Small cancellation theory is one of representative geometric techniques in group theory initiated from Dehn in 1911. In order to understand what small cancellation theory is and how it can be applied, we will first consider an easy question concerning the fundamental group of an orientable closed surface of genus 2. In contrast with a topological proof to the question using universal covering space introduced by Gyuntae Kim, a group theoretic proof using small cancellation theory will be introduced. Also some recent results obtained jointly with Makoto Sakuma by applying small cancellation theory to 2-bridge link groups and even Heckoid groups will be briefly discussed.

Not only does the study of positive definite matrices remain a flourishing area of mathematical investigation, but positive definite matrices have become fundamental computational objects in many areas of engineering, statistics, quantum information, and applied mathematics. A variety of metric-based computational algorithms for positive definite matrices have arisen for approximations, interpolation, filtering, estimation, and averaging, the last being the concern of this talk. A natural and attractive candidate of averaging procedures is the least squares mean (or Karcher mean, center of mass) for the Riemannian trace metric. The Strong Law of Large Number, established by T. Sturm on Hadamard spaces, plays a crucial role for the monotonicity conjecture and gives rise to a problem of finding deterministic approximation to the Karcher mean, namely the no dice conjecture. We provide an affirmative answer to the no dice conjecture on the general setting of Hadamard (or CAT(0), NPC) spaces.

In this talk we will consider some aspects of abstract harmonic analysis on compact groups and its quantum extension, namely operator amenabilities of the L^1 algebras. Basic concepts on compact groups and quantum groups will be briefly reviewed focusing on the examples of SU(2) and its q-deformations.