We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, ho...

The representation of curves by integral invariant signatures is an important step inshape recognition and classification. Integral invariants are preferred over their differential counter-parts due to their robustness with respect to noise. However, in contrast to differential invariants ofcurves, it is currently unknown whether integral signatures offer unique representations of c...

Braid Floer homology is an invariant of proper relative braid classes [12]. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid classes is non-trivial, then additional closed integral curves of the Hamiltonian equations are forced via a Morse type theory. In this article we show t...

In this paper, we give a constant $C$ in \cite[Theorem 1.2]{sha2014bounding} by using an explicit Baker's inequality, hence have bound of the integral points on modular curves.