Speaker: Sam Buss (Univ. of Calif., San Diego) Title: Polynomial Local Search higher in the Polynomial Hierarchy and Bounded Arithmetic Abstract: The talk will discuss provably total functions of bounded arithmetic, and recent characterization

of the b i denable functions of S k+1 2 (or T k 2 ), for all i k . The main tool is extensions of polynomial local search problems to higher levels of the polynomial time hierarchy, where the feasible set is dened by a b k predicate but the cost and neighborhood functions are denable by polynomial time terms. These higher level PLS problems can be used to determine the truth of b k properties and also allow \witness doubling". These results can be formalized and then Skolemized with a weak base theory such as S 1 2 | the stronger theory S k+1 2 (or T k 2 ) is needed only to prove the existence of a solution. The Skolemization allows us to dene sets of clauses that are refutable in a depth m propositional refutation system (a Tait style system for propositional logic), but are conjectured not to be provable in a depth m 1=2 system. We discuss open problems and future directions for research. This work is joint with Arnold Beckmann. Abstract: We will discuss some results on the innite combinatorics of words and their connection to idempotent ultralters. Speaker: Douglas Cenzer (University of Florida) Title: Space Complexity of Abelian Groups Abstract: We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups Z, Z(p 1 ), and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the eective