Dr Martin Gallauer
Martin GallauerAssistant Professor  D0.03  martin.gallauer@warwick.ac.uk

papers and preprints
all  none  abstract homotopy theory Mackey functors modular representation theory motives motivic homotopy theory nonarchimedean analytic geometry periods permutation modules six operations ttgeometry
Exponentiation of coefficient systems and exponential motives
with Simon Pepin Lehalleur arxiv
 Description: We construct new sixfunctor formalisms capturing cohomological invariants of varieties with potentials. Starting from any sixfunctor formalism C, encoded as a coefficient system, we associate a new sixfunctor formalism C_{exp}. This requires in particular constructing the convolution product symmetric monoidal structure at the ∞categorical level. We study C_{exp} and how it relates to C. We also define motives in C_{exp} attached to varieties with potential and study their properties.
 Comments: This is a preliminary version, to be updated soon.
 motives motivic homotopy theory six operations
The ttgeometry of permutation modules. Part I: Stratification
with Paul Balmer arxiv
 Description: This is the first in a series of three papers in which we study the derived category of permutation modules in modular representation theory (equivalently, the triangulated category of Artin motives in algebraic geometry). In this first part we focus on finite groups. (In part 3 we will also discuss profinite groups.) In this setting we completely classify the localizing ideals. And as part of the proof we determine the ttspectrum of the compact objects as a set. The topology (and hence the classification of thick ideals) is the subject of part 2.
 modular representation theory permutation modules ttgeometry
An introduction to sixfunctor formalisms
 arxiv, most recent (28 December 2022)
 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.
 motivic homotopy theory motives six operations
Permutation modules, Mackey functors, and Artin motives
with Paul Balmer Proceedings of ICRA2020 (accepted)
 arxiv
 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
 Doc. Math. (accepted)
 arxiv
 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 the reconstruction theorems from this paper one reconstructs even a large class of schemes.
 ttgeometry motives
The sixfunctor formalism for rigid analytic motives
with Joseph Ayoub and Alberto Vezzani Forum Sigma (2022)
 journal, arxiv
 Description: The original goal of this project was to establish a full sixfunctor formalism for rigidanalytic motives over general rigidanalytic spaces. We achieve this as an application of a powerful technique that we also develop in this (long) paper. It allows reducing certain questions about rigidanalytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well.
 Comments: An informal account of the main construction in the last chapter of this paper is given in paper 14.
 nonarchimedean analytic geometry motives motivic homotopy theory six operations
The universal sixfunctor formalism
with Brad Drew Ann. Ktheory (accepted)
 arxiv
 Description: We prove that MorelVoevodsky’s stable 𝔸^{1}homotopy theory affords the universal sixfunctor formalism.
 motivic homotopy theory abstract homotopy theory six operations
Permutation modules and cohomological singularity
with Paul Balmer Comment. Math. Helv. (2022)
 journal, arxiv
 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 Duke Math. J. (accepted)
 arxiv, most recent (30 January 2022)
 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. (2021)
 journal, arxiv
 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 Adv. Math. (2022)
 journal, arxiv
 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
Tensor triangular geometry of filtered modules
 Algebra Number Theory (2018)
 journal, arxiv
 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. In the meantime it has been generalized here.
 ttgeometry
Homotopy theory of dg sheaves
with Utsav Choudhury Comm. Algebra (2019)
 journal, arxiv
 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 had appeared in the literature before. In the meantime, this has been generalized to nondg contexts.  abstract homotopy theory
An isomorphism of motivic Galois groups
with Utsav Choudhury Adv. Math. (2017)
 journal, arxiv
 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. (2014)
 journal, arxiv
 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

The spectrum of Artin motives over finite fields
with Paul Balmer announcement
 most recent (5 May 2022)
 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.

Traces, homotopy theory, and motivic Galois groups
 PhD thesis (2015)
 thesis
 Description: This consists essentially of 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 (2011)
 thesis
 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, 2010)
 thesis
 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
 22/23: Algebraic Topology MA3H6