Tobias Barthel

I am a W2 Research Group Leader at the Max Planck Institute of Mathematics in Bonn, interested in homotopy theory, tensor-triangular geometry, and their various interactions with other areas of mathematics. My work is supported by an ERC Starting Grant. I finished my PhD thesis under the supervision of Mike Hopkins in 2014, then spent the following years as a postdoc in Bonn as well as at the University of Copenhagen,  partly supported by an EU Marie Curie Individual Fellowship. 

Email: barthel.tobi@gmail.com

Together with Paul Balmer, John Greenlees, Henning Krause, and Julia Pevtsova, we recently organized a HIM trimester program in Bonn on Spectral Methods in Algebra, Geometry, and Topology. More details are available on the webpage of the program. In 2026, there will be a follow-up workshop.

List of preprints.

1. Homological stratification and descent, joint with Drew Heard, Beren Sanders, and Changhan Zou (2024)

2. On Hopkins' Picard group, joint with Tomer Schlank, Nat Stapleton, and Jared Weinstein (2024)

3. The spectrum of excisive functors, joint with Gregory Arone, Drew Heard, and Beren Sanders (2024)

4. On the rationalization of the K(n)-local sphere, joint with Tomer Schlank, Nat Stapleton, and Jared Weinstein (2024)

5. Profinite equivariant spectra and their tensor-triangular geometry, joint with Scott Balchin and Dave Barnes (2024)

6. Prismatic decompositions and rational G-spectra, joint with Scott Balchin and J.P.C. Greenlees (2023)

7. Lattices over finite group schemes and stratification, joint with Dave Benson, Srikanth Iyengar, Henning Krause, and Julia Pevtsova (2023)

8. Cosupport in tensor triangular geometry, joint with Natalia Castellana, Drew Heard, and Beren Sanders (2023)

9. Stratifying integral representations via equivariant homotopy theory (2022)

10. Stratifying integral representations of finite groups (2021)

List of publications. 

1. Quillen stratification in equivariant homotopy theory, joint with Natalia Castellana, Drew Heard, Niko Naumann, and Luca Pol, accepted for publication in Inventiones Mathematicae (2024), arXiv link

2. On surjectivity in tensor triangular geometry, joint with Natalia Castellana, Drew Heard, and Beren Sanders, Math. Z. (2024), arXiv link

3. Descent in tensor triangular geometry, joint with Natalia Castellana, Drew Heard, Niko Naumann, Luca Pol, and Beren Sanders, Proceedings of the Abel Symposium 2022 (2024), arXiv link

4. The chromatic Fourier transform, joint with Shachar Carmeli, Tomer Schlank, and Lior Yanovski, Forum of Mathematics, Pi (2024), arXiv link

5. Morava K-theory and Filtrations by Powers, joint with Piotr Pstragowski, Journal of the Institute of the Mathematics of Jussieu (2023), arXiv link

6. Stratification in tensor triangular geometry with applications to spectral Mackey functors, joint with Drew Heard and Beren Sanders, Cambridge Journal of Mathematics (2023), arXiv link

7. Stratification and the comparison between homological and tensor triangular support, joint with Drew Heard and Beren Sanders, Q. J. Math (2023), arXiv link

8. Power operations in the Stolz--Teichner program, joint with Daniel Berwick-Evans and Nathaniel Stapleton, Geometry & Topology (2022), arXiv link

9. On stratification for spaces with Noetherian mod p cohomology, joint with Natalia Castellana, Drew Heard, and Gabriel Valenzuela, Amer. J. Math. (2022), arXiv link

10. On conjectures of Hovey--Strickland and Chai, joint with Drew Heard and Niko Naumann, Selecta Math (N.S.) (2022), arXiv link

11. Constructing the determinant sphere using a Tate twist, joint with Agnès Beaudry, Paul G. Goerss, and Vesna Stojanoska, Math. Z. (2022), arXiv link

12. Chromatic fracture cubes, joint with Omar Antolin-Camarena, Proceedings of the 2019 Equivariant Topology and Derived Algebra Conference (2022), arXiv link

13. Monochromatic homotopy theory is asymptotically algebraic, joint with Tomer Schlank and Nathaniel Stapleton, Advances in Mathematics (2021), arXiv link

14. A Whitehead theorem for periodic homotopy groups, joint with Gijs Heuts and Lennart Meier, Israel. J. Math. (2021), arXiv link

15. Local Gorenstein duality for cochains on spaces, joint with Natalia Castellana, Drew Heard, and Gabriel Valenzuela, J. Pure Appl. Algebra (2021), arXiv link

16. Appendices A and B (joint with Krause) to Completing perfect complexes by Henning Krause,  with another appendix by Bernhard Keller, Math. Z. (2020), arXiv link

17. Chromatic structures in stable homotopy theory, joint with Agnès Beaudry, The Handbook of Homotopy Theory (2020), arXiv link

18. Transfer ideals and torsion in the Morava E-theory of abelian groups, joint with Nathaniel Stapleton,  J. Homotopy Relat Str. (2020), arXiv link

19. A short introduction to the telescope and chromatic splitting conjectures, Surveys around Ohkawa's theorem on Bousfield classes, Springer Proc. Math. Stat. (2020), arXiv link

20. Chromatic homotopy theory is asymptotically algebraic, joint with Tomer Schlank and Nathaniel Stapleton, Invent. Math.  (2020), arXiv link

21. Derived completion for comodules, joint with Drew Heard and Gabriel Valenzuela,  Manuscripta Math (2020), arXiv link

22. On the Balmer spectrum for compact Lie groups, joint with J.P.C. Greenlees and Markus Hausmann, Compos. Math. (2020), arXiv link

23. Stratifications and duality for homotopical groups, joint with Natalia Castellana, Drew Heard, and Gabriel Valenzuela, Advances in Mathematics (2019), arXiv link

24. Gross--Hopkins duals of higher real K-theory spectra, joint with Agnès Beaudry and Vesna Stojanoska, Trans.  Amer. Math. Soc. (2019), arXiv link

25. The Balmer spectrum of the equivariant homotopy category of a finite abelian group, joint with Markus Hausmann, Niko Naumann, Thomas Nikolaus, Justin Noel, and Nathaniel Stapleton, Invent. Math. (2019), arXiv link

26. On the comparison of stable and unstable p-completion, joint with A. K. Bousfield, Proc. Amer. Math. Soc. (2019), arXiv link

27. A simple universal property of Thom ring spectra, joint with Omar Antolin-Camarena, J. Topol. (2019), arXiv link

28. Algebraic chromatic homotopy theory for BP_*BP-comodules, joint with Drew Heard, Proc. Lond. Math. Soc. (2018), arXiv link

29. The algebraic chromatic splitting conjecture for Noetherian commutative ring spectra, joint with Drew Heard and Gabriel Valenzuela, Math. Z. (2018), arXiv link

30. Local duality in algebra and topology, joint with Drew Heard and Gabriel Valenzuela, Advances in Mathematics (2018), arXiv link

31. Excellent rings in transchromatic homotopy theory, joint with Nathaniel Stapleton, Homology Homotopy Appl. (2018), arXiv link

32. On localization sequences in the algebraic K-theory of ring spectra, joint with Ben Antieau and David Gepner, J. Eur. Math. Soc. (2018), arXiv link

33. Local duality for structured ring spectra, joint with Drew Heard and Gabriel Valenzuela, J. Pure Appl. Algebra (2018), arXiv link

34. Brown--Peterson cohomology from Morava E-theory, joint with Nathaniel Stapleton and with an appendix by Jeremy Hahn, Compos. Math. (2017), arXiv link

35. Auslander--Reiten sequences, Brown--Comenetz duality, and the K(n)-local generating hypothesis, Algebr. Represent. Theory (2017), arXiv link

36. The character of the total power operation, joint with Nathaniel Stapleton, Geom. Topol. (2017), arXiv link

37. Centralizers in good groups are good, joint with Nathaniel Stapleton, Algebr. Geom. Topol. (2016), arXiv link

38. The E_2-term of the K(n)-local E_n-based Adams spectral sequence, joint with Drew Heard, Topology Appl. (2016), arXiv link

39. Chromatic completion, Proc. Amer. Math. Soc. (2016), arXiv link

40. Completed power operations for Morava E-theory, joint with Martin Frankland, Algebr. Geom. Topol. (2015), arXiv link

41. Six model structures for DG-modules over DGAs, joint with Peter May and Emily Riehl, New York J. Math. (2014), arXiv link

42. On the constructions of functorial factorizations for model categories, joint with Emily Riehl, Algebr. Geom. Topol. (2013), arXiv link

Presentations

• Representation theory in intermediate characteristics, Representation Theory and Triangulated Categories, a conference in honour of Henning Krause, Paderborn (Sep 2022). The image on the list slide is: Moses Harris, Prismatic Colour Wheel (1785); From: The natural system of colours, Royal Academy of Arts, London.

The Transatlantic Transchromatic Homotopy Theory Workshop II

This was a five-day workshop on transchromatic homotopy theory in Regensburg, organized by Drew Heard, Niko Naumann, Nat Stapleton, and myself.  Due to the covid-19 pandemic, the main workshop had been postponed to August 2023. For more information, please see the workshop's homepage. 

The European Talbot Workshop

Together with Sean Tilson, we started the European Talbot Workshop in 2014. The current organizers are: Bertram Arnold,  Luciana Basualdo Bonatto, Jack Davies, and Alice Hedenlund.  

Masterclasses

In 2018, I was involved in the organization of two Masterclasses in Copenhagen, together with Jesper Grodal and  Markus Hausmann.

• Tensor triangular geometry and equivariant stable homotopy theory, mentored by Paul Balmer and Justin Noel, March 2018.

• Rigidity and algebraic models in stable homotopy theory, mentored by John Greenlees and Stefan Schwede, April 2018.

In March 2017, Jesper Grodal and I organized a Masterclass on Stratifications and duality in modular representation theory. 

My collaborators: Ben Antieau, Omar Antolin-Camarena, Ko Aoki, Greg Arone, Scott Balchin, Dave Barnes, Miguel Barrero, Agnès Beaudry, Dave Benson, Dan Berwick-Evans, A.K. Bousfield, Shachar Carmeli, Natalia Castellana, Anish Chedalavada, Martin Frankland, David Gepner, Paul Goerss, Frank Gounelas, J.P.C. Greenlees, Jesper Grodal, Jeremy Hahn, Peter Haine, Markus Hausmann, Drew Heard, Gijs Heuts, Kaif Hilman, Joshua Hunt, Srikanth Iyengar, Bernhard Keller, Nikolay Konovalov, Achim Krause, Henning Krause, Peter May, Lennart Meier, Niko Naumann, Thomas Nikolaus, Justin Noel, Eric Peterson, Julia Pevtsova, Luca Pol, Piotr Pstragowski, Emily Riehl, Beren Sanders, Tomer Schlank, Nathaniel Stapleton, Greg Stevenson, Vesna Stojanoska, Neil Strickland, Gabriel Valenzuela, Marco Volpe, Jared Weinstein, Jordan Williamson, Lior Yanovski, Changhan Zou. 

The above image (Projet de clavier ultrachromatique, 1943) is taken from the book of writings by Wyschnegradsky (Libération du son: écrits 1916-1979, ed. Pascale Criton. Lyon: Symétrie 2013), Ivan Wyschnegradsky Collection, Paul Sacher Stiftung, Basel.