We introduce a variant of horocompactification which takes “directions” into account. As an application, we construct a compactification of the Teichmüller spaces via the renormalized volume of quasi-Fuchsian manifolds. Although we observe that the renormalized volume itself does not give a distance, the compactification allows us to define a new distance on the Teichmüller space. We show that the translation length of pseudo-Anosov mapping classes with respect to this new distance is precisely the hyperbolic volume of their mapping tori. A similar compactification via the Weil–Petersson metric is also discussed.

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.

In this paper, we show that for a given finitely presented group $G$, there exist integers $h_G \geq 0$ and $n_G \geq 4$ such that for all $h \geq h_G$ and $n \geq n_G$, and for all $0 \leq i \leq 2n - 2$, there exists a genus-$(2h + n - 1)$ Lefschetz fibration on a minimal symplectic 4-manifold with $(\chi, c_{1}^{2}) = (n, i)$ whose fundamental group is isomorphic to $G$. We also prove that such a fibration cannot be decomposed as a fiber sum for $1 \leq i \leq 2n - 2$ if $h > (5n - 3)/2$. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.

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.

We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for generalized pairs. We also discuss the theory of complements for generalized pairs.

A stochastic fractionally dissipative quasi-geostrophic equation with stochastic damping is considered in this paper. First, we show that the null solution is exponentially stable in the sense of $q^{-}$-th moment of $\|\cdot\|_{L^{q}}$, where $q > 2/(2\alpha - 1)$ and $q^{-}$ denotes the number strictly less than $q$ but close to it, and from this fact we further prove that the sample paths of solutions converge to zero almost surely in $L^{q}$ as time goes to infinity. In particular, a simple example is used to interpret the intuition. Then the uniform boundedness of pathwise solutions in $H^{s}$ with $s \geq 2 - 2\alpha$ and $\alpha \in (1/2, 1)$ is established, which implies the existence of non-trivial invariant measures of the quasi-geostrophic equation driven by nonlinear multiplicative noise.

In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homology and torsion in tensor products of modules, and obtain new results that are of independent interest.

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.

We shall propose a new proof scheme for Planar Cover Conjecture, focusing on the rotation systems of planar coverings of connected graphs. We shall introduce the notion of “rotation compatible coverings” and show that a rotation compatible covering of $G$ embedded on the sphere can be covered by a regular covering of $G$ embedded on an orientable closed surface on which its covering transformation group acts. The surface may not be homeomorphic to the sphere in general, but its quotient becomes either the sphere or the projective plane which contains $G$. As an application of our theory, we shall prove that if a 3-connected graph $G$ has a 3-connected finite planar covering such that the pre-images of each vertex has sufficiently large distance, then $G$ can be embedded on the projective plane.

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.

Time-dependent free surface problem for the incompressible Navier–Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We obtain global well-posedness of the problem for a small initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^{1}$-regularity of the corresponding Stokes problem in the half-space and special structures of the quasi-linear term appearing from the Lagrangian transform of the coordinate.

We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where ‘irreducible nontrivial’ roughly means to be not constructed from Lévy families nor box convergent sequences.

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $\mathfrak{g}$ naturally arises, with the zero component $\mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\mathfrak{g}_{0}$: the “maximal one” $\mathsf{P}_{\max}$ and the “minimal one” $\mathsf{P}_{\min}$. Furthermore, the parabolic BGG category arising from $\mathsf{P}_{\max}$ essentially turns out to be a subcategory of the one arising from $\mathsf{P}_{\min}$. Such a priority of $\mathsf{P}_{\min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $\mathcal{O}^{\min}$ associated with $\mathsf{P}_{\min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows.

(1) We classify and obtain a precise description of the blocks of $\mathcal{O}^{\min}$.

(2) We investigate indecomposable tilting and indecomposable projective modules in $\mathcal{O}^{\min}$, and compute their character formulas.

We consider a continuum percolation built over stationary ergodic point processes. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies volume regularity and isoperimetric condition, we prove a quenched invariance principle for reflecting diffusions on the cluster.

In this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.

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.

For every fixed $h \geq 1$, we construct an infinite family of simply connected symplectic 4-manifolds $X'_{g,h}[i]$, for all $g > h$ and $0 \leq i < 2p - 1$, where $p = \lfloor \frac{g+1}{h+1} \rfloor$. Each manifold $X'_{g,h}[i]$ is the total space of a symplectic genus $g$ Lefschetz pencil constructed by an explicit monodromy factorization. We then show that each $X'_{g,h}[i]$ is diffeomorphic to a complex surface that is a fiber sum formed from two standard examples of hyperelliptic genus $h$ Lefschetz fibrations, here denoted $Z_{h}$ and $H_{h}$. Consequently, we see that $Z_{h}$, $H_{h}$, and all fiber sums of them admit an infinite family of explicitly described Lefschetz pencils, which we observe are different from families formed by the degree doubling procedure.

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $\theta$-twisted $V_{L}$-module. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$. In this paper (Part 2), we show that there exists an irreducible $M(1)^{+}$-submodule in any non-zero weak $V_{L}^{+}$-module and we compute extension groups for $M(1)^{+}$.

This paper deals with links and braids up to link-homotopy, studied from the viewpoint of Habiro's clasper calculus. More precisely, we use clasper homotopy calculus in two main directions. First, we define and compute a faithful linear representation of the homotopy braid group, by using claspers as geometric commutators. Second, we give a geometric proof of Levine's classification of 4-component links up to link-homotopy, and go further with the classification of 5-component links in the algebraically split case.

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi–Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell–Lang conjecture and verify the Kawaguchi–Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.

Let $n$ be an integer such that $n = 5$ or $n \geq 7$. In this article, we introduce a recipe for a certain infinite family of non-singular plane curves of degree $n$ which violate the local-global principle. Moreover, each family contains infinitely many members which are not geometrically isomorphic to each other. Our construction is based on two arithmetic objects; that is, prime numbers of the form $X^{3} + NY^{3}$ due to Heath-Brown and Moroz and the Fermat type equation of the form $x^{3} + Ny^{3} = Lz^{n}$, where $N$ and $L$ are suitably chosen integers. In this sense, our construction is an extension of the family of odd degree $n$ which was previously found by Shimizu and the author. The previous construction works only if the given degree $n$ has a prime divisor $p$ for which the pure cubic fields $\mathbb{Q}(p^{1/3})$ or $\mathbb{Q}((2p)^{1/3})$ satisfy a certain indivisibility conjecture of Ankeny–Artin–Chowla–Mordell type. In this time, we focus on the complementary cases, namely the cases of even degrees and exceptional odd degrees. Consequently, our recipe works well as a whole. This means that we can unconditionally produce infinitely many explicit non-singular plane curves of every degree $n = 5$ or $n \geq 7$ which violate the local-global principle. This gives a conclusion of the classical story of searching explicit ternary forms violating the local-global principle, which was initiated by Selmer (1951) and extended by Fujiwara (1972) and others.