K-theory of Teichmüller spaces

The Theory of Teichmüller Spaces – A View Towards Moduli Spaces of Kähler Manifolds

An analog of a modular functor from quantized Teichmüller theory

We review the quantization of Teichmüller spaces as initiated by Kashaev and Checkov/Fock. It is shown that the quantized Teichmüller spaces have factorization properties similar to those required in the definition of a modular functor.

Some Properties of Continuous $K$-frames in Hilbert Spaces

The theory of  continuous frames in Hilbert spaces is extended, by using the concepts of measure spaces, in order to get the results of a new application of operator theory.  The $K$-frames were  introduced by G$breve{mbox{a}}$vruta (2012) for Hilbert spaces to study atomic systems with respect to a bounded linear operator. Due to the structure of  $K$-frames, there are many differences between...

Teichmüller spaces and holomorphic dynamics

One fundamental theorem in the theory of holomorphic dynamics is Thurston’s topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston’s theorem (the marked Thurston’s theorem). We also mention some applications and related results...

On the Relation between Quantum Liouville Theory and the Quantized Teichmüller Spaces

— We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verificat...