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.

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.

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.

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.

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 $M(1)^{+}$.

In this paper, we formulate a new conjecture concerning Kato's Euler system for elliptic curves $E$ over $\mathbb{Q}$. This ‘Generalized Perrin-Riou Conjecture’ predicts a precise congruence relation between a Darmon-type derivative of the zeta element of $E$ over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the $L$-function of $E$ over $\mathbb{Q}$. We prove the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithms of zeta elements, and also that, under mild technical hypotheses, the ‘order of vanishing’ part of the conjecture is unconditionally valid in arbitrary rank. This approach also allows us to prove a natural higher-rank generalization of Rubin's formula concerning derivatives of $p$-adic $L$-functions and to establish an explicit connection between the $p$-part of the classical Birch and Swinnerton-Dyer formula and the Iwasawa main conjecture in arbitrary rank and for arbitrary reduction at $p$. In a companion article we prove that the approach developed here also provides a new interpretation of the Mazur–Tate conjecture that leads to the first (unconditional) theoretical evidence in support of this conjecture for curves of strictly positive rank.

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 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.

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.