research interests
I am interested in things of algebraic, geometric, arithmetic, or homotopical nature, including motives, motivic homotopy theory, periods, Dmodules, tensortriangular geometry, higher categories, derivators, nonarchimedean analytic geometry, modular representation theory.
papers and preprints
Permutation modules, Mackey functors, and Artin motives
with Paul Balmer Proceedings of ICRA2020 (accepted)
 arXiv:2107.11797 (pdf, source)
 Description: This is a companion to the papers 0809. It discusses the `big' derived category of permutation modules, and describes the beautiful connections with cohomological Mackey functors and Artin motives. The note is more expository than those papers.
 permutation modules Mackey functors motives
Supports for constructible systems
 Preprint, submitted (29 pages)
 arXiv:2107.03731 (pdf, source). Last updated version (19 October 2021)
 Description: Assigning to a constructible sheaf (or holonomic Dmodule, mixed Hodge module, ...) its support is shown to have a convenient universal property which leads to their classification up to the tensor triangulated structure at the level of derived categories. A refinement of this result allows the systematic reconstruction of the Zariski topological space underlying an algebraic variety from these derived categories. Together with arXiv:2003.04847 one reconstructs even a large class of schemes.
 ttgeometry motives
The sixfunctor formalism for rigid analytic motives
with Joseph Ayoub and Alberto Vezzani Preprint, submitted (182 pages)
 arXiv:2010.15004 (pdf, source)
 Description: We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their sixfunctor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud's approach to rigid analytic geometry.
 nonarchimedean analytic geometry motives motivic homotopy theory
The universal sixfunctor formalism
with Brad Drew Preprint, submitted (48 pages)
 arXiv:2009.13610 (pdf, source). Last updated version (19 July 2021)
 Description: We prove that MorelVoevodsky’s stable 𝔸^{1}homotopy theory affords the universal sixfunctor formalism.
 motivic homotopy theory abstract homotopy theory
Permutation modules and cohomological singularity
with Paul Balmer Preprint, submitted (14 pages)
 arXiv:2009.14093 (pdf, source)
 Description:
This is the sequel to paper 08.
The question we investigate in both of them is How and to what extent are general representations controlled by permutation ones?
In the first paper we settled the How?, and in this paper we do the same for the To what extent?. For this we construct an invariant, using cohomology and singularity categories, that detects which representations are controlled by permutation modules.  modular representation theory permutation modules
Finite permutation resolutions
with Paul Balmer Preprint, submitted (20 pages)
 arXiv:2009.14091 (pdf, source). Last updated version (4 March 2021)
 Description: Modular representation theory is wellknown to be `wild' for most groups, whereas permutation representations with their finitely many isomorphism types of indecomposables seem relatively `tame'. In this paper and its sequel 09 we investigate how and to what extent the former is controlled by the latter. For example we prove that, contrary to what one might expect, every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by ppermutation modules.
 modular representation theory permutation modules
A note on Tannakian categories and mixed motives
 Bull. Lond. Math. Soc. 53 (2021) 119129. 10.1112/blms.12405
 arXiv:1912.12483 (pdf, source)
 Description: Assuming "all" motivic conjectures, the triangulated category of mixed motives over a field F is the derived category of a Tannakian category. I explain why one should therefore expect this category to be simple in the tensortriangular sense. In other words, why every nonzero motive generates the whole category up to the tensortriangulated structure. Under the same assumptions, I also completely classify triangulated étale motives over F with integral coefficients, up to the tensortriangulated structure, in terms of the characteristic and the orderings of F.
 motives ttgeometry
Three real ArtinTate motives
with Paul Balmer Preprint, submitted (54 pages)
 arXiv:1906.02941 (pdf, source)
 Description: We classify mixed ArtinTate motives over real closed fields up to the tensortriangular structure. Compared to paper 05, the additional difficulty lies at the prime 2 where we are required to solve some problems in "filtered modular representation theory".
 motives ttgeometry modular representation theory
ttgeometry of Tate motives over algebraically closed fields
 Compositio Math. 155 (2019) 18881923. 10.1112/S0010437X19007528
 arXiv:1708.00834 (pdf, source)
 Description: I classify mixed Tate motives over algebraically closed fields up to the tensortriangular structure.
 Comments: The description of the spectrum of étale motives with finite coefficients was completed in paper 07.
 motives ttgeometry
Tensor triangular geometry of filtered modules
 Algebra Number Theory 12 (2018), no. 8, pp. 1975–2003. 10.2140/ant.2018.12.1975
 arXiv:1708.00833 (pdf, source)
 Description: A classical result of Hopkins, Neeman, and Thomason classifies the thick subcategories of the category of perfect complexes over a (commutative) ring. Here I prove an analogous result for perfect filtered complexes, taking into account the tensor structure.
 Comments: This result was used in paper 05.
 ttgeometry
Homotopy theory of dg sheaves
with Utsav Choudhury Comm. Algebra 47 (2019), no 8, pp. 320228. 10.1080/00927872.2018.1554744
 arXiv:1511.02828 (pdf, source)
 Description: This is a careful study of the homotopy theory of sheaves of complexes on a site, in the language of model categories.
 Comments: This corresponds to the third chapter of my PhD thesis. Several of the results here were used in paper 02.
After completing this note we learned that our description of the fibrant objects also appeared in doi:10.1016/j.aim.2004.07.007. In the meantime, this has been generalized to nondg contexts in arXiv:1801.10129.  abstract homotopy theory
An isomorphism of motivic Galois groups
with Utsav Choudhury Adv. Math. 313 (2017), pp. 470536. 10.1016/j.aim.2017.04.006
 arXiv:1410.6104 (pdf, source). Last updated: 18 May 2017
 Description: In characteristic 0 there are two approaches to the conjectural theory of mixed motives: Nori motives and Voevodsky motives. Here we prove that their associated motivic Galois groups are canonically isomorphic, thereby providing some evidence that the two approaches are essentially equivalent.
 Comments: This corresponds to the fourth chapter of my PhD thesis.
 motives periods
Traces in monoidal derivators, and homotopy colimits
 Adv. Math. 261 (2014), pp. 2684. 10.1016/j.aim.2014.03.029
 arXiv:1303.0153 (pdf, source). Last updated: 16 July 2014
 Description: I define and study traces and Euler characteristics in abstract homotopy theory (using the language of derivators). As an application I prove a formula for the trace of the homotopy colimit of endomorphisms over finite categories in which all endomorphisms are invertible. This generalizes the additivity of traces in triangulated categories proved by May.
 Comments: This corresponds to the second chapter of my PhD thesis. In the meantime the same result has been obtained independently here.
 abstract homotopy theory
other documents
An introduction to sixfunctor formalisms
 Lecture notes (work in progress). Last updated: 15 November 2021
 Description: These are notes for a minicourse given at the summer school and conference SixFunctor Formalism and Motivic Homotopy Theory in Milan 9/2021. They provide an introduction to the formalism of Grothendieck’s six operations and end with an excursion to rigidanalytic motives. The notes do not correspond precisely to the lectures delivered but provide a more selfcontained accompaniment for the benefit of the audience. No originality is claimed.
The spectrum of Artin motives over finite fields
with Paul Balmer Announcement (2 pages). Last updated: 30 September 2020
 Description: In this short announcement we describe the spectrum of Artin motives over a finite field, and thereby classify them up to the tensor triangulated structure. Proofs will appear as part of forthcoming work on the tensortriangular geometry of ArtinTate motives.
Statistical mechanics of kinks on a gliding screw dislocation
with M. Boleininger, S. L. Dudarev, T. D. Swinburne, D. R. Mason, D. Perez Phys. Rev. Research 2 (2020) 10.1103/PhysRevResearch.2.043254
 arXiv:2005.13336 (pdf, source)
Traces, homotopy theory, and motivic Galois groups
 PhD thesis (146 pages), 2015.
 Description: This consists of essentially papers 0103, bundled together and prefaced with an introduction.
The LefschetzVerdier trace formula and a generalization of a theorem of Fujiwara
after Y. Varshavsky Master's thesis (101 pages), 2011.
 pdf, source
 Description: This is a study of trace maps in algebraic geometry, including their additivity, commutation with many natural operations, and their computation in good local situations. As an application one obtains a proof of Deligne's conjecture regarding the LefschetzVerdier trace formula in positive characteristic.
 Comments: In comparison to the original article by Varshavsky, this document is mainly more detailed.
Beweise und mathematisches Wissen (Proofs and mathematical knowledge)
 Philosophy master's thesis (German, 92 pages), 2010.
 Description: The first part consists of a critique of some conceptions of proofs rather popular in the philosophy of mathematics. Common to these conceptions is that they reduce the role of proofs to justifying theorems. This leads to the second part, a discussion of how proofs convey implicit knowledge: often called "methods", "techniques", "ideas" etc. Finally, some examples are presented in which making such implicit knowledge explicit led to tangible mathematical progress.
teaching
In Trinity Term 2021, I am coorganizing a learning seminar on Étale Cohomology, see here for details.
 @ oxford (tutor at Keble college):

 HT 21:
 Topology
 Rings and Modules
 MT 20:
 Geometry
 @ oxford (as class tutor):

 HT 20:
 C3.7: Elliptic Curves
 MT 19:
 C2.7: Introduction to Category Theory
 MT 18:
 B2.1: Introduction to Representation Theory
 @ ucla (as principal instructor):

 Spring 18:
 MATH 116
 Winter 18:
 MATH 32A
 Fall 17:
 MATH 61
 Spring 17:
 MATH 110B
 Fall 16:
 MATH 32A
 Spring 16:
 MATH 33A
 Fall 15:
 MATH 131A
 @ uzh (as teaching assistant):

 SS15:
 Algebraic Geometry 2
 FS14:
 Algebraic Geometry 1
 SS14:
 "Sabbatical"
 FS13:
 Algebra
 SS13:
 Analysis 2
 FS12:
 Analysis (for future teachers)
 SS12:
 Grundlagen der Mathematik (for future teachers)
 FS11:
 Analysis (for future teachers)