Level Up: Pythagorean triples in model sets - part 3

For 100+ years physicists believed that only solids which are ordered in a periodic way can produce a clear diffraction pattern. This belief was shattered by Dan Shechtman's discovery of quasicrystals, a discovery for which he was awarded the 2011 Nobel Prize in Chemistry. The clear diffraction of quasicrystals shows that they must be highly ordered but in a non-periodic way. The precise order present in them is still mysterious, and not understood. In this project we investigate mathematical aperiodic orders. Namely we will study the existence of infinitely many solutions to Pythagoras's equation in aperiodic model sets. Such solutions are known to be present in periodic crystals. This is new mathematical research that will be carried out by two MacEwan undergraduate research assistants under the supervision of Dr. Chris Ramsey and Dr. Nicolae Strungaru. Stage 3:  Solutions in Fibonacci In this stage, the research assistants extend the approach to various other model sets such as the Silver Model set or the noble means model sets. For each model set they need to first find solutions to and then solve the Pythagoras equation in Z[lambda], where lambda is the largest eigenvalue of the corresponding matrix, and then by using this general solution to solve the question in that model set. If time permits, the research assistants will study various cases of Fermat's Last Theorem or other Diophantine equations in the above-mentioned model sets.

Level Up: Pythagorean triples in model sets - part 2

For 100+ years physicists believed that only solids which are ordered in a periodic way can produce a clear diffraction pattern. This belief was shattered by Dan Shechtman's discovery of quasicrystals, a discovery for which he was awarded the 2011 Nobel Prize in Chemistry. The clear diffraction of quasicrystals shows that they must be highly ordered but in a non-periodic way. The precise order present in them is still mysterious, and not understood. In this project we investigate mathematical aperiodic orders. Namely we will study the existence of infinitely many solutions to Pythagoras's equation in aperiodic model sets. Such solutions are known to be present in periodic crystals. This is new mathematical research that will be carried out by two MacEwan undergraduate research assistants under the supervision of Dr. Chris Ramsey and Dr. Nicolae Strungaru. Stage 2:  General solution in Z[tau] The research assistants search for solutions to the Pythagoras's equation in Z[tau]. Continuing with the particular solutions found in Stage 1 they go on to find the general solution, a description of all solutions. They complete this stage by using the general solution in Z[tau] to solve the problem in the Fibonacci model set.

Level Up: Pythagorean triples in model sets - part 1

For 100+ years physicists believed that only solids which are ordered in a periodic way can produce a clear diffraction pattern. This belief was shattered by Dan Shechtman's discovery of quasicrystals, a discovery for which he was awarded the 2011 Nobel Prize in Chemistry. The clear diffraction of quasicrystals shows that they must be highly ordered but in a non-periodic way. The precise order present in them is still mysterious, and not understood. In this project we investigate mathematical aperiodic orders. Namely we will study the existence of infinitely many solutions to Pythagoras's equation in aperiodic model sets. Such solutions are known to be present in periodic crystals. This is new mathematical research that will be carried out by two MacEwan undergraduate research assistants under the supervision of Dr. Chris Ramsey and Dr. Nicolae Strungaru. Stage 1: Background research and Z[tau] The research assistants search the literature and read about the Fibonacci substitution. They get familiar with the geometric realization, find the eigenvalues/eigenvectors of the substitution matrix. They read and get familiar with the ring Z[tau], where tau = (1+squareroot(5))/2 is the largest eigenvalue of the Fibonacci matrix. They also read about solving the Pythagoras's equation in integers and find particular solutions in Z[tau].

Investigating semi-Dirichlet matrix operator algebras - Phase 2

Operator algebras are collections of linear functions on geometric spaces of vectors called Hilbert spaces. Operator algebras represent collections of physical observables in quantum mechanics, and have also found application to many other mathematical fields. In the last 50 years, the study of operator algebras which are "non-selfadjoint", such as algebras of triangular matrices, has been especially fruitful. Of special interest in modern research-level mathematics are non-selfadjoint operator algebras which are "semi-Dirichlet". In this project, a MacEwan undergraduate research assistant will instigate a theoretical and/or computer classification of those finite-dimensional operator algebras which are semi-Dirichlet. In the most basic cases, such algebras can be classified using directed graphs. A celebrated theorem of Burnside describes the structure of any finite-dimensional algebra, and a classification of the semi-Dirichlet property will combine graph theory and Burnside's Theorem. By the end of this project, the student will produce an article suitable for publication in an undergraduate or research-level journal. This is new mathematical research carried out by an undergraduate researcher at MacEwan University, supervised by Drs. Adam Humeniuk and Chris Ramsey.