Mathematical Sciences: Geometric Stability Theory

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9400894 Hrushovski The investigator would like to study the model theory of the Frobenius automorphism. This work has already started, on two fronts. On the one hand, he formulated a conjecture analogous to the Lang-Weil estimates, but more general, that would determine the first order theory of the Frobenius. On the other hand, together with several coworkers, he is in the process of carrying out the fundamental model theoretic analysis of the theory of difference fields in question, along the lines of Shelah and Zilber. Put together, these should throw considerable light on the issue; there will be applications to finite simple groups and perhaps to finite groups in general. The work described in the above paragraph forms part of a general trend in the area of mathematical logic known as model theory. For several decades, this subject developed autonomously. The lack of external pressure allowed it to refine distinct points of view on algebraic structures, and to reach a degree of technical maturity. However, this was done as it were in laboratory conditions, under assumptions ("stability") that are too restrictive in practical applications. The time seems ripe to test the ideas developed in this period against more complex structures, in particular, number theoretic and geometric ones. This requires identifying structures appropriate for model theoretic analysis and performing an algebro-geometric analysis needed to prepare for the model theoretic tools. It also requires generalizing the model theoretic ideas beyond the "stable" context in which they were conceived. Some success has been had with these methods, including the investigation of finite structures with a high degree of symmetry, and the proof of certain conjectures in number theory. One hopes that this is only the beginning of the road. ***