Download On One Dimensional Dynamical Systems and Commuting

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
Mälardalens University
First Network Meeting for Sida- and ISP-funded PhD Students in
Mathematics
Stockholm 7–8 March 2017
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My Supervisors
S. D. Silvestrov
Johan Richter
Vincent Ssembatya
Main supervisor
Mälardalens University
Co-supervisor
Mälardalens University
Co-supervisor
Makerere University
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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.
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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.
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Tack så mycket!
Thank you!
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