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Non-Archimedean Analysis and Applications in Physics -
Prof. K. M. Shamseddine

My research interests and activities include various areas of non-Archimedean Analysis, which is analysis on field extensionsof the real numbers that contain also infinitely small numbers (positive numbers smaller than any positive real number) and infinitely large numbers (numbers that are larger than any real number.) Besides its importance from an abstract mathematical point of view, research in this area has many potential applications in Physics as well as other fields of science and engineering.

Research projects at the undergraduate and graduate levels are available for students with a strong background in Theoretical Physics and/or Mathematics. Interested students who wish to familiarize themselves with this research area are encouraged to visit my publications web site: www.physics.umanitoba.ca/~khodr/#Publications and contact me at khodr@physics.umanitoba.ca for more information.

Quantum Phase Space and Semiclassical Dynamics -
Prof. T.A. Osborn
This research program in mathematical physics aims to achieve a unification of classical and quantum mechanics in a common mathematical framework. The theory that emerges (quantum phase space, QPS) is an altered version of classical phase space in which the usual communtative product of functions is deformed (as Planck's constant varies away from zero) into a noncommuntative (star) product. With this one structural modification it is possible to state the full content of quantum mechanics as a noncommutative phase space theory. In this setting, the Schrodinger wave function never arises, Hilbert space operators are represented by phase space (Wigner) distributions, and quantum expectation values are given by integrals over phase space. The unification via QPS provides an alternate, autonomous statement of quantum mechanics that is purely geometrical in character. It clarifies the interpretations of quantum mechanics and at the same time provides a new computational platfrom that has many parallels to that of classical mechanics.

Several active projects are: 1) In recent work on charged particle systems in electromagnetic fields we have developed a QPS representation (called a perfect quantization) that is both gauge and geometrically covariant and has an exact star product determined by a symplectic area phase. Perfect quantization provides an ideal platform for studying the semiclassical charged particle dynamics. We aim to extend this perfect quantization to general non-Abelian gauge theories, i.e., to include Riemannian manifolds and arbitrary spin structure. 2) Develop effective methods to treat long time dynamics in QPS and to rigorously define quantum chaos and the quantum Lyapunov exponent. 3) Study completely integrable systems in QPS and obtain their energy spectrums via higher order semiclassical Bohr-Sommerfeld methods.
Scientific Studies of Magical Squares - Prof. P.D. Loly
Explorations of the moment of inertia of semi-magic squares as well as the inertia tensor and electric multipole expansion for semi-magic cubes have lead to several general theorems. The eigenproperties of highly constrained magic squares and a newly discovered class of non-magic pandiagonal squares reveal some highly singular matrices. Compounding methods for generating very large magical squares have been shown to preserve global characteristics of several interesting varieties. All 8th order Franklin squares with distinct elements 1..64 have been constructed by a backtracking computation. The count of 1,105,920 extends a handful of extant examples, whilst being rather lower than an upper bound of some 228 trillion. Some applications of magical hypercubes already made include Chinese patterns and binary classification, while elsewhere there are possibilities in cryptography and image reproduction.
Dark matter and string theory - Prospective Adjunct Prof. A. Frey (University of Winnipeg)
My main research interest is in the intersection of cosmology (the history and composition of the universe) with high energy physics (particle physics and string theory). In my research, I answer questions about dark energy, dark matter, inflation, and the Big Bang. In this work, I use tools from particle physics, string theory, and general relativity; as I am a theoretical physicist, that involves analytical (paper and pencil) as well as numerical calculations. Please see my homepage for more details! Students interested in working for me should also drop me a line!