Given a totally real number field $F$ and its adèle ring $\mathbb{A}_{F}$, let $\pi$ vary in the set of irreducible cuspidal automorphic representations of $\operatorname{PGL}_{2}(\mathbb{A}_{F})$ corresponding to primitive Hilbert modular forms of a fixed weight. We determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s, \operatorname{Sym}^{r}(\pi))$ weighted by special values of the symmetric square $L$-functions $L((z+1)/2, \operatorname{Sym}^{2}(\pi))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \leq 1$, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power $L$-functions for $F = \mathbb{Q}$ with harmonic weight. Hence our result is regarded as a $z$-interpolation of Ricotta and Royer's result. If $z = 0$, the density of low-lying zeros weighted by central values is of a different type only when $r = 2$.

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension 3 turns out to be purely algebraic. One of our main results is the complete and explicit description of integrable to second order infinitesimal Einstein deformations on the complex 2-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $\mathrm{Gr}_{2}(\mathbb{C}^{n+2})$ for $n$ odd is rigid.

We present a new notion, the upper Aikawa codimension, and establish its equivalence with the upper Assouad codimension in a metric space with a doubling measure. To achieve this result, we first prove a variant of a local fractional Hardy inequality.

Honda, Izawa and Suwa define Čech–Dolbeault representation of hyperfunctions and an embedding of distributions to the space of hyperfunctions. With this embedding, we can regard $C^{\infty}$ functions as hyperfunctions in the framework of Čech–Dolbeault cohomology.

This article aims to characterize a Čech–Dolbeault representative which corresponds to the image of the embedding of a $C^{\infty}$ function, and also to construct the inverse map of the embedding of $C^{\infty}$ functions.

In this paper, we derive the generalized hypergeometric functions (period integrals) used in mirror computation of Calabi–Yau hypersurface in $CP^{N-1}$ as generating functions of intersection numbers on the moduli space of quasimaps from $CP^{1}$ with $2 + 1$ marked points to $CP^{N-1}$.

Höhn and Mason classified the groups acting symplectically on an irreducible holomorphic symplectic (IHS) manifold of K3$^{[2]}$-type, finding that $\mathbb{Z}_{3}^{4} : \mathcal{A}_{6}$ is the one with the largest order. In this paper we study IHS manifolds of K3$^{[2]}$-type with a symplectic action of $\mathbb{Z}_{3}^{4} : \mathcal{A}_6$ which also admit a non-symplectic automorphism. We characterize such IHS manifolds and prove their existence. We also prove that the order of a finite group acting on an IHS manifold of K3$^{[2]}$-type is bounded by 174960, this bound is sharp and there is a unique IHS manifold of K3$^{[2]}$-type acted by a group of this order, which is the Fano variety of lines of the Fermat cubic fourfold.

Let $k \geq 2$ be a given integer. We study the set of 3-fold canonical thresholds $\mathrm{ct}(X;S)$ with $\frac{1}{k} < \mathrm{ct}(X;S) < \frac{1}{k-1}$ where $S$ is a $\mathbb{Q}$-Cartier prime divisor of a projective 3-fold $X$. Express $\mathrm{ct}(X;S)$ as the rational number $\frac{a}{m}$ where $a$ (resp. $m$) denotes the weighted discrepancy (resp. weighted multiplicity). We conclude that if $a \geq 54k^{4}$, then we may choose positive integers $p$ and $q$ satisfying $\mathrm{ct}(X;S) = \frac{a}{m} = \frac{1}{k} + \frac{q}{p}$ and $q < 6k^{3}$. As a consequence, the set of accumulation points of the set of 3-fold canonical thresholds consists of $\{ 0 \} \cup \{ \frac{1}{k} \}_{k \in \mathbb{Z}_{\geq 2}}$. Moreover, we generalize the ACC for the set of 3-fold canonical thresholds to pairs.

We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second author [Proc. Amer. Math. Soc., 151 (2023), 3645–3653] and the solution of Hirst's conjecture [B.-W. Wang and J. Wu, Bull. London Math. Soc., 40 (2008), 18–22], both previously obtained for the regular continued fraction. To prove the result, we construct non-autonomous iterated function systems well-adapted to the given restrictions and growth conditions on partial quotients, estimate the associated pressure functions, and then apply Bowen's formula.

Let $D$ be a strongly self-absorbing $C^{*}$-algebra. In previous work, we showed that locally trivial bundles with fibers $\mathcal{K} \otimes D$ over a finite CW-complex $X$ are classified by the first group $E^{1}_{D}(X)$ in a generalized cohomology theory $E^{*}_{D}(X)$. In this paper, we establish a natural isomorphism $ E^{1}_{D \otimes \mathcal{O}_{\infty}}(X) \cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{D}(X)$ for stably-finite $D$. In particular, $E^{1}_{\mathcal{O}_{\infty}}(X) \cong H^{1}(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{\mathcal{Z}}(X)$, where $\mathcal{Z}$ is the Jiang–Su algebra. The multiplication operation on the last two factors is twisted in a manner similar to Brauer theory for bundles with fibers consisting of graded compact operators. The proof of the isomorphism described above made it necessary to extend our previous results on generalized Dixmier–Douady theory to graded $C^{*}$-algebras. More precisely, for complex Clifford algebras $\mathbb{C}\ell_{n}$, we show that the classifying spaces of the groups of graded automorphisms of $\mathbb{C}\ell_{n} \otimes \mathcal{K} \otimes D$ possess compatible infinite loop space structures. These structures give rise to a cohomology theory $\hat{E}^{*}_{D}(X)$. We establish isomorphisms $\hat{E}^{1}_{D}(X) \cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{D}(X)$ and $\hat{E}^{1}_{D}(X) \cong E^{1}_{D \otimes \mathcal{O}_{\infty}}(X)$ for stably finite $D$. Together, these isomorphisms represent a crucial step in the integral computation of $E^{1}_{D \otimes \mathcal{O}_{\infty}}(X)$.

In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $\pi : M \to S^{1}$ with fiber $N$ and structure group $\varGamma$ and $r \in \mathbb{Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(\pi, r) \in \mathbb{Z}_{\geq 0}$ and construct a function $\nu^{\wedge} : \mathrm{Diff}^{r}_{\pi}(M)_{0} \to \mathbb{R}_{k}$. When $k \geq 1$, it is shown that the bundle diffeomorphism group $\mathrm{Diff}^{r}_{\pi}(M)_{0}$ is bounded and $clb_{\pi}d\, \mathrm{Diff}^{r}_{\pi}(M)_{0} \leq k + 3$, if $\mathrm{Diff}^{r}_{\varrho,c}(E)_{0}$ is perfect for the trivial fiber bundle $\varrho : E \to \mathbb{R}$ with fiber $N$ and structure group $\varGamma$. On the other hand, when $k = 0$, it is shown that $\nu^{\wedge}$ is a unbounded quasimorphism, so that $\mathrm{Diff}^{r}_{\pi}(M)_{0}$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $\varphi$ for a mapping torus $\pi : M_{\varphi} \to S^{1}$ and give some explicit examples of (un)bounded groups.

In this article, we will investigate a generalization of the Dirichlet form associated with a one-dimensional diffusion process. In this generalization, the scale function, which determines the expression of the Dirichlet form, is only required to be non-decreasing. While this generalized form is almost a Dirichlet form, it does not satisfy regularity in general. Consequently, it cannot be directly associated with a process in probability theory. To tackle this issue, we adopt Fukushima's regular representation method, which enables to find a family of strong Markov processes that are homeomorphic to each other and related to the generalized form in a certain sense. Additionally, this correspondence reveals the connection between this generalized form and a quasidiffusion. Moreover, we interpret the probabilistic implications behind the regular representation through two intuitive transformations. These transformations offer us the opportunity to obtain another symmetric non-strong Markov process with continuous sample paths. The Dirichlet form of this non-strong Markov process is precisely the non-regular generalized form we previously analyzed. Furthermore, the strong Markov process obtained from the regular representation is its Ray–Knight compactification.

In this paper, we consider infinite length versions of multiple zeta-star values. We give several explicit formulas for the infinite length versions of multiple zeta-star values. We also discuss analytic properties of the map from indices to the infinite length versions of multiple zeta-star values.

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.

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.

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.