SR1

Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.
One can do better and compute the cohomology in terms of this function using the so-called Witten complex.
Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.
I will explain Witten's (heuristic) arguments and describe the Witten complex.

A topological space is called rigid if its only autohomeomorphism is the identity map. Using the Axiom of Choice it is easy to construct rigid subsets of the real line R, but sets constructed in this way always have size continuum. I will explore the question of whether it is possible to have rigid subsets of R that are small, meaning that their cardinality is smaller than that of the continuum. On the one hand, we will see that forcing can be used to produce models of ZFC in which such small rigid sets abound. On the other hand, I will introduce a combinatorial axiom that can be used to show the consistency with ZFC of the statement "CH fails but every rigid subset of R has size continuum". Only a working knowledge of basic set theory (roughly what one might remember from C1.2b) and topology will be assumed.

"Among the first successes of the h-cobordism theorem was the classification of simply connected closed 5-manifolds. Dimension five is sufficiently large to be able to implement the tools of surgery theory, yet low enough to allow an explicit classification of the manifolds. These traits make dimension five interesting in terms of existence results of geometric structures, like Riemannian metrics of positive Ricci/nonnegative sectional/positive sectional curvature, Einstein metrics, contact structures, Sasakian structures, among others. The talk will be a limited survey of the five-dimensional symbiosis between topology and geometry"

I will be taking us on a journey through low dimensional topology, starting in 2 dimensions motivating handles decompositions in a dimension that we can visualize, moving onto to a brief of note of what this means in 3 dimensions and then moving onto the wild world of 4 manifolds. I will be showing a way in which we can actually try and view a 4 manifold before moving onto a way of manipulating these diagrams to give diffeomorphic 4 manifolds. Hopefully, I will have time to go into some ways in which Kirby calculus has been used to show that certain potential exotic 4 spheres are not exotic and some results on stable diffeomorphims of 4 manifolds.