Sets, groups, and fields definable in vector spaces with a
There is a long history of study of algebraic objects definable in
classical mathematical structures. As a prominent example, by results of Weil, Hrushovski, and van den Dries, it is known that the groups definable in an algebraically closed field K are precisely the algebraic groups over K, and the only infinite field definable in K is the field K itself. The talk will be a report on my work on dimension, definable groups, and definable fields in vector spaces over algebraically closed [real closed] fields equipped with a non-degenerate alternating bilinear form or a non-degenerate [positive-definite] symmetric bilinear form. The main result states that every definable group is (algebraic-by-abelian)-by-algebraic [(semialgebraic-by-abelian)-by-semialgebraic], which, in particular, answers a question of N. Granger. It follows that every definable field is definable in the field of scalars, hence either finite or definably isomorphic to it [finite or algebraically closed or real closed]. If time permits, I will very briefly discuss some model-theoretic phenomena in the considered structures, including an observation from a recent joint work with D. Hoffmann describing dividing of formulas in the algebraically closed case, which answers
another question of Granger.