Email: christian.urech 'at' math.ethz.ch
Office: HG G66.3
Address: Group 1, Department of Mathematics, Rämistrasse 101, 8092 Zurich, Switzerland
I am a Senior Lecturer at ETH Zürich. Before that, I worked as a Bernoulli Instructor at EPFL and as a research associate at Imperial College London. I completed my PhD in 2017 at the University of Basel and the University of Rennes 1 under the supervision of Jérémy Blanc and Serge Cantat. Here you find my CV.
My research lies at the intersection of algebraic geometry, geometric group theory, and dynamics. I apply tools from geometric group theory to study Cremona groups. But I am excited by everything related to groups of birational transformations or polynomial automorphisms. Recently, I have also worked on braided Thompson’s groups and their finiteness properties.
Group Theoretical Characterizations of Rationality, with A. Regeta, and I. van Santen, ArXiv.
The Structure of Algebraic Families of Birational Transformations, with A. Regeta, and I. van Santen, ArXiv.
On a Theorem by Lin and Shinder through the Lens of Median Geometry, with A. Genevois and A. Lonjou, ArXiv.
Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded, with A. Lonjou and P. Przytyzki, ArXiv, to appear in Journal de l’École polytechnique.
Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature, with A. Genevois and A. Lonjou, ArXiv, to appear in Transactions of the AMS.
Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes, with A. Genevois and A. Lonjou, ArXiv, International Mathematics Research Notices, Vol. 2024, Issue 1, January 2024, 554–596.
Characterization of affine toric varieties by their automorphism groups, with A. Liendo and A. Regeta, ArXiv, Ann. Sc. Norm. Sup. Pisa-Cl. Sci. (2023), Vol. XXIV, 249-289.
Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson’s and Houghton’s groups, with A. Genevois and A. Lonjou, ArXiv, Geom. Topol. 26 (2022) 1385–1434.
Continuous automorphisms of Cremona groups, with S. Zimmermann, ArXiv, Int. J. Math., Vol. 32 (2021), No. 04, 2150019.
Actions of Cremona groups on CAT(0) cube complexes, with A. Lonjou, ArXiv, Duke Math. J. 170.17 (2021): 3703-3743.
On the characterization of Danielewski surfaces by their automorphism group, with A. Liendo and A. Regeta, ArXiv, Transform. Groups 27, 181–187 (2022).
Simple groups of birational transformations in dimension two, ArXiv, Comment. Math. Helv., Vol. 95, Issue 2, 2020, pp. 211–246.
Subgroups of elliptic elements of the Cremona group, ArXiv, J. fur Reine Angew. Math., 2021, Issue 770, pp. 27-57.
A new presentation of the plane Cremona group, with S. Zimmermann, ArXiv, Proc. Amer. Math. Soc. 147 (2019), no. 7, 2741–2755
Remarks on the degree growth of birational transformations, ArXiv, Math. Res. Lett., Vol. 25, No. 1 (2018), pp. 291-308.
On homomorphisms between Cremona groups, ArXiv, Ann. Inst. Fourier, Vol. 68, No. 1 (2018), pp. 53-100.
Actions of Cremona groups on CAT(0) cube complexes, contribution to the ZAG Volume.
Lecture Notes for a Mini-course on median graphs with applications to Cremona groups, given together with A. Genevois and A. Lonjou at the University of Rennes 1 in May 2023.
Subgroups of elliptic elements of the plane Cremona group, in: Oberwolfach Report 28/2018.
Subgroups of Cremona groups, PhD thesis at the University of Basel and the University of Rennes 1, defended in September 2017.
I enjoy teaching mathematics on all levels, be it in person or online. During the last years I got interested in evidence-based teaching strategies and I have started teaching my courses in the mode of a flipped classroom (integrating peer instruction). This has been received very positively by the students.
Courses I have taught as an instructor:
Between 2010 and 2017 I worked as a teaching assistant at the University of Basel for around a dozen different courses in various fields of mathematics, which included grading homework and giving weekly tutorials.
In Summer 2018 I taught two courses at the 2018 TMD Undergraduate and Graduate Summer School at the Nesin Math Village, Turkey - a wonderful place!
Do not hesitate to get in touch, if you want to do a project with me. At the intersection of algebraic geometry, group theory, and topology there are many beautiful and fun subjects on all levels.
Supervised student projects:
I thoroughly enjoy all kinds of performing arts (and I think a good math talk might very well fall into this category). I have cofounded and I still coorganize the following charming project in Basel: Station Circus - a venue for contemporary circus performances, as well as space for creation and community.