Abstract: Smith theory is the homological theory of transformation groups, relating mod p cohomological information to actions of a p-group. Rationally (or even away from the divisors of the order of the group), most of the information one has comes from the Lefshetz fixed point theorem (theorems of Jones and Oliver from the 1970's make this precise). In this talk, I will try to describe some pieces of the non-simply connected theory that I have been working on with Sylvain Cappell (NYU) and Min Yan (HKUST).

Abstract: The s-cobordism theorem gives a complete classification of h-cobordisms from a fixed manifold of dimension at least five, but it does not say much about the manifolds "at the other end" -- something important for applications. In this talk I will discuss this problem and give a survey of both old results and more recent joint work with Kwasik and Hausmann.

Abstract: There exist topological spaces which have multiple decompositions into Cartesian products. There exist such examples for continua, polyhedra and manifolds. We also have a lot of theorems where with additional conditions the decomposition is unique. This is a survey of the results in this field.

Abstract: We introduce a new stable range invariant for the classification of closed, oriented topological 4--manifolds (up to s-cobordism), after stabilization by connected sum with a uniformly bounded number of copies of $S^2\times S^2$.

Abstract: Given a self-map of a compact, connected topological space we consider the problem of determining bounds for the fixed point indices of the map. In general, to obtain bounds one needs to restrict attention to the class of spaces under consideration, and possibly, the class of self-maps. Motivated by an elementary result in the case of a 1-dimensional complex and the analogous result for pseudo-Anosov surface homeomorphisms this talk will focus attention to the setting of 2-complexes and what are referred to as the hyperbolic index bounds. Some past results and related examples will be presented, leading to some current joint work with D. L. Goncalves (Univ. Sao Paulo, Brasil).

Abstract: I will discuss the problem of existence of positive scalar curvature on manifolds with fibered singularities. It turns out there are necessary and sufficient conditions for a psc-metric to exist on such objects. There is a particular case of manifolds with fibered singularities when the fiber is a circle. This case leads to psc-metrics spinc manifolds with special conditions near the singular locus. In particular, I describe some results concerning metrics on spinc manifolds with positive ``twisted scalar curvature,'' where the twisting comes from the curvature of the spinc line bundle.
This work is joint with Jonathan Rosenberg and Paolo Piazza.

Abstract: We shall discuss 10 open questions about finite group actions on 4-manifolds. The answers to many of these questions are beyond the reach by the current techniques, so hopefully these questions may stimulate inventions of new ideas or methods. On the other hand, we choose these questions in order to explore the differences between the locally-linear, smooth, symplectic, and holomorphic categories in the study of finite group actions in dimension 4, and to understand the subtleties of some of the issues in group actions when being considered under the different categories.

Abstract: The configuration space $F_2(M)$ of ordered pairs of distinct points in a manifold $M$, also known as the deleted square of $M$, is not a homotopy invariant of $M$: Longoni and Salvatore produced examples of homotopy equivalent lens spaces $M$ and $N$ of dimension three for which $F_2(M)$ and $F_2(N)$ are not homotopy equivalent. We study the conjecture, proposed by S{\l}awomir Kwasik, that two arbitrary 3-dimensional lens spaces $M$ and $N$ must be homeomorphic in order for $F_2(M)$ and $F_2(N)$ to be homotopy equivalent. Among our tools are the Cheeger–Simons differential characters of deleted squares and the Massey products of their universal covers. This is a joint work with Kyle Evans-Lee.