Mathematical Sciences: Stable Groups

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This project is about the Cherlin-Zil'ber Conjecture, which states that a simple group of finite Morley rank is an algebraic group over an algebraically closed field. It has two parts: (1)Disprove the conjecture by finding a potential counter-example (a bad group) in SO3(R); (2)Prove the conjecture for good groups (i.e., for simple groups with involutions). This is too ambitious, and the main emphasis will be on BN-pairs of ranks 1 and 2. The investigator expects to be able to extend methods of finite group theory and come up with new methods (as was done in joint work with Borovik and DeBonis). Groups are basic structures of algebra, the mathematical embodiment of symmetry. Model theory is one of the main branches of the foundations of mathematics. The investigator will bring methods from model theory to bear upon an outstanding conjecture in group theory. The conjecture is interesting in itself but also as a test case to see how far methods of this type will avail.