Valuative stability of polarised varieties

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s $$\beta $$ β -invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the $$\delta $$ δ -invariant plays in the study of valuative stability and K-stability of polarised varieties.

On alternating closed braids

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.

Families of vector fields with many numerical invariants

<p style='text-indent:20px;'>We study bifurcations in finite-parameter families of vector fields on <inline-formula><tex-math id="M1">\begin{document}$S^2$\end{document}</tex-math></inline-formula>. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable <inline-formula><tex-math id="M2">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of <inline-formula><tex-math id="M3">\begin{document}$(2D+1)$\end{document}</tex-math></inline-formula>-parameter families such that the topological classification of these families has at least <inline-formula><tex-math id="M4">\begin{document}$D$\end{document}</tex-math></inline-formula> numerical invariants and used those examples to construct families with functional invariants of topological classification.</p><p style='text-indent:20px;'>In this paper, we construct locally generic <inline-formula><tex-math id="M5">\begin{document}$4$\end{document}</tex-math></inline-formula>-parameter families with any prescribed number of numerical invariants and use them to construct <inline-formula><tex-math id="M6">\begin{document}$5$\end{document}</tex-math></inline-formula>-parameter families with functional invariants. We also describe a locally generic class of <inline-formula><tex-math id="M7">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families with a tail of an infinite number sequence as an invariant of topological classification.</p>

Algebraic Program Analysis

This paper is a tutorial on algebraic program analysis. It explains the foundations of algebraic program analysis, its strengths and limitations, and gives examples of algebraic program analyses for numerical invariant generation and termination analysis.

Braid group and leveling of a knot

Any knot [Formula: see text] in genus-[Formula: see text] [Formula: see text]-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the [Formula: see text]-length. We show that the [Formula: see text]-length equals the level number. We then find braid descriptions for [Formula: see text]-positions of all [Formula: see text]-bridge knots providing upper bounds for their level numbers and also show that the [Formula: see text]-pretzel knot has level number two.

Infants’ intermodal perception of numerosity in an experimental study with objects and socially salient stimuli

In the present cross-sectional experimental study we investigated infants’ early ability to intermodally detect numerosity of visual-auditory object-like and social stimuli. We assumed that presentation of face – voice stimuli would distract infants’ attention from detection of numerical invariant. Seventy-eight infants (aged 5, 7 and 9 months) participated in four experimental Conditions (simultaneously projected pairs of identical objects, non-identical objects, objects projected together with familiar face and objects projected together with unfamiliar face). Visual stimuli in each trial varied in numerosity (1 -2 / 1-3 / 2 -3) and they were accompanied by piano sounds or voice sounds also varying in numerosity (one, two or three sounds in La tonality). By means of preferential looking technique, we measured infants’ fixation of attention to the visual stimulus that numerically matched with the sound. When object-like stimuli were projected, infants –except 5-month-old boys –tended to intermodally detect numerical invariant. Shape similarity of the objects facilitated infants’ intermodal detection of numerosity. When socially salient stimuli were co-presented with object-like stimuli, infants preferred to look at the face, ignoring numerosity of the auditory stimulus. Nor sound quality (piano vs. voice) neither familiarity of the face (mother’s face vs. stranger woman’s face) affected infants’ perception. Although intermodal detection of perceptual cues is a primary function of both face and number perception, each one of these perceptual systems seems to follow a different developmental path.

Monotonic invariants under blowups

We prove that the numerical invariant [Formula: see text] of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

On Graph-Theoretical Invariants of Combinatorial Manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed $n$-dimensional PL manifold. Here we consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low-dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013.