Related papers: The mathematics of taffy pullers
The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional…
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on…
We describe a circle of ideas relating the dynamics of 2-dimensional homeomorphisms to that of 1-dimensional endomorphisms. This is used to introduce a new class of maps generalizing that of Thurston's pseudo-Anosov homeomorphisms.
A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…
The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) We estimate the probability of a trivial knot formation on the lattice…
Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit…
We review the polynomial parameterization of classical knots and prove the analogous results for long $2$ knots. We also construct polynomial parameterizations for certain classes of knotted spheres (such as spun and twist spun of the…
Notes on the Spinpossible puzzle game. We give a mathematical description of the game, prove some elementary bounds on the length of optimal solutions, and consider variations of the game which place restrictions on the set of permitted…
Optical tweezers are powerful tools based on focused laser beams. They are able to trap, manipulate and investigate a wide range of microscopic and nanoscopic particles in different media, such as liquids, air, and vacuum. Key applications…
Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these…
We establish Conway's thrackle conjecture in the case of spherical thrackles; that is, for drawings on the unit sphere where the edges are arcs of great circles.
We study types of mapping classes which arise as a product of a given mapping class and powers of certain pure mapping classes. We derive an explicit constant depending only on a surface such that almost all above pure mapping classes give…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…
A classical way for exploring the scattering behavior of a small sphere is to approximate Mie coefficients with a Taylor series expansion. This ansatz delivered a plethora of insightful results, mostly for small spheres supporting localized…
Optical tweezers have found widespread application in many fields, from physics to biology. Here, we explain in detail how optical forces and torques can be described within the geometrical optics approximation and we show that this…
We introduce a family of reconfiguration puzzles arising from ideas in geometry and topology. We present their construction from square-tiled shapes, discuss some of the underlying mathematics and describe how they are naturally associated…