Tobias Barthel

News: I have been awarded an ERC Starting Grant.

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. 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.

List of preprints.

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

2. Quillen stratification in equivariant homotopy theory, joint with Drew Heard, Natalia Castellana, Niko Naumann, and Luca Pol (2023)

3. The chromatic Fourier transform, joint with Shachar Carmeli, Tomer Schlank, and Lior Yanovski (2022)

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

5. Morava K-theory and Filtrations by Powers, joint with Piotr Pstragowski (2021)

6. Stratifying integral representations of finite groups (2021)

List of publications. 

1. Stratification in tensor triangular geometry with applications to spectral Mackey functors, joint with Drew Heard and Beren Sanders, accepted for publication in  the Cambridge Journal of Mathematics (2022), arXiv link

2. Stratification and the comparison between homological and tensor triangular support, joint with Drew Heard and Beren Sanders, accepted for publication in Q. J. Math (2022), arXiv link

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

4. Constructing the determinant sphere using a Tate twist, joint with Agnès Beaudry, Paul G. Goerss, and Vesna Stojanoska, accepted for publication in Math Z. (2021), arXiv link

5. On stratification for spaces with Noetherian mod p cohomology, joint with Natalia Castellana, Drew Heard, and Gabriel Valenzuela, accepted for publication in Amer. J. Math. (2021), arXiv link

6. Power operations in the Stolz--Teichner program, joint with Daniel Berwick-Evans and Nathaniel Stapleton, accepted for publication in  Geometry & Topology (2021), arXiv link

7. Chromatic fracture cubes, joint with Omar Antolin-Camarena, accepted for publication in the Proceedings of the 2019 Equivariant Topology and Derived Algebra Conference (2021), arXiv link

8. Monochromatic homotopy theory is asymptotically algebraic, joint with Tomer Schlank and Nathaniel Stapleton, accepted for publication in Advances in Mathematics (2020), arXiv link

9. Local Gorenstein duality for cochains on spaces, joint with Natalia Castellana, Drew Heard, and Gabriel Valenzuela, accepted for publication in J. Pure Appl. Algebra (2020), arXiv link

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

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

12. Chromatic homotopy theory is asymptotically algebraic, joint with Tomer Schlank and Nathaniel Stapleton, accepted in Invent. Math.  (2019), arXiv link

13. A Whitehead theorem for periodic homotopy groups, joint with Gijs Heuts and Lennart Meier, accepted for publication in Israel. J. Math. (2019), arXiv link

14. A short introduction to the telescope and chromatic splitting conjectures, accepted for publication in Surveys around Ohkawa's theorem on Bousfield classes, Springer Proc. Math. Stat. (2019), arXiv link

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

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

17. Chromatic structures in stable homotopy theory, joint with Agnès Beaudry, The Handbook of Homotopy Theory (2019), arXiv link

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

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

20. 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

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

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

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

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

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

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

27. 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

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

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

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

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

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

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

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

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

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

37. 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

There will be 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 has 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. 

The next workshop will take place in 2021. The topic is Operads in deformation theory and homotopical algebraa, and the mentors are Geoffroy Horel and Bruno Valette.

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, Agnès Beaudry, Dave Benson, Dan Berwick-Evans, A.K. Bousfield, Shachar Carmeli, Natalia Castellana, Dominic Culver, Martin Frankland, David Gepner, Paul Goerss, Frank Gounelas, J.P.C. Greenlees, Jesper Grodal, Jeremy Hahn, Markus Hausmann, Drew Heard, Gijs Heuts, Joshua Hunt, Srikanth Iyengar, Bernhard Keller, Achim Krause, Henning Krause, Peter May, Lennart Meier, Niko Naumann, Thomas Nikolaus, Justin Noel, Eric Peterson, Julia Pevtsova, Viet-Cuong Pham, Luca Pol, Piotr Pstragowski, Emily Riehl, Beren Sanders, Tomer Schlank, Nathaniel Stapleton, Greg Stevenson, Vesna Stojanoska, Gabriel Valenzuela, Lior Yanovski. 

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.