Speaker: Dr. Jerzy Szulga

Title: Algebraic structures induced by operators of random chaos. Part II.

 

Abstract: The term "random chaos" refers to a highly organized and heavily investigated system of functionals of Brownian motion. In the 1980s K. R. Parthasarathy introduced certain important operators, originated in quantum probability, known as "creation," "annihilation," and "conservation" operators.  In the 1990s P-A. Meyer adopted these quantum concepts to the framework of "discrete random chaos," which in analytic terms refers to Walsh series and discrete Fourier analysis.

 

That was the subject of my seminar talk 4 week ago. It wasn't enough time to finish it. Here it is:

 

The algebras spanned by these three basic noncommutative operators (or even just by two symmetries) yield simple yet quite sophisticated structures that go beyond the limits of typical probabilistic endeavors and enter areas usually governed by coding theory, theory of finite fields (especially of characteristic 2), tensor products, Clifford algebras, etc.

 

In signed multiplicative systems all elements either commute or anticommute and are either positive or negative (as defined by their squares). Quaternions or Pauli matrices are elementary examples. There is a universal model that contains all (finite or countable) signed systems.