NSERC grants at Laurentian University Subventions du CRSNG `a l ...
NSERC grants at Laurentian University Subventions du CRSNG `a l ...
NSERC grants at Laurentian University Subventions du CRSNG `a l ...
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93Stephanie CzaporApplic<strong>at</strong>ions of Symbolic Algebraic Comput<strong>at</strong>ion in AppliedM<strong>at</strong>hem<strong>at</strong>icsA grant of $8,200 per year.Une subvention de 8 200 $ par année.Discovery Grant – Subvention à la découverteMy research involves the use of systems for symbolicalgebraic comput<strong>at</strong>ion (e.g. Maple, M<strong>at</strong>hem<strong>at</strong>ica) insolving comput<strong>at</strong>ionally challenging problems in M<strong>at</strong>hem<strong>at</strong>icalPhysics and Applied M<strong>at</strong>hem<strong>at</strong>ics.One such problem concerns the validity of Huygens’Principle (HP; from his “Tre<strong>at</strong>ise on Light”, 1690), orr<strong>at</strong>her its precise formul<strong>at</strong>ion <strong>du</strong>e to J. Hadamard (1923),for the wave equ<strong>at</strong>ion on curved spacetimes. Roughlyspeaking, this st<strong>at</strong>es th<strong>at</strong> waves do not exhibit “ripples”behind the wave front. While this problem is wellunderstoodin two and three dimensions, it becomes far more difficult inspaces involving four dimensions (e.g. as in General Rel<strong>at</strong>ivity). The modernformul<strong>at</strong>ion of the problem is in terms of the so-called Cauchy problem, andas such its role is r<strong>at</strong>her central to the theory of partial differential equ<strong>at</strong>ions.My present goal is to precisely determine the validity of HP for thenon-self-adjoint form of the wave equ<strong>at</strong>ion, when considered in an underlyingfour-dimensional space of a particular algebraic type. This forms thenext logical step in a program to eventually tre<strong>at</strong> all of the possible cases.Another (rel<strong>at</strong>ed) problem concerns investig<strong>at</strong>ing the so-called conformalstructure of space-time. A conformal transform<strong>at</strong>ion is one of a particularkind imposed on the metric tensor (which measures distances) of a space,which preserves angles but not distances. Despite its simplicity, its exactphysical significance is difficult to understand. Still, such transform<strong>at</strong>ionshave played an important role in rel<strong>at</strong>ivity (e.g. in studying conformallyinvariant wave propag<strong>at</strong>ion in HP, as above). The particular problem I amnow investig<strong>at</strong>ing concerns finding the precise conditions under which a givenspace-time is conformally rel<strong>at</strong>ed to one which is empty (in the sense th<strong>at</strong>the Ricci part of its curv<strong>at</strong>ure vanishes).
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