On One Dimensional Dynamical Systems and Commuting Elements in Non Commutative Algebras Alex Behakanira Tumwesigye Department of Mathematics, Makere University Department of Mathematics and Applied Mathematics M¨ alardalens University First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017 1 / 6
My Supervisors S. D. Silvestrov Johan Richter Vincent Ssembatya Main supervisor Co-supervisor Co-supervisor M¨ alardalens University M¨ alardalens University Makerere University 2 / 6
Brief Introduction My research is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. 3 / 6
Brief Introduction My research is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. Definition Two processes commute if the order of application of the processes does not matter. 3 / 6
Brief Introduction My research is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. Definition Two processes commute if the order of application of the processes does not matter. Examples include: Ordering of bills when paying for an item at the counter by cash. (Commutative ) 3 / 6
Brief Introduction My research is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. Definition Two processes commute if the order of application of the processes does not matter. Examples include: Ordering of bills when paying for an item at the counter by cash. (Commutative ) Opening and going through a door. (Non-Commutative ) 3 / 6
Brief Introduction My research is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. Definition Two processes commute if the order of application of the processes does not matter. Examples include: Ordering of bills when paying for an item at the counter by cash. (Commutative ) Opening and going through a door. (Non-Commutative ) In Mathematics, matrices (and in general linear and non linear operators) do not commute under multiplication. 3 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. 4 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. Already done! 4 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. Already done! We treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. 4 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. Already done! We treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. Already done! 4 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. Already done! We treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. Already done! We give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras. 4 / 6
My research We treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems where derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. Already done! We treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. Already done! We give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras. Partially done. 4 / 6
Impact and Applications of My Research Commutation relations considered here have applications in various areas in Mathematics, Physics and Engineering such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others. 5 / 6
Impact and Applications of My Research Commutation relations considered here have applications in various areas in Mathematics, Physics and Engineering such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others. Description of commutative subalgebras and commutative subrings allows one to relate properties of non-commutative algebras (representation theory, ideals and subalgebras, homological ) to properties naturally associated with commutative algebras (spectral theory, duality, algebraic geometry and topology). 5 / 6
Impact and Applications of My Research Commutation relations considered here have applications in various areas in Mathematics, Physics and Engineering such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others. Description of commutative subalgebras and commutative subrings allows one to relate properties of non-commutative algebras (representation theory, ideals and subalgebras, homological ) to properties naturally associated with commutative algebras (spectral theory, duality, algebraic geometry and topology). In representation theory, crossed products (or semi-direct products) play a central role in the construction and classification of representations using the method of induced representations. 5 / 6
Impact and Applications of My Research Commutation relations considered here have applications in various areas in Mathematics, Physics and Engineering such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others. Description of commutative subalgebras and commutative subrings allows one to relate properties of non-commutative algebras (representation theory, ideals and subalgebras, homological ) to properties naturally associated with commutative algebras (spectral theory, duality, algebraic geometry and topology). In representation theory, crossed products (or semi-direct products) play a central role in the construction and classification of representations using the method of induced representations. 5 / 6
Tack s˚ a mycket! Thank you! 6 / 6
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