Aarhus University

Global singularity theory is a classical subject which classifies singularities of maps between manifolds, and describes topological reasons for their appearance. I will start with explaining a central problem of the subject regarding multipoint and multisingularity loci, then give an introduction into some recent major developments by Kazarian, Rimanyi, Szenes and myself.

The presence of glaciers and ice sheets leaves a significant imprint
on Earth's surface.  We steadily improve our physical understanding of
the involved processes, from the erosion of kilometer-deep fjords in
crystalline bedrock to the broad ice-marginal deposition of sediments.
This talk will highlight observations of landforms, sedimentary deposits,
laboratory experiments, and models that aim to capture the interplay
between ice and substratum.  I show how the interplay may play a role in
the future evolution of the West Antarctic Ice Sheet in a warming climate.

The Witten-Reshetikhin-Turaev invariant Z(X,K) of a closed oriented three-manifold X containing a knot K, was originally introduced by Witten in order to extend the Jones polynomial of knots  in terms of Chern-Simons theory. Classically, the Jones polynomial is defined for a knot inside the three-sphere in  a combinatorial manner. In Witten's approach, the Jones polynomial J(K) emerge as the expectation value of a certain observable in Chern-Simons theory, which makes sense when K is embedded in any closed oriented three-manifold X. Moreover; he proposed that these invariants should be extendable to so-called topological quantum field theories (TQFT's). There is a catch; Witten's ideas relied on Feynman path integrals, which made them unrigorous from a mathematical point of view. However; TQFT's extending the Jones polynomial were subsequently constructed mathematically through combinatorial means by Reshetikhin and Turaev. In this talk, I shall expand slightly on the historical motivation of WRT invariants, introduce the formalism of TQFT's, and present some of the open problems concerning WRT invariants. The guiding motif will be the analogy between TQFT and quantum field theory.

I will introduce star products and formal connections and describe approaches to the problem of finding a trivialization of the formal Hitchin connection, using graph-theoretical computations.

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,

Andersen and Kasheav have proposed a new state-integral model for the

Andersen--Kashaev TQFT, where the circle valued state variables live on

the edges of oriented levelled shaped triangulations. I will look at a

couple of examples which give an idea of how the theories are coupled.

In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.

Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.

We shall introduce complex projective structures on a surface, and discuss a new result that relates grafting, which are certain geometric deformations of these structures, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any Fuchsian representation of a surface-group, the set of projective structures with that as holonomy, is dense in moduli space.

In this talk we show how Teichmüller curves can be used to compute

quantum invariants of certain Pseudo-Anasov mapping tori. This involves

computing monodromy of the Hitchin connection along closed geodesics of

the Teichmüller curve using iterated integrals. We will mainly focus on

the well known Teichmüller curve generated by a pair of regular

pentagons. This is joint work with J. E. Andersen.