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No abstract

We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed. (2000). 12H05. Mathematics Subject Classification

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.