Office: Ernst-Zermelo-Str. 1, 79104 Freiburg i. Br., Room 433

Office hours: Di, 8-10

E-Mail: lukas.braun@math.uni-freiburg.de

Phone: 0761 / 203 5550

I am currently a Walter-Benjamin-fellow at the University of Freiburg with my project 'Fundamentalgruppe und Divisorenklassengruppe: Endlichkeit und Zusammenspiel' funded by the DFG.

Before that, I was a postdoc at the University of Freiburg in the Arithmetic Geometry group of Prof. Annette Huber-Klawitter.

Even before that, on November 08th, 2019, I defended my PhD thesis entitled "Quotient presentations of Mori Dream Spaces" with *summa cum laude* under the supervision of Prof. Jürgen Hausen.

I am interested in complex algebraic varieties with klt singularities. In particular, I am working on Mori Dream Spaces and Cox rings, questions surrounding the Minimal Model Program, varieties with group actions, and also on Invariant Theory.

You can find a copy of my CV here.

Also have a look at my private homepage!

- This spring, I will give talks in the Online Algebraic Geometry Seminar at the University of Nottingham and in the Oberseminar Differentialgeometrie at the Universität Freiburg.
- April 9-14th 2023, I will be at the Oberwolfach Workshop 'Komplexe Analysis — Differential and Algebraic methods in Kähler spaces'
- 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.

- ODD Metrics; submitted.
- Orbifold Kähler-Einstein metrics on projective toric varieties; submitted.
- Reductive covers of klt varieties, with J. Moraga; submitted.
- Reductive quotients of klt singularities, with D. Greb, K. Langlois, J. Moraga; submitted.
- Iteration of Cox rings of klt singularities, with J. Moraga; submitted.

- The Jordan property for local fundamental groups, with S. Filipazzi, J. Moraga, and R. Svaldi;
*Geometry and Topology***26**(2022), no. 1, 283-319. - Gorensteinness and iteration of Cox rings for Fano type varieties;
*Mathematische Zeitschrift***301**(2022), 1047–1061. - The local fundamental group of a Kawamata log terminal singularity is finite;
*Inventiones Mathematicae***226**(2021), no. 3, 845–896. - Invariant rings of sums of fundamental representations of SLn and colored hypergraphs,
*Advances in Mathematics***389**(2021), 107929; 34 pp. - Completing the classification of representations of SLn with complete intersection invariant ring,
*Transformation Groups***26**(2021), no. 1, 115-144. - Canonical threefold singularities with a torus action of complexity one and k-empty polytopes, with D. Hättig;
*Rocky Mountain Journal of Mathematics***50**(2020), no. 3, 881-939. - Hilbert series of the Grassmannian and k-Narayana numbers,
*Communications in Mathematics***27**(2019), no. 1, 27-41. - Log terminal singularities, platonic tuples and iteration of Cox rings, with I. Arzhantsev, J. Hausen, and M. Wrobel;
*European Journal of Mathematics***4**(2018), no. 1, 242-312.

- Finiteness and klt singularities, Oberwolfach Report 24/2020, Komplexe Analysis - Algebraicity and Transcendence (2020), 18-23.
- Covariants of SL
_{4}(C) and incidence geometry of P_{3}(C),*J. Phys.: Conf. Ser.***1194**(2019), 012015.

- Long-term monitoring (1953-2019) of geomorphically active sections on LIA lateral moraines under changing meteorological conditions, with M. Altmann, M. Pfeiffer, F. Haas, J. Rom, F. Fleischer, T. Heckmann, L. Piermattei, M. Wimmer, M. Stark, S. Betz-Nutz, and M. Becht; EGUsphere [preprint], https://doi.org/10.5194/egusphere-2022-1512, 2023.

- 11/2022:
*"Orbifold Kähler-Einstein metrics on toric non-K-polystable Fano varieties "*, Oberseminar Algebra, University of Jena. - 10/2022:
*"Riemann und der gekrümmte Raum"*, outreach talk for highschool students, University of Freiburg. - 08/2022:
*"Reductive quotients of klt varieties"*at the Simons conference "Stability and Moduli", Simons Centre at Stony Brook - 06/2022:
*"Reductive quotients of klt varieties"*at the Oberseminar Algebraische Geometrie, Universität des Saarlandes - 06/2022:
*"Reductive quotients of klt varieties"*at the Birational Geometry Conference, EPFL Lausanne - 06/2022:
*"Reductive quotients of klt varieties"*at Universität Bayreuth - 05/2022:
*"Reductive quotients of klt varieties"*at the seminar Algèbre et Géométries at Institut Fourier/Université Grenoble-Alpes - 03/2022:
*"Finiteness of local fundamental groups"*at the MMP Learning Seminar, Princeton University - 02/2022:
*"Kähler metrics, orbifold structures, and fundamental groups on resolutions of klt spaces"*at the Conference on Subvarieties and Foliations of Complex Projective Varieties, Osaka City University. - 06/2021:
*"Local Cox rings for klt singularities"*at Politecnico/Università degli studi di Milano - 06/2021:
*"Local Cox rings for klt singularities"*at the Oberseminar Algebra, Zahlentheorie und algebraische Geometrie, Universität Freiburg - 06/2021:
*"Local Cox rings for klt singularities"*at the Oberseminar Algebraische Geometrie und Arithmetik, Universität Duisburg-Essen - 08/2020:
*"Finiteness and klt singularities"*at the Oberwolfach Workshop "Komplexe Analysis - Algebraicity and Transcendence" - 06/2020:
*"Some finiteness results on klt singularities in characteristic zero"*at the Groups, Arithmetic & Algebraic Geometry Seminar, EPFL Lausanne - 04/2020:
*"Fundamental groups of smooth loci of klt spaces"*at the Yale Algebraic Geometry online seminar - 06/2019:
*"Gorensteinness and iteration of Cox rings"*at the Basel-Dijon-EPFL joint conference, Universität Basel - 05/2019:
*"Gorensteinness and iteration of Cox rings"*at the Oberseminar Algebra, Zahlentheorie und algebraische Geometrie, Universität Freiburg - 05/2019:
*"Gorensteinness and iteration of Cox rings"*at the Oberseminar Zahlentheorie und Arithmetische Geometrie, Leibniz Universität Hannover - 08/2018:
*"Invariants of fundamental representations of SLn and colored hypergraphs"*, Short Communication at the ICM 2018 in Rio de Janeiro - 07/2018:
*"Invariants of fundamental representations of SLn and colored hypergraphs"*at the GROUP32 Conference at Technical University of Prague - 04/2018:
*"Can the Big Bang be embedded somewhere and is it a quotient?"*at the Doctoral club seminar series, Forum Scientiarum, Universität Tübingen - 08/2017:
*"Classification of compound du Val and canonical threefold singularities with torus action of complexity one"*at the workshop Experimental Classification of Fano varieties, Universität Tübingen