Welcome
I am Lukas, a mathematician, but also a travel and sports enthusiast, and a nature-lover. In one sentence, I like to explore the world in as many facets as possible.
I am a complex algebraic geometer. In particular, I am interested in the geometry and topology of varieties with klt singularities. I also like to think about algebraic group actions and invariant theory.
I do lots of different sports, such as mountaineering, ski touring, mountain running, snowkiting, downhill mountainbiking, kitesurfing, spearfishing, apnoe diving. Previously, I also played waterpolo.
During my travels, I explored a few quite extraordinary places, such as the mountains and seas of Papua New Guinea, the far east of Turkey, Albania, remote Greek islands, and parts of the Ho Chi Minh trail.
If you want to find out more, just check out the rest of my page!
Math
I am currently a PI with my research project "Degenerate orbifold Kähler metrics and fundamental groups" funded by FWF (Austrian Science Fund) at the University of Innsbruck. Here, I am part of the Algebra group of Prof. Tim Netzer.
Before that, I was a Walter-Benjamin fellow in the Algebraic Geometry group of Stefan Kebekus at the University of Freiburg and principal investigator in my research project 'Fundamental and Divisor Class Group: Finiteness and Interplay' funded by the DFG (German research foundation).
Before that, I was a postdoc in the Arithmetic Geometry group of Annette Huber-Klawitter also at the University of Freiburg.
Even before that, I obtained my PhD in Nov. 2019 with summa cum laude from the University of Tübingen. My advisor was Jürgen Hausen.
I am interested in varieties with klt singularities, in particular their geometry and topology. My main tools are Cox rings, group actions, and orbifolds. I aim to generalize existing notions and fields of application of Cox rings and investigate their interplay with fundamental groups.
You can find my CV here.
Upcoming
----> 25/01/2024, I will give a talk at the Oberseminar Algebraische Geometrie und Arithmetik at the Universität Duisburg-Essen
Research
Preprints
16.
Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs
Joint with Fernando Figueroa, arXiv:2401.01315, submitted.
15.
arXiv:2211.12088, submitted.
14.
Reductive covers of klt varieties
Joint with Joaquin Moraga, arXiv:2210.10095, submitted.
13.
Reductive quotients of klt singularities
Joint with Daniel Greb, Kevin Langlois, and Joaquin Moraga, arXiv:2111.02812, submitted.
publications
12.
Iteration of Cox rings of klt singularities
Joint with Joaquin Moraga, J. Topology 17 (2024), no. 1, 71 pp.
11.
Orbifold Kähler-Einstein metrics on projective toric varieties
Bull. LMS, doi:10.1112/blms.12892.
10.
The Jordan property for local fundamental groups
Joint with Stefano Filipazzi, Joaquin Moraga, and Roberto Svaldi, Geometry and Topology 26 (2022), no. 1, 283-319.
9.
Gorensteinness and iteration of Cox rings for Fano type varieties
Mathematische Zeitschrift 301 (2022), 1047–1061.
8.
The local fundamental group of a Kawamata log terminal singularity is finite
Inventiones Mathematicae 226 (2021), no. 3, 845-896.
7.
Invariant rings of sums of fundamental representations of SLn and colored hypergraphs
Advances in Mathematics 389 (2021), 107929.
6.
Completing the classification of representations of SLn with complete intersection invariant ring
Transformation Groups 26 (2021), no. 1, 115-144.
5.
Canonical threefold singularities with a torus action of complexity one and k-empty polytopes
Joint with D. Hättig, Rocky Mountain J. of Mathematics 50 (2020), no. 3, 881-939.
4.
Hilbert series of the Grassmannian and k-Narayana numbers
Communications in Mathematics 27 (2019), no. 1, 27-41.
3.
Covariants of SL4(C) and incidence geometry of P3(C)
Journal of Physics: Conference Series 1194, 012015 (2019).
Log terminal singularities, platonic tuples and iteration of Cox rings
Joint with Ivan Arzhantsev, Jürgen Hausen, and Milena Wrobel, European Journal of Mathematics 4 (2018), no.1, 242-312.
Other Writings
1.
Finiteness and klt singularities
Oberwolfach Report 24/2020, Komplexe Analysis - Algebraicity and Transcendence (2020), 18-23.
TRavel
Mountain
In the mountains, I do sports such as mountaineering, downhill mountainbiking, trail/mountain running, road cycling and hiking - in summer. In winter, I like to go ski mountaineering and snowkiting.
I also enjoy searching for mushrooms, which happens mostly in the mountains!