**Title:** Pseudo-reductive groups

**Speaker:** Professor Brian Conrad

**Speaker Info:** Stanford

**Brief Description:**

**Special Note**:

**Abstract:**

One of the most beautiful topics in pure mathematics is the structure theory of connected reductive algebraic groups over general fields. This is especially nice over separably closed fields, where it is given in terms of root systems. Over general perfect fields (such as Q) one can often reduce questions involving general smooth connected affine algebraic groups to the reductive case, but this is not at all possible over imperfect fields, such as function fields of curves over finite fields. This leads one to seek a weakening of reductivity (equivalent to it over perfect fields) for which one can nonetheless develop a useful structure theory. The right weakening, called pseudo-reductivity, was introduced and studied by Borel and Tits, but they were unable to develop a classification suitable for arithmetic applications.In joint work with O. Gabber and G. Prasad we have established such a classification (away from characteristic 2 for now). I will explain some concrete problems that motivate the desire to study pseudo-reductive groups, and show a number of examples to illustrate various aspects of the classification. The precise form of the classification theorem will also be given.

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