Martin Gallauer


portrait

I am a Titchmarsh Research Fellow at the Mathematical Institute at the University of Oxford. Before, I was a Hedrick postdoc at UCLA and, prior to that, a graduate student at UZH. See my CV for more details.

email address:
ku.ca.xo.shtam@reuallag

office:
N02.25

postal address:
Mathematical Institute
University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
Woodstock Road
Oxford
OX2 6GG

research interests

I am interested in things of algebraic, geometric, arithmetic, or homotopical nature, including motives, motivic homotopy theory, periods, D-modules, tensor-triangular geometry, higher categories, derivators, non-archimedean analytic geometry, modular representation theory.

papers and preprints

The six-functor formalism for rigid analytic motives

with Joseph Ayoub and Alberto Vezzani
  • Preprint (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 six-functor 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.
  • non-archimedean analytic geometry motives motivic homotopy theory

The universal six-functor formalism

with Brad Drew
  • Preprint, submitted (48 pages)
  • arXiv:2009.13610 (pdf, source)
  • Description: We prove that Morel-Voevodsky’s stable 𝔸1-homotopy theory affords the universal six-functor 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

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 well-known 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 p-permutation modules.
  • modular representation theory

A note on Tannakian categories and mixed motives

  • Bull. Lond. Math. Soc. 53 (2021) 119-129. 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 tensor-triangular sense. In other words, why every non-zero motive generates the whole category up to the tensor-triangulated structure. Under the same assumptions, I also completely classify triangulated étale motives over F with integral coefficients, up to the tensor-triangulated structure, in terms of the characteristic and the orderings of F.
  • motives tt-geometry

Three real Artin-Tate motives

with Paul Balmer
  • Preprint, submitted (54 pages)
  • arXiv:1906.02941 (pdf, source)
  • Description: We classify mixed Artin-Tate motives over real closed fields up to the tensor-triangular 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 tt-geometry modular representation theory

tt-geometry of Tate motives over algebraically closed fields

  • Compositio Math. 155 (2019) 1888-1923. 10.1112/S0010437X19007528
  • arXiv:1708.00834 (pdf, source)
  • Description: I classify mixed Tate motives over algebraically closed fields up to the tensor-triangular structure.
  • Comments: The description of the spectrum of étale motives with finite coefficients was completed in paper 07.
  • motives tt-geometry

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.
  • tt-geometry

Homotopy theory of dg sheaves

with Utsav Choudhury
  • Comm. Algebra 47 (2019), no 8, pp. 3202-28. 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 non-dg contexts in arXiv:1801.10129.
  • abstract homotopy theory

An isomorphism of motivic Galois groups

with Utsav Choudhury
  • Adv. Math. 313 (2017), pp. 470-536. 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. 26-84. 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

Permutation modules, Mackey functors, and Artin motives

with Paul Balmer
  • Note (33 pages). Last updated: 9 February 2021
  • pdf
  • Description: This is a companion to the papers 08-09. 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.

The spectrum of Artin motives over finite fields

with Paul Balmer
  • Announcement (2 pages). Last updated: 30 September 2020
  • pdf
  • 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 tensor-triangular geometry of Artin-Tate motives.

Statistical mechanics of kinks on a gliding screw dislocation

with M. Boleininger, S. L. Dudarev, T. D. Swinburne, D. R. Mason, D. Perez

Traces, homotopy theory, and motivic Galois groups

  • PhD thesis (146 pages), 2015.
  • pdf
  • Description: This consists of essentially papers 01-03, bundled together and prefaced with an introduction.

The Lefschetz-Verdier 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 Lefschetz-Verdier 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.
  • pdf
  • 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 co-organizing 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
Winter 17:
MATH 116
MATH 214A
Fall 16:
MATH 32A
Spring 16:
MATH 33A
Winter 16:
MATH 131A
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)