Abstract: An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

Abstract: We show that the category of coefficients for Hopf cyclic cohomology has two proper subcategories where one of them is the category of stable anti Yetter-Drinfeld modules. Generalizations of suitable coefficients for Hopf cyclic cohomology are introduced. The notion of stable anti Yetter-Drinfeld modules is extended based on underlying symmetries. We show that the new introduced categories for coefficients of Hopf cyclic cohomology and the category of stable anti-Yetter-Drinfeld modules are all different. (This is joint work with Bahram. Rangipour and Dan. Kucerovsky )

Abstract: Roughly speaking, the Hall algebra $H(A)$ of a (small) Abelian category $A$ is the algebra of finitely supported functions on the moduli space of objects of $A$ (i.e. the set of isoclasses of objects of $A$ with the discrete topology). Interest in Hall algebras exploded in the early 1990's when Ringel discovered that the Hall algebra associated to the category of $F_q$-representations of a Dynkin quiver $Q$ provides a realization of the positive part of the (quantized) enveloping algebra of the (simple) complex Lie algebra associated to the same Dynkin diagram. To\"{e}n and Bergner have used the theory of model categories to obtain Hall algebras on triangulated categories. In this talk we will survey these constructions and, time permitting, explain some open problems in this area which are being studied via homotopy theory.