Abstract: I will explain the construction of Kontsevich-Vishik canonical trace on non-integer order classical pseudodifferential operators. This construction has it roots in the old methods of extracting a finite part from a divergent sum or integral (infra-red and ultra-violet divergence), used by mathematicians and physicists.

If time permits I will explain some of the results on generalizations of this construction to noncommutative setting.

Abstract: We survey the real normed division algebras (the real numbers, the complex numbers, the quaternions, and the octonions) and discuss how a result of M. Franz over the complex numbers was effectively generalized to all four of these number systems in the summer of 2014. The recommended background for this talk is second year linear algebra.