## POSITION

- Since April 2023 I am DFG (Eigene Stelle) fellow at the University of Jena.

- Since October 2019 until March 2023 I was a postdoc at the University of Jena.

- Starting from January 2019 until September 2019 I was a postdoc at the University of Tübingen.

During January - March 2019 I was a visitor in MPI Bonn.

- Starting from October 2017 until December 2018 I was a postdoc at the University of Cologne.

- Starting from April 2016 until September 2017 I was an SNSF (Early Postdoc Mobility) fellow at the University of Grenoble.

- Starting from April 2012 until March 2016 I was a PhD student at the University of Basel.

Conferences which I co-organise:

Regional Workshop in Algebraic Geometry, University of Jena, March 2023 (co-organized with R. Púček).

Workshop on Algebraic Transformation Groups, Monte Verità, Switzerland, May 5 to May 8, 2024

(co-organized with J. Blanc and I. van Santen).

Teaching in the winter semester 2022/2023 (at the University of Jena)

Seminar: The geometry of T-varieties

Teaching in the summer semester 2022 (at the University of Jena)

Lecture: Linear Algebra

Teaching in the winter semester 2021/2022 (at the University of Jena)

Exercise classes: Linear Algebra

Teaching in the summer semester 2021 (at the University of Jena)

Lecture: Introduction to Algebraic Geometry

Teaching in the winter semester 2020/2021 (at the University of Jena)

Exercise classes: Linear Algebra

Teaching in the summer semester 2020 (at the University of Jena)

Lecture: Reflection Groups and Invariant Theory

Earlier I thought other courses at the University of Jena, at the university of Tübingen,

at the university of Cologne and at the University of Basel.

Master student

January-July 2022: Leif Jakob, title: Characterization of smooth Danielewski surfaces by their auto-

morphism groups. Since Fall 2022 L. Jacob is a PhD student under the supervision of Hendrik Süß.

Publications

1. Automorphisms of the Lie algebra of vector fields on affine n-Space (with H. Kraft), J. Eur. Math. Soc, Vol. 19, Issue 5, 2017,

2. Is the affine space determined by its automorphism group? (with H. Kraft and I. van Santen), Int. Math. Res. Not., Vol. 2021,

Issue 6, pp. 4280–4300.

3. Characterizing quasi-affine spherical varieties via the automorphism group (with I. van Santen), Journal de l’École poly-

technique — Mathématiques, Vol. 8 (2021) pp. 379-414.

4. Bracket width of simple Lie algebras (with A. Dubouloz and B. Kunyavskii), Doc. Math. 26, 1601-1627 (2021).

(B. Kunyavskii gave a recorded talk on this research work at the conference "Affine Algebraic Groups, Motives and Co-

homological Invariants" in Banff.)

5. On the characterization of Danielewski surfaces by their automorphism group (with A. Liendo and C. Urech), Transform.

Groups, Vol. 25, No. 4, 2022.

6. Vector Fields and Automorphism Groups of Danielewski Surfaces (with M. Leuenberger), Int. Math. Res. Not., Vol. 2022,

7. Characterisation of n-dimensional normal affine SL_n-varieties, Transform. Groups, Vol. 27, pp. 271–293 (2022).

8. When is the automorphism group of an affine variety nested? (with A. Perepechko), Transf. Groups, Vol. 28, pp. 401-412 (2023).

(Sasha gave a recorded talk on an this work at the conference Algebraic Geometry - Mariusz Koras in memoriam").

9. Characterisation of affine surfaces by their automorphism groups (with A. Liendo and C. Urech) arXiv:1805.03991

Ann. Sc. Norm. Super. di Pisa, 2023: VOL. XXIV, ISSUE 1, 249-289, doi:10.2422/2036-2145.201905_009

(C. Urech gave a recorded talk on an this work at the conference "Algebraic Geometry - Mariusz Koras in memoriam").

10. Families of commuting automorphisms, and a characterization of the affine space (with S. Cantat and J. Xie)

American Journal of Mathematics 145, no. 2 (2023): 413-434. muse.jhu.edu/article/885815.

11. Bracket width of the Lie algebra of vector fields on a smooth affine curve (with I. Makedonskyi),

Journal of Lie Theory 33 (2023), No. 3, 919--923.

12. Lie subalgebras of vector fields on affine 2-space and the Jacobian conjecture, to appear in Ann. Inst. Fourier.

13. Small G-varieties (with H. Kraft and S. Zimmermann), Canad. J. of Math., doi:10.4153/S0008414X22000682

(H. Kraft gave a recorded talk on this work at the "Quadratic forms, Linear algebraic groups and Beyond" Seminar).

14. Automorphism groups of affine varieties without non-algebraic elements (with A. Perepechko) arXiv:2203.08950

to appear in Proc. Amer. Math. Soc, DOI: https://doi.org/10.1090/proc/16759

Preprints

15. Maximal commutative unipotent subgroups and a characterization of affine spherical varieties (with I. van Santen)

(I gave a recorded talk on this research work at the "Seminar on Algebraic Transformation Groups".)

16. Characterization of affine G_m-surfaces of hyperbolic type, arXiv:2202.10761 (submitted).

17. When is the automorphism group of an affine variety linear? (submitted).

18. On the characterization of affine toric varieties by their automorphism group (with R. Díaz and A. Liendo)

arXiv:2308.08040 (submitted).

19. Bracket width of current Lie algebras (with B. Kunyavskii and I. Makedonskyi) arXiv:2404.06045 (submitted)

20. On the annihilators of rational functions in the Lie algebra of derivations of K[x,y] (with O. Iena and A. Petravchuk)

arXiv:0910.4465 (this text is not and will not be submitted to any peer review journal).

##### Poster

Automorphism groups without non-algebraic elements, link to the poster (with Perepechko).

Other texts

Questions/Tasks

1. Does Tits alternative hold for the automorphism/birational transformation groups of affine varieties?

2. Is it true that the projective space is determined by its group of birational transformations (up to birational

equivalence) in the category of all irreducible projective varieties?

3. Describe/characterize all maximal subgroups of the Cremona group that consists of algebraic elements

(for the automorphism groups of affine (maybe even all algebraic) varieties this problems seems to be

doable, but there is no tool at the moment for the groups of birational transformations).

4. Is there an example of an affine variety that has a non-discrete group of automorphisms which does not

contain an algebraic subgroup of positive dimension?

5. Is SAut( A^n) simple? Is Aut(A^n) generated by connected algebraic subgroups?

6. Is there an example of a non-simple ind-group of automorphsims that has a simple Lie algebra?

Useful Links

Seminars

Seminar on Algebraic Transformation Groups

Conferences in Algebraic Geometry

International Centre for Mathematics in Ukraine

Problems in affine algebraic geometry (by H.Kraft)

A first glimpse at the minimal model program (by Cadman et al.)