Gordan Savin is continuing his work on algebraic aspects of analysis, with applications in number theory. The main tools of this research are minimal representations (discovered by Kazhdan and Savin) which have been successful in dealing with certain aspects of Langlands conjectures. Indeed, several instances of Langlands conjectures, out of reach of standard methods, can be obtained through use of minimal representations.
Analysis, broadly defined, deals with functions satisfying certain differential equations. Analysis of differential equations, in general, is a very hard problem. For example, differential equations coming from the dynamics of fluids are notoriously difficult. One possible reason for this is a lack of symmetries. On the other hand, if there are plenty of symmetries to work with, then the corresponding analysis is easier, since algebraic tools can be used.
This research deals precisely with situations when a large group of symmetries is present. The main purpose of this research is to construct small representations of groups of Lie type. Smallness, roughly speaking, refers to the fact that it is possible to combine together many symmetries acting on a relatively small space. Importance of having small representations can be illustrated with the following example. One of the most famous (and most notorious) groups is the Reisz-Fischer Monster. Yes, this group is huge and hence its name, but the size is not the main difficulty here. The main problem is the lack of small representations. Indeed, the group of permutations of 60 letters - nobody would consider this group a ``monster'' - is in fact bigger then the Riesz-Fischer Monster. However, this group of permutations can be written down (i.e. represented) by means of 60 by 60 matrices. The monster, on the other hand, requires use of (roughly) 2000 by 2000 matrices. It is our hope that the small representations will be a fruitful tool that can be used to answer some relevant questions in analysis and number theory.
