On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS)21(9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1, e^{h_{\mathrm {top}}})$ .

Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

Ontogeny in the steinmanellines (Bivalvia: Trigoniida): an intra- and interspecific appraisal using the Early Cretaceous faunas from the Neuquén Basin as a case study

Despite the paleontological relevance and paleobiological interest of trigoniid bivalves, our knowledge of their ontogeny—an aspect of crucial evolutionary importance—remains limited. Here, we assess the intra- and interspecific ontogenetic variations exhibited by the genus Steinmanella Crickmay (Myophorellidae: Steinmanellinae) during the early Valanginian–late Hauterivian of Argentina and explore some of their implications. The (ontogenetic) allometric trajectories of seven species recognized for this interval were estimated from longitudinal data using 3D geometric morphometrics, segmented regressions, and model selection tools, and then compared using trajectory analysis and allometric spaces. Our results show that within-species shell shape variation describes biphasic ontogenetic trajectories, decoupled from ontogenetic changes shown by sculpture, with a gradual decay in magnitude as ontogeny progresses. The modes of change characterizing each phase (crescentic growth and anteroposterior elongation, respectively) are conserved across species, thus representing a feature of Steinmanella ontogeny; its evolutionary origin is inferred to be a consequence of the rate modification and allometric repatterning of the ancestral ontogeny. Among species, trajectories are more variable during early ontogenetic stages, becoming increasingly conservative at later stages. Trajectories’ general orientation allows recognition of two stratigraphically consecutive groups of species, hinting at a potentially higher genus-level diversity in the studied interval. In terms of functional morphology, juveniles had a morphology more suited for active burrowing than adults, whose features are associated with a sedentary lifestyle. The characteristic disparity of trigoniids could be related to the existence of an ontogenetic period of greater shell malleability betrayed by the presence of crescentic shape change.

Padé approximants on Riemann surfaces and KP tau functions

The paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Non-negativity of BMN two-point functions with three string modes

Recently, we proposed a novel entry of the pp-wave holographic dictionary, which equated the Berenstein-Maldacena-Nastase (BMN) two-point functions in free $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory with the norm squares of the quantum unitary transition amplitudes between the corresponding tensionless strings in the infinite curvature limit, for the cases with no more than three string modes in different transverse directions. A seemingly highly non-trivial conjectural consequence, particularly in the case of three string modes, is the non-negativity of the BMN two-point functions at any higher genus for any mode numbers. In this paper, we further perform the detailed calculations of the BMN two-point functions with three string modes at genus two, and explicitly verify that they are always non-negative through mostly extensive numerical tests.

Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables $$q=e^{iu}$$ q = e iu , refined floor diagrams for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of $${\mathbb {P}}^1$$ P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of $${\mathbb {F}}_0$$ F 0 and $${\mathbb {F}}_2$$ F 2 are related by the Abramovich–Bertram formula.