Small Representations and Applications

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Gordan Savin is continuing his work on algebraic aspects of analysis,with applications in number theory. The main tools of this research areminimal representations (discoveredby Kazhdan and Savin) which have been successful in dealing with certainaspects of Langlands conjectures.Indeed, several instances of Langlands conjectures, out ofreach of standard methods, can be obtained throughuse of minimal representations.Analysis, broadly defined, deals withfunctions satisfying certain differential equations. Analysis ofdifferential equations, in general, is a very hard problem. For example,differential equations coming from the dynamics of fluids are notoriouslydifficult. One possible reason for this is a lack of symmetries. On theother hand, if there are plenty of symmetries to work with, thenthe corresponding analysis is easier, since algebraic tools can be used.This research deals precisely with situationswhen a large group of symmetries is present. The main purposeof this research is to construct small representations ofgroups of Lie type. Smallness, roughly speaking, refers to the fact thatit is possible to combine together many symmetries acting on arelatively small space.Importance of having small representationscan be illustrated with the following example.One of the most famous (and most notorious) groups is the Reisz-FischerMonster. Yes, this group is huge and hence its name,but the size is not the main difficultyhere. The main problem is the lack of small representations.Indeed, the group of permutations of 60 letters - nobody would considerthis group a ``monster'' - is in fact bigger then the Riesz-FischerMonster. However, this group of permutations can be written down(i.e. represented) by means of60 by 60 matrices. The monster, on the other hand, requires use of(roughly) 2000 by 2000 matrices. It is our hope that the smallrepresentations will be a fruitful tool that can be used to answersome relevant questions in analysis and number theory.