A search for an algebraic equivalence analogue of motivic theories

Abstract: We reflect on mathematical efforts made years ago, initiated by Blaine Lawson and much influenced by Vladimir Voevodsky's work. In work with Lawson, Mazur, Walker, Suslin, and Haesemyer, a "semi-topological theory" for cohomology and K-theory of complex (or real) varieties has evolved which has led to a few computations and many conjectures. Over the intervening years, Haesemeyer, Walker, and I have sought to formulate such a theory over more general fields, a theory in which algebraic equivalence replaces the role played by rational equivalence in motivic cohomology and algebraic K-theory. This talk will report on challenges and conjectures.