I will describe a general procedure to homotope an immersed positive genus surface in a simply connected 4-manifold to a locally flat embedding. This is a special case of a surface embedding theorem, joint with Daniel Kasprowski, Mark Powell, and Peter Teichner.
I will introduce and study relations of 4-manifolds up to connected sum with copies of $S^2\times S^2$ and their relations. This includes stable diffeomorphism and homotopy equivalence. The talk is based on joint work with Johnny Nicholson and Simona Veselá.
Vector spaces having “duals” are automatically finite dimensional, and this is the case for those appearing as values of TFTs. However, if we assume that the vector spaces are pointed, they are automatically one dimensional (lines). When constructing extended n-dimensional TFTs, a natural family of targets (replaceing Vect) naturally has the feature that pointings are built in. This is due to...
String topology can be seen as a form of 2d field theory on the homology of the free loop spaces of manifolds. I’ll describe this field theory, and exhibit some of its interesting features.
In the first part of the talk, we give some background on the geometry of diamond cuts, and the special optical properties that make them so captivating to look at. In the second part, we discuss how the classical mathematics of the Maxwell-Cremona correspondence can assist in the enumeration of possible diamond cuts.
A basic problem in low dimensional topology is to understand the 2-complexes with a given fundamental group G. I will explain how this can be studied using a division algorithm in the group ring of G, and describe some instances in which such an algorithm is available.
