From an August 2016 combination summer school and conference at Tohoku University in Japan, a survey article and 11 research articles explore

operator algebras and mathematical physics.

Skew PBW extensions include rings and algebras coming from mathematical physics such PBW extensions, group rings of polycyclic-by-finite groups, Ore algebras,

operator algebras, diffusion algebras, some quantum algebras, quadratic algebras in three variables, some 3-dimensional skew polynomial algebras, some quantum groups, some types of Auslander-Gorenstein rings, some Koszul algebras, some Calabi-Yau algebras, some Artin-Schelter regular algebras, some quantum universal enveloping algebras, and others.

Sakai,

Operator algebras in dynamical systems, CAMBRIDGE UNIVERSITY PRESS, 1991.

Takesaki, Theory of

operator algebras. I, Springer, New York, 1979.

Very early, in the beginnings of the theory of partial actions of groups in the context of

operator algebras, many unexpected connections were explored.

Zhu, "Modular invariance of characters of vertex

operator algebras," Journal of the American Mathematical Society, vol.

Meng, "Continuity of ([alpha], [beta])-derivations of

operator algebras," Journal of the Korean Mathematical Society, vol.

We have considered how primes (or prime numbers) act on

operator algebras (e.g., see [9], [10] and [12]).

The results will have a lasting impact on and connect further the theories of non-commutative geometry,

operator algebras, Lie theory, quantum group theory and partly quantum physics.

Vertex

operator algebras with central charge 1/2 and -68/7 ...

He presents the theory of Krichever-Nobikov algebras, Lax

operator algebras, their interaction, elements of their representation theory, relations to moduli spaces of Riemann surface and holomorphic vector bundles of them and to Lax integrable systems and conformal field theory.

Other topics of the 17 papers include nonself-adjoint

operator algebras for dynamical systems, noncommutative geometry as a functor, examples of mases in C*-algebras, simple group graded rings, and classifying monotone complete algebras of operators.