Just as one can generate an exact, symplectic map by an implicit generating function, exact volume-preserving maps can be obtained by implicit generating forms. Hector Lomeli and I have shown recently how to generalize this idea, originally due to Carroll, to obtain generating forms of different "types" analogous to the four standard canonical generators.
A first step in a theory of transport is computation of the flux though lobes of turnstiles. For the area preserving case, these fluxes are given by differences between the actions of the heteroclinic orbits that bound the lobes. We show how to generalize this to the exact volume-preserving case. In this case the generating forms give rise to an "action" given by an integral along heteroclinic curves.
The class of groups of finite Morley rank consists of groups that can be equipped with a (model theoretic) notion of dimension, called Morley rank, that applies to the definable subsets of the group in much the same way that the notion of dimension of an algebraic group applies to its constructible sets. In fact, much effort has been put into the Cherlin-Zil'ber conjecture which states that a group of finite Morley rank which is simple is isomorphic (as an abstract group) to an algebraic group over an algebraically closed field. We will discuss how BN-pairs, which have been very helpful in understanding simple algebraic groups, may be helpful in approaching this conjecture.
