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. 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

News: Together with Paul Balmer, John Greenlees, Henning Krause, and Julia Pevtsova, we are organizing a HIM trimester program in Bonn on Spectral Methods in Algebra, Geometry, and Topology. More details to follow soon on the webpage of the program.

List of publications.

  1. 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

  2. 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

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

  4. 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

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

  6. 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

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

  8. 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

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

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

  11. 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

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

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

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

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

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

  17. 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

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

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

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

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

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

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

  24. 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

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

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

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

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

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

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

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

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

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

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

List of preprints.

  1. Stratification and the comparison between homological and tensor triangular support, joint with Drew Heard and Beren Sanders (2021)

  2. Stratification in tensor triangular geometry with applications to spectral Mackey functors, joint with Drew Heard and Beren Sanders (2021)

  3. On conjectures of Hovey--Strickland and Chai, joint with Drew Heard and Niko Naumann (2020)

Here is a video of me speaking about ultrachromatic homotopy theory at Ravenel's 70th birthday conference, Reed College, August 2017

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 2022, with a second warm-up event on 9 August 2021. 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.

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, Scott Balchin, Dave Barnes, Agnès Beaudry, 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, Bernhard Keller, Achim Krause, Henning Krause, Peter May, Lennart Meier, Niko Naumann, Thomas Nikolaus, Justin Noel, Eric Peterson, Viet-Cuong Pham, 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.