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 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
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April 9-14 2023, I will be at the Oberwolfach Workshop 'Komplexe Analysis — Differential and Algebraic methods in Kähler spaces'
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April 20th, 2023, I give an online talk at Nottingham University about 'Reductive quotients of klt singularities'
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May 8th, 2023, I give a talk in the Differential Geometry Seminar at the University of Freiburg about 'Reductive quotients of klt singularities'
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The fall of 2023, together with Roberto Svaldi and Calum Spicer, we participate in the Junior Trimester Program Algebraic geometry: derived categories, Hodge theory, and Chow groups at the Hausdorff Research Institute for Mathematics in Bonn.
Research
Preprints
15.
arXiv:2211.12088, submitted.
14.
Orbifold Kähler-Einstein metrics on projective toric varieties
arXiv:2211.00404, submitted.
13.
Reductive covers of klt varieties
Joint with Joaquin Moraga, arXiv:2210.10095, submitted.
12.
Reductive quotients of klt singularities
Joint with Daniel Greb, Kevin Langlois, and Joaquin Moraga, arXiv:2111.02812, submitted.
11.
Iteration of Cox rings of klt singularities
Joint with Joaquin Moraga, arXiv:2103.13524, submitted.
publications
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!