave fronts. Moreover, we determine $L$-complete null wave fronts whose singular sets are compact.

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn–Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we prove Gårding inequalities for three global symbolic pseudodifferential calculi on groups.

We develop a variational principle for mean dimension with potential of $\mathbb{R}^{d}$-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term. A basic strategy of the proof is the same as the case of $\mathbb{Z}$-actions. However measure theoretic details are more involved because $\mathbb{R}^{d}$ is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for $\mathbb{R}^{d}$-actions.

In this paper, for each $d > 0$, we study the minimum integer $h_{3,2d} \in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^{2} = 2d$ and Picard number $\rho(X) := \mathrm{rank} \operatorname{Pic} X = h_{3,2d}$ admitting an automorphism of order 3. We show that $h_{3,2} \in \{4, 6\}$ and $h_{3,2d} = 2$ for $d > 1$. Analogously, we study the minimum integer $h^{*}_{3,2d} \in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X, H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h^{*}_{3,2d}$ is equal to 2 if $d > 1$ and equal to 6 if $d = 1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds.

Let $f$ be a polynomial map from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ with $m > n > 0$ and $t_{0}$ be a regular value of $f$. For a small open ball $D_{t_{0}}$ centered at $t_{0}$, we show that the map $f : f^{-1}(D_{t_{0}}) \to D_{t_{0}}$ is a Serre fibration if and only if $f$ is a Serre fibration over a finite number of certain simple arcs starting at $t_{0}$. We characterize the fibration $f : f^{-1}(D_{t_{0}}) \to D_{t_{0}}$ by relative homotopy groups defined for these arcs and use it to prove the assertion.

For a flat proper morphism of finite presentation between schemes with almost coherent structural sheaves (in the sense of Faltings), we prove that the higher direct images of quasi-coherent and almost coherent modules are quasi-coherent and almost coherent. Our proof uses Noetherian approximation, inspired by Kiehl's proof of the pseudo-coherence of higher direct images. Our result allows us to extend Abbes–Gros' proof of Faltings' main $p$-adic comparison theorem in the relative case for projective log-smooth morphisms of schemes to proper ones, and thus also their construction of the relative Hodge–Tate spectral sequence.

Borel's stability and vanishing theorem gives the stable cohomology of $\operatorname{GL}(n, \mathbb{Z})$ with coefficients in algebraic $\operatorname{GL}(n, \mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild–Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\operatorname{Aut}(F_{n})$ of the free group $F_{n}$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\operatorname{IA}_{n}$ of $F_{n}$. We propose a conjectural algebraic structure of the stable rational cohomology of $\operatorname{IA}_{n}$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.

A system of nonlinear differential equations $x^{1+\gamma} (dY/dx) = F_{0}(x) + A(x)Y + F(x, Y)$ $(\gamma \geq 1)$ is considered. The origin $x = 0$ is irregular singular. There exist pioneering works about them. We study more precisely than preceding works, the meaning of asymptotic expansion of transformations and solutions by using Borel summable functions in asymptotic analysis and construct exponential series solutions often called transseries.

By a lot of previous work, it is known that the zeros of the period polynomial for a newform $f \in S_{k}(\Gamma_0(N))$ all lie on the circle $|z| = 1/\sqrt{N}$. In this paper we show that these zeros satisfy various interlacing properties for fixed $N$ and varying $k$ when either $k$ or $N$ is large. We also present a complete result when $N = 1$. Lastly, we establish the interlacing properties when $k$ is fixed and $N$ varies.

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D}^{\mathsf{b}}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we study the structure of the Verdier quotient $\operatorname{\mathsf{D}^{\mathsf{b}}}(R)/\operatorname{\mathsf{thick}}(R \oplus k)$. We give necessary and sufficient conditions for it to admit an additive generator.

In this note, we establish Guan–Zhou's unified version of optimal $L^{2}$ extension theorem for holomorphic vector bundles with smooth Hermitian metrics on weakly pseudoconvex Kähler manifolds. Combining with previous work of Guan–Mi–Yuan, we generalize Guan–Zhou's unified version of optimal $L^{2}$ extension theorem to weakly pseudoconvex Kähler manifolds.

Briançon and Speder gave an example of a $\mu$-constant family of weighted homogeneous polynomials for which $\mu^{*}$ is not constant. In this note we analyze this example. We study similar weighted homogeneous polynomials and determine the number of possible different topology of curves which are obtained as generic plane sections.

We consider a generalized Gambaudo–Ghys construction on bounded cohomology and prove its injectivity. As a corollary, we prove that the third bounded cohomology of the group of area-preserving diffeomorphisms on the 2-disk is infinite-dimensional. Additionally, we establish similar results for the 2-sphere, the 2-torus, and the annulus.

Every $a$ in the torus $A = \mathbb{R}^{3}/2\mathbb{Z}^{3}$ determines a unique spherical, Euclidean or hyperbolic triangle $T(a)$ with angles $(\pi a_{i})$. In this paper we study the Galois orbits $\operatorname{Gal}(a)$ of torsion points $a \in A$, focusing on the ramification density $$ \rho(a) = \frac{|\{b \in \operatorname{Gal}(a) \: : \: T(b) \;\text{is spherical}\}|}{|\operatorname{Gal}(a)|} \cdot $$ We show that the closure $\overline{R}$ of the set of values of $\rho(a)$ is a countable subset of $[0, 1]$, with 0 and 1 as isolated points. The spectral gaps at 0 and 1 lead to general finiteness statements for the classical triangle groups $\Delta(p,q,r) \subset \operatorname{SL}_{2}(\mathbb{R})$. For example, we obtain a conceptual proof, based on equidistribution, that the set of arithmetic triangle groups is finite.

We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of the flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable $\mathscr{R}$-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.

For virtual knot theory, the virtual braid group was defined by generalizing the braid group. It was proved that any virtual link can be obtained by the closure of a virtual braid. On the other hand, due to work by Jones et al., it is known that any (oriented) link is constructed from an element of Thompson's group $F$. In this paper, we define the “virtual version” of Thompson's group $F$ and prove that any virtual link is constructed from an element of the group.

Let $S$ be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle $T_{S}$ is pseudo-effective if and only if the canonical divisor $K_{S}$ is nef and the second Chern class vanishes, i.e., $c_{2}(S) = 0$. Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.

Assume an almost huge cardinal with Mahlo target exists. We construct a model of ZFC in which a small cardinal carries a strongly centered filter by forcing with an iteration of two nested products of Levy collapses.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. The asymptotic behavior of the quotient modules $M/I^{n} M$ of $M$ is an actively studied subject in commutative algebra. The main result of this paper shows that for large integers $n > 0$, the depth of the localizations of $(M/I^{n} M)_{\mathfrak{p}}$ are stable uniformly for all prime ideals $\mathfrak{p}$ of $R$ in each of the following cases: (1) $R$ is CM-excellent, (2) $R$ is semi-local, (3) $M$ or $M/I^{n} M$ for some $n > 0$ is Cohen–Macaulay.

It was conjectured that there are no hyperharmonic integers $h_n^{(r)}$ except 1. In 2020, a disproof of this conjecture was given by showing the existence of infinitely many hyperharmonic integers. However, the corresponding proof does not give any general density results related to hyperharmonic integers. In this paper, we first get better error estimates for the counting function of the pairs $(n, r)$ that correspond to non-integer hyperharmonic numbers using sums on gaps between consecutive prime numbers. Then, based on a plausible assumption on prime powers with restricted digits, we show that there exist positive integers $n$ such that the set of positive integers $r$ where $h_{n}^{(r)} \in \mathbb{Z}$ has positive density. Apart from that, we also obtain exact densities of sets $\{r \in \mathbb Z_{>0} : h_{33}^{(r)} \in \mathbb{Z}\}$ and $\{r \in \mathbb Z_{>0} : h_{39}^{(r)} \in \mathbb{Z}\}$. Finally, we give the smallest hyperharmonic integer $h_n^{(r)}$ greater than 1, which is obtained when $n = 33$ and $r = 10\, 667\, 968$.

A minimal group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, we prove that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary.