A note on Tannakian categories and mixed motives
 Preprint (13 pages)
 arXiv:1912.12483
(pdf, source)
 Description.
We explain why every nontrivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures.
In other words, every nonzero motive generates the whole category up to the tensortriangulated structure.
Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensortriangulated structure, in terms of the characteristic and the orderings of F.
Three real ArtinTate motives
with
Paul Balmer
 Preprint (54 pages)
 arXiv:1906.02941
(pdf, source)
 Description.
We classify mixed ArtinTate motives over real closed fields up to the tensortriangular structure. Compared to the paper below on Tate motives, the additional difficulty lies at the prime 2 where we are required to solve some problems in "filtered modular representation theory".
ttgeometry of Tate motives over algebraically closed fields
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 the paper above on Tate motives over algebraically closed fields.
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 the paper below on motivic Galois groups.
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.
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.
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.
Traces, homotopy theory, and motivic Galois groups
 PhD thesis (146 pages), 2015.
 pdf
 Description. This consists of essentially the last three papers listed above, 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.
 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.
 @ oxford (as class tutor):

 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)