Ph.D Student Ph.D Thesis Krishnan Sunder Ram Some Problems in Random Topology Department of Electrical and Computer Engineering PROFESSOR EMERITUS Robert Adler

Abstract

We are interested in solving some theoretical problems arising from the nascent field of Random Topology. Broadly, this field combines methods from Probability and Topology to study topological/geometric properties of random structures. In the first part of this thesis, the random structures are real Riemannian manifolds, whereas they are knots for the latter half. For the random manifolds, we are interested firstly in studying a functional referred to as the reach or critical radius. This is an important quantity characterising the curvature and global geometry of the manifold such as closeness to self intersection. Subsequently, we focus on a class of intrinsic functionals that depend solely on the Riemannian metric. In the second part, we analyse knotting probabilities in a random knot model, prove that they decay asymptotically, and also provide a distribution of knots according to their crossing numbers.

Specifically, in the first two chapters, we generate random manifolds via random embeddings defined by means of a Gaussian field  on a compact, boundaryless manifold .  Suppose  denotes the embedding of  into the sphere  with  large enough. We are interested in the reach of the random submanifold  of the sphere  as  tends to infinity. It is proved that the global reach converges almost surely (a.s.) to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory. We also establish a fluctuation theory for the local reaches.

Building on these results, we show that the special embedding map we have is almost isometric, and as a consequence, several intrinsic functionals of  also converge a.s. to those of  equipped with a special Riemannian metric coming from the process .

In the next part of the thesis, we study random Chebyshev knots using billiard table diagrams. A billiard knot starts with the trajectory of a ball traveling in a 3D domain on a straight line, reflecting perfectly off the walls at rational angles. Chebyshev knots are examples of generalized billiard knots in an  rectangular prism. These can be projected onto billiard table diagrams denoted by

We introduce randomness in the model by choosing the `crossing type', determined by one portion of the ball's trajectory passing over another, randomly, in a uniform manner. This  model for studying random knots is quite general because it is known that every knot has a projection that is a Chebyshev plane curve.

We choose  (corresponding to 2-bridge Chebyshev knots), and study the probability of a knot as  tends to infinity. It is easy to see from the random model described above that finding the probabilities of knots essentially amounts to looking for certain patterns of heads and tails in a sequence of independent Bernoulli trials. We identify specific reduction moves that serve to simplify a string, and then use these to obtain explicit probabilities for any knot appearing in the model.

It is seen that the probability of any knot (including the unknot) goes to zero as  tends to infinity. A distribution of knots in terms of crossing numbers is also derived.