Concepts of polymer statistical topology

Abstract

I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called statistical (or probabilistic) topology. Contents I. Introduction: What we are talking about? II. Milestones A. Abelian epoch B. Non-Abelian epoch 1. Methods of algebraic topology in polymer statistics. Topology as quenched disorder 2. How to define the knot complexity? 3. Conformal methods in statistics of entangled random walks 4. Group-theoretic methods in statistics of entangled random walks 5. Conditional Brownian bridges in Hyperbolic spaces C. Crumpled globule: Topological correlations in collapsed unknotted rings III. Conclusion A. The King is dead, long live The King! B. Where to go? References

Non-Euclidean Geometry in Nature

“…Some conceptual steps in the analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, emerging at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects have been reviewed in Ref. [1]. This new branch is called statistical (or probabilistic) topology.…”

“…In this review I will not discuss anymore questions dealing with non-Euclidean geometry in statistics of knots, since this subject has deserved much attention in Ref. [1]. Rather I will concentrate on other challenging manifestations of hyperbolic geometry in physics where the "conflict" between the hyperbolic protocol of growth of natural tissues (like plants) with the restricted geometry of the three-dimensional space leads to high variety of self-similar shapes.…”