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.

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form a two, three or fourfold lattice tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form fivefold lattice tilings. This paper characterizes all the convex domains which can form five or sixfold lattice tilings of the Euclidean plane. They are parallelograms, centrally symmetric hexagons, three types of centrally symmetric octagons and three types of centrally symmetric decagons.

The joint distribution of the first hitting time of a Brownian motion with constant drift to a sphere and the hitting place is studied. Explicit formulae by means of spherical harmonics for the joint density are presented. The result is applied to a study on the asymptotic behavior of the distribution function.

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite dimensional spaces. Then smooth rearrangement of multiple eigenvalues is explicitly given. This result follows from an abstract theory, applicable to general perturbations of symmetric bilinear forms.

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.

G. Lupton conjectured that the group of self-homotopy equivalences of an $F_{0}$-space inducing the identity on the homotopy groups is finite. Thus, the aim of this paper is to establish this conjecture.

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.

Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p > 0$. The first main result in this paper is to show that support $\tau$-tilting $k\tilde{G}$-modules with some properties are support $\tau$-tilting modules as $kG$-modules, too. As the second main result, we give equivalent conditions for support $\tau$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $\tau$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules as posets. As an application, we show that the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules is isomorphic to the set of support $\tau$-tilting $k\tilde{G}$-modules in the case that the index of $G$ in $\tilde{G}$ is a $p$-power. As a further application, we give a feature of vertices of indecomposable $\tau$-rigid $k\tilde{G}$-modules. Finally, we give block versions of the above results.

Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ be a unipotent group defined over $k$, and $\sigma$ a $k$-rational involution of $U$ with fixed points $U^{+}$. As a consequence of the results of Moore, the space $L^{2}(U(k) \backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+$ to be the period integral attached to $\sigma$ on the space of smooth vectors of $L^{2}(U(k) \backslash U_{\mathbb{A}})$, we prove that if $\Pi$ is a topologically irreducible subspace of $L^{2}(U(k) \backslash U_{\mathbb{A}})$, then $p^+$ is nonvanishing on the subspace of smooth vectors in $\Pi$ if and only if $\Pi^{\vee} = \Pi^{\sigma}$. This is a global analogue of local results of Benoist and the author, on which the proof relies.

Let $f_{1} : (\mathbb{R}^{n}, \mathbf{0}_{n}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ and $f_{2} : (\mathbb{R}^{m}, \mathbf{0}_{m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ be real analytic map germs of independent variables, where $n, m \geq 2$. Then the pair $(f_{1}, f_{2})$ of $f_{1}$ and $f_{2}$ defines a real analytic map germ from $(\mathbb{R}^{n+m}, \mathbf{0}_{n+m})$ to $(\mathbb{R}^{4}, \mathbf{0}_{4})$. We assume that $f_{1}$ and $f_{2}$ satisfy the $a_{f}$-condition at $\mathbf{0}_{2}$. Let $g$ be a strongly non-degenerate mixed polynomial of 2 complex variables which is locally tame along vanishing coordinate subspaces. A mixed polynomial $g$ defines a real analytic map germ from $(\mathbb{C}^{2}, \mathbf{0}_{4})$ to $(\mathbb{C}, \mathbf{0}_{2})$. If we identify $\mathbb{C}$ with $\mathbb{R}^{2}$, then $g$ also defines a real analytic map germ from $(\mathbb{R}^{4}, \mathbf{0}_{4})$ to $(\mathbb{R}^{2}, \mathbf{0}_{2})$. Then the real analytic map germ $f : (\mathbb{R}^{n} \times \mathbb{R}^{m}, \mathbf{0}_{n+m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ is defined by the composition of $g$ and $(f_{1}, f_{2})$, i.e., $f(\mathbf{x}, \mathbf{y}) = (g \circ (f_{1}, f_{2}))(\mathbf{x}, \mathbf{y}) = g(f_{1}(\mathbf{x}), f_{2}(\mathbf{y}))$, where $(\mathbf{x}, \mathbf{y})$ is a point in a neighborhood of $\mathbf{0}_{n+m}$.

In this paper, we first show the existence of the Milnor fibration of $f$. We next show a generalized join theorem for real analytic singularities. By this theorem, the homotopy type of the Milnor fiber of $f$ is determined by those of $f_{1}, f_{2}$ and $g$. For complex singularities, this theorem was proved by A. Némethi. As an application, we show that the zeta function of the monodromy of $f$ is also determined by those of $f_{1}, f_{2}$ and $g$.

In this work, we translate at the level of decorated trees some of the crucial arguments which have been used by P. Linares et al. in their recent paper to propose a diagram-free approach for the convergence of the model in regularity structures. This allows us to broaden the perspective and enlarge the scope of singular SPDEs covered by this approach. It also sheds new light on algebraic structures introduced in the foundational paper of M. Hairer on regularity structures which was used later for recursively described renormalised models.

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 $r = 10\, 667\, 968$.

The paper applies the JSJ decomposition and Koda–Ozawa's annulus classification to analyze the annulus configuration in a handlebody-knot exterior. We introduce the notion of the annulus diagram, to pack the configuration into a labeled graph, and classify genus two handlebody-knots in terms of their annulus diagrams. Applications to handlebody-knot symmetries are discussed; methods to produce handlebody-knots with various types of annulus diagrams are also presented.

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.

Let $G$ be a finite group. If $n \leq 5$ then any $n$-dimensional homotopy sphere never admits a smooth action of $G$ with exactly one fixed point. Let $A_n$ and $S_n$ denote the alternating group and the symmetric group on some $n$ letters. If $n \geq 6$ then the $n$-dimensional sphere possesses a smooth action of $A_5$ with exactly one fixed point. Let $V$ be an $n$-dimensional real $G$-representation with exactly one fixed point. It is interesting to ask whether there exists a smooth $G$-action with exactly one fixed point on the $n$-dimensional sphere such that the associated tangential $G$-representation is isomorphic to $V$. In this paper, we study this problem for nonsolvable groups $G$ and real $G$-representations $V$ satisfying certain hypotheses. Applying a theory developed in this paper, we can prove that the $n$-dimensional sphere has an effective smooth action of $S_5$ with exactly one fixed point if and only if $n = 6$, 10, 11, 12, or $n \geq 14$ and that the $n$-dimensional sphere has an effective smooth action of $A_5 \times Z$ with exactly one fixed point if $n$ satisfies $n \geq 6$ and $n \neq 9$, where $Z$ is a group of order 2.

In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian subgroup. By applying the refinement of a generalized Dedekind's theorem, we determine all possible values of integer group determinants of the direct product group of the cyclic group of order 8 and the cyclic group of order 2.

One of the important research subjects in the study of multiple zeta functions is to clarify the linear relations and functional equations among them. The Schur multiple zeta functions are a generalization of the multiple zeta functions of Euler–Zagier type. Among many relations, the duality formula and its generalization are important families for both Euler–Zagier type and Schur type multiple zeta values. In this paper, following the method of previous works for multiple zeta values of Euler–Zagier type, we give an interpolation of the sums in the generalized duality formula, called Ohno relation, for Schur multiple zeta values. Moreover, we prove that the Ohno relation for Schur multiple zeta values is valid for complex numbers.