# The public defense of Alex Behakanira Tumwesigye's doctoral thesis in Mathematics/Applied mathematic

Doctoral thesis and Licentiate seminars

The public defense of Alex Behakanira Tumwesigye's doctoral thesis in Mathematics/Applied mathematics will take place at Mälardalen university, room Kappa at 13.00 on May 29, 2018.

Title: “Dynamical systems and commutants in non-commutative algebras”.

The faculty examiner is Docent Olga Liivapuu, Estonian University of Life Sciences, and the examining committee consists of Professor Viktor Abramov, University of Tartu; Professor Melanija Mitrović, University of Niš; Assoociate Professor John Mango, Makerere University.

Reserve; Professor Dr Manuel Joaquim Alves, Eduardo Mondlane University

The doctoral thesis has serial number 258

## Summary

The thesis treats commutativity which is a fundamental topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the result of the processes does not depend on the order in which the processes are applied. Commutativity of addition can be observed when paying for items at the counter by cash. No matter the order in which bills are handed over, they always give the same total, whereas washing and then drying clothes, and drying and then washing produce markedly different results.

Another example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged.

Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations. This means that the operators do not actually commute but there is an explicit relation for the difference of the two possible products of the operators.

The first part of this thesis treats commutativity of monomials of pairs of operators satisfying certain commutation relations. I consider products of powers of the operators, called monomials, and derive commutativity conditions of the said monomials. I show that this is related to the existence of periodic points of certain one-dimensional dynamical systems.

In the second part of the thesis, I treat maximal commutative subalgebras of crossed products of algebras of piecewise constant functions with the integers. By the crossed product of an algebra with the integers I mean a generalization of Laurent polynomials with coefficients from the algebra. I describe commutants (set of elements that commute with a given set) and the center (set of elements that commute with the whole algebra) in a number of cases.

Finally, I turn attention to Ore extensions. By an Ore extension of a ring, I mean a generalization of polynomials with coefficients from the ring. I describe the commutant of the coefficient algebra for the Ore extension of the algebra of functions on a set.