Non-semisimple constructions in quantum topology produce strong invariants and TQFTs with unprecedented properties. The first family of non-semisimple quantum invariants of 3-manifolds was defined by Hennings in 1996. The construction enabled Lyubashenko to build mapping class groups representations out of every finite-dimensional factorizable ribbon Hopf algebra. Further attempts at extending these constructions to TQFTs only produced partial results, as the vanishing of Hennings invariants in many crucial situations made it impossible to treat non-connected surfaces.
SEMINARI DI GEOMETRIA
Since the volume of a hyperbolic manifold is often regarded as a measure of its complexity, it is interesting to study the examples of minimal volume.
The 2 and 3-dimensional cases are well understood.
In dimension four there is an abundance of examples, but we are far from classifying them. In this talk, we will survey all known examples of minimal volume hyperbolic 4-manifolds and prove that these fall into three commensurability classes. This is joint work with Stefano Riolo.
Given a closed 3-manifold Y with the action of a finite group G, we show how to find a closely related hyperbolic G-manifold Y’. The two manifolds are related by an invertible equivariant homology cobordism; a cobordism from Y to Y’ is invertible if there is a second cobordism from Y’ to Y such that the union of the two along Y’ is a product cobordism. I will give a collection of applications in 3 and 4-dimensional topology. (Joint work with Dave Auckly, Hee Jung Kim, and Paul Melvin.)
Most applications of gauge theory in 4-dimensional topology are concerned with simply-connected manifolds with non-trivial second homology. I will discuss the opposite situation, first describing the classical Rohlin invariant for manifolds with first homology = Z and vanishing second homology. I will give an interpretation in terms of a Seiberg-Witten theory, with an unusual index-theoretic correction term. I will discuss recent work with Jianfeng Lin and Nikolai Saveliev giving a new formula for this invariant in terms of monopole homology.
A cork is a contractible Stein domain that gives arise to exotic pairs of 4-manifolds. The first example was found by Akbulut. It is known that any two exotic, simply-connected, closed 4-manifolds are related by a cork twist. We show that there are no corks having shadow-complexity zero. We also show that there are infinitely many corks having shadow-conplexity 1 and 2.
Kodaira fibrations are surfaces of general type with a non-isotrivial fibration,
which are differentiable fibre bundles. First examples were constructed by
Kodaira and Atiyah to show that the signature need not be multiplicative in
We review approaches to construct such surfaces with focus on controlling the
signature. From this we obtain new examples with signature 16. (Joint work
with Ju A Lee and Michael Lönne)
In his PhD thesis, Paul Seidel found examples of symplectomorphisms which are smoothly isotopic to the identity, but not isotopic to the identity within the symplectomorphism group.
The problems of conjugacy rigidity and of entropy rigidity have a long history, with major breakthroughs in the last twenty years, for negatively curved compact manifolds and symmetric spaces. On the other hand, little is known for finite volume manifolds. I will survey the state of the art of these problems, and explain what can be generalized to finite volume manifolds, pointing out the main difficulties that arise in the non-compact case.
Kazhdan's property (T) is a strong negation of amenability, relevant in the study of algebraic properties of groups, in the construction of expander graphs, in dynamics and in connection to the Baum-Connes conjectures. Various strengthened versions of property (T) have been formulated in recent years.