arXiv subjects

Steve Smale

Publications and source records attributed to Steve Smale.

5 recordsLinked to original sources

arXiv2026

Global analysis of the Kuramoto flow

Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It is also used in power system control. The simplest possible model assumes that all oscillators are identical and connected to each other with equal pairwise attraction. In this paper, we give a full geometric description of its global dynamics in terms of Morse theory and dynamical systems. Most of this description is stable in the sense that it is topologically preserved under small perturbations of the parameters.

math.DS↗

arXiv2016

Emergence of Function

This work gives a mathematical study of tissue dynamics. We combine within-cell genome dynamics and diffusion between cells, where the synthesis of the two gives rise to the emergence of function. We introduce a concept of monotonicity and prove that monotonicity together with hardwiring, defined as all cells of the same tissue having the same genome dynamics, is sufficient for the global convergence of the tissue dynamics.

q-bio.OT↗

arXiv2016

The Pitchfork Bifurcation

We give development of a new theory of the Pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry.

math.DS↗

arXiv2016

Mathematics of the Genome

This work gives a mathematical foundation for bifurcation from a stable equilibrium in the genome. We construct idealized dynamics associated with the genome. For this dynamics we investigate the two main bifurcations from a stable equilibrium. Finally, we give a mathematical framework for the Repressilator and the Toggle gene circuits.

math.DS↗

arXiv2009

Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology.

math.KT↗