Steven A. Bleiler

Spherical space forms and Dehn filling

Abstract This paper concerns those Dehn fillings on a torally bounded 3-manifold which yield manifolds with a finite fundamental group. The focus will be on those torally bounded 3-manifolds which either contain an essential torus, or whose interior admits a complete hyperbolic structure. While we give several general results, our sharpest theorems concern Dehn fillings on manifolds which contain an essential torus. One of these results is a sharp “finite surgery theorem.” The proof incl udes a characterization of the finite fillings on “generalized” iterated torus knots with a complete classification for the iterated torus knots in the 3-sphere. We also give a proof of the so-called “2π” theorem of Gromov and Thurston, and obtain an improvement (by a factor of two) in the original estimates of Thurston on the number of non-negatively-curved Dehn fillings on a torally bounded 3-manifold whose interior admits a complete hyperbolic structure.

OCTONIONIZATION OF THREE PLAYER, TWO STRATEGY MAXIMALLY ENTANGLED QUANTUM GAMES

We develop an octonionic representation of the payo function for a three player, two strategy, maximally entangled quantum game.

A game theoretic approach to quantizing Markov processese

In the context of quantum information theory, “quantization” of various mathematical constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done “properly”, a faithful copy of the original construction is required to exist within the new “quantum” one, just as is required when a function is extended to a larger domain. Here, Markov processes that serve as mathematical models of history dependent Parrondo games are quantized properly.

Little big knots

Abstract For knots with ‘little’ groups, yet ‘big’ topologies, there is a surprisingly strong connection between certain presentations of the group and the knot exteriors Heegaard structure.