This is joint work with Angus Macintyre. I will discuss new developments in
our work on the model theory of adeles concerning model theoretic
properties of adeles and related issues on adelic geometry and number theory.
In this talk, two concepts are brought together: Algebras with triangular decomposition (as studied by Bazlov & Berenstein) and pointed Hopf algebra. The latter are Hopf algebras for which all simple comodules are one-dimensional (there has been recent progress on classifying all finite-dimensional examples of these by Andruskiewitsch & Schneider and others). Quantum groups share both of these features, and we can obtain possibly new classes of deformations as well as a characterization of them.
We take a look at diamond and use it to build interesting
mathematical objects.
I will show how to construct an infinite family of totally geodesic surfaces in the figure eight knot complement that do not remain totally geodesic under certain Dehn surgeries. If time permits, I will explain how this behaviour can be understood via the theory of quadratic forms.
We will discuss TQFTs (at a basic level), then higher categorical extensions, and see how these lead naturally to the notion of Segal spaces.
We will finish presenting Nyikos' counterexample to
Bozyakova's conjecture: If e(Y) = L(Y) for every subspace Y of X, must X
be hereditarily D?
Last week in the Kinderseminar I talked about a rough estimate on volumes of certain hyperbolic 3-manifolds. This time I will describe a different approach for similar estimates (you will not need to remember that talk, don't worry!), which is, in some sense, complementary to that one, as it regards mapping tori. A theorem of Jeffrey Brock provides bounds for their volume in terms of how the monodromy map acts on the pants graph (a relative of the better known curve complex) of the base surface. I will describe the setting and the relevance of this result (in particular the one it has for me); hopefully, I will also tell you part of its proof.