# Yannick Krifka

I am a PhD student at ETH Zürich. My advisor is Prof. Alessandra Iozzi .

J 16.5
ETH Zürich
Hauptgebäude
Rämistrasse 101
CH-8092 Zürich
Switzerland

yannick.krifka@math.ethz.ch

# Curriculum Vitae

You can find my CV here.

# Teaching Activities

• HS 2017:

Organization and Teaching Assistant for "Introduction to Lie Groups" by Prof. Alessandra Iozzi (lecture homepage).

• FS 2017:

Organization of "Analysis II (D-BAUG)" by Dr. Menashe-hai Akka Ginosar (moodle-course).

• HS 2016:

Organization of "Analysis I (D-BAUG)" by Dr. Menashe-hai Akka Ginosar (moodle-course).

# Miscellaneous

Here are some miscellaneous documents:
• Lecture Notes for Dr. Anne Thomas' course on "Geometric and topological aspects of Coxeter groups and buildings" at the ETH Zürich during the spring term 2016; last lecture included: Lecture 7 from 20.04.2016.
• My master's thesis "On Volume Rigidity of Lattices" at the University of Heidelberg. Advisor: Prof. Anna Wienhard.

+ Abstract

Our goal in this master's thesis is to give a detailed proof of the volume rigidity theorem due to Bucher, Burger, and Iozzi, following the lines of the article [BBI13]. To every lattice embedding $$i : \Gamma \to \textrm{Isom}^+(\mathbb{H}^n)$$ and any representation $$\rho : \Gamma \to \textrm{Isom}^+(\mathbb{H}^n)$$ we may associate a real number $$\textrm{Vol}( \rho )$$, the so called volume of $$\rho$$. The definition of $$\textrm{Vol}(\rho)$$ relies on techniques from bounded cohomology and is reminiscent of the definition of the Toledo invariant for surface group representations as in [BIW03], [BI07]. If $$n \geq 3$$, the volume rigidity theorem asserts that $$| \textrm{Vol}(\rho) | \leq | \textrm{Vol}(i) | = \textrm{Vol}(M)$$, where $$M = i(\Gamma) \backslash \mathbb{H}^n$$. Moreover equality holds if and only if $$\rho$$ is conjugated to $$i$$ by an isometry. This may be considered as a generalization of Mostow's rigidity theorem for finite volume hyperbolic manifolds of dimension at least three.

Along the way, background information on hyperbolic geometry and in particular on continuous (bounded) cohomology is provided, introducing the reader to the subject. We also prove a version of de Rham's theorem for relative de Rham cohomology in the appendix. Further a detailed discussion of Douady-Earle's barycenter construction for probability measures on $$\partial \mathbb{H}^n$$ with no atoms of mass $$\geq 1/2$$ is included.

• My bachelor's thesis "Fenchel-Nielsen Koordinaten auf Teichmüllerräumen" at the University of Heidelberg. Advisor: Prof. Winfried Kohnen.

+ Abstract

The aim of this bachelor's thesis is to give an elementary introduction to the topic of Fenchel-Nielsen coordinates on Teichmüller spaces. Therefore we explain some basic notions of topology and Riemann surfaces in chapter one. In chapter two we study the upper half-plane. Thereby we consider Möbius transformations and outline their connection to the geometry of the upper half-plane. In chapter three we prove a classification theorem for compact Riemann surfaces based on the uniformization theorem, which is crucial for our further examinations. In chapter four we first consider the moduli space of tori and show that it can be identified with $$\mathbb{H} / \textrm{PSL}(2; \mathbb{Z})$$. Later on we study the Teichmüller spaces $$T_g$$ of compact Riemann surfaces with genus $$g \geq 2$$ and introduce the so called Fenchel-Nielsen coordinates. These will provide us with a bijection $$\Lambda : \mathbb{R}_+^{3g - 3} \times \mathbb{R}^{3g-3} \to T_g$$.