How should homotopy groups with coefficients in an abelian group be defined? There are three criteria. They should be functors on the homotopy category of pointed spaces. They should satisfy a universal coefficient theorem. They should have long exact sequences related to fibrations.

For coefficients in finitely generated abelian groups, such functors exist and are corepresentable. For rational coefficients, such functors exist but it is a theorem of Kan and Whitehead that they are not corepresentable.

In the case of finite coefficients, one would like that the homotopy groups have a global exponent which is the same as that of the coefficient group. The question reduces to the so-called co H-space exponents of Moore spaces. In dimensions 4 and higher, these exponent questions are easy but the answer can be surprising. For example, groups with mod 2 coefficients can have exponent 4.

The case of the exponent of the 3 dimensional homotopy group has some subteties which are addressed by application of a variation of the classical Hopf invariants introduced by Hopf.

The most famous open problem in the area of nil algebras is the Köthe Conjecture, first posed in 1930, which asserts that if a ring has no nonzero nil ideals then it has no nonzero nil one-sided ideals. This is a fundamental question about the general structure of rings, and a thorough understanding of nil and nilpotent rings is necessary for any serious attempt to understand general rings. The most famous problem about algebraic algebras is the Kurosh Problem, which is of a similar vintage and asks whether the knowledge that a finitely generated algebra is algebraic over a base field is sufficient to ensure that the algebra is finite dimensional. This is untrue in general, as demonstated by Golod and Shafarevich in 1964. However, many partial positive results are known, and the borderline between positive and negative solutions of the Kurosh Problem is still being investigated. There are close connections between these two general themes; for example, the Golod-Shafarevich algebras are infinite dimensional finitely generated nil algebras that are not nilpotent. My talk will be a short survey of the current state of these two themes, including more on their relationship.

We review slope stability for coherent sheaves on algebraic curves and then discuss Tom Bridgeland's generalization of stability to triangulated categories. For a certain triangulated category associated to a Kleinian singularity, Bridgeland conjectured that a connected component of the space of stability conditions should be the universal cover of a $K(G,1)$ for $G$ a generalized braid group and showed this is the case for $G$ the classical braid group. We generalize this to braid groups of types ADE. This is joint work with Hugh Thomas.