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Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. For associative algebras, the symmetric homology theory  was introduced by Z. Fiedorowicz (1991) based on his earlier work with J.-L. Loday. In this talk, after giving a survey of known results on symmetric homology, we show that, for algebras over a field of characteristic 0, this homology theory is naturally equivalent to the (one-dimensional) representation homology. Then, using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz (2007), including their main conjecture on topological realization of symmetric homology of polynomial algebras.

Frank Sottile: "Galois groups in Enumerative Geometry"

Abstract:  In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.

Aldo Conca: "Two bounds on Castelnuovo-Mumford regularity"

Abstract:  I will report on bounds on the Castelnuovo-Mumford regularity for ideals with polynomial parametrization  (joint work with F.Cioffi) and for ideals associated with general subspace arrangements (joint work with M.Tsakiris).