Homotopy theory of dg sheaves
joint work with
Utsav
Choudhury
 Submitted (29 pages)
 arXiv:1511.02828
(pdf, source)
 Abstract. In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
 Comments. After completing this note we learned that this description of the fibrant objects also appeared in doi:10.1016/j.aim.2004.07.007 and in the Ph.D. thesis of Brad Drew.
An isomorphism of motivic Galois groups
joint work 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
 Abstract. In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
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
 Abstract. A variant of the trace in a monoidal
category is given in the setting of closed monoidal derivators,
which is applicable to endomorphisms of fiberwise dualizable
objects. Functoriality of this trace is established. As an
application, an explicit formula is deduced for the trace of the
homotopy colimit of endomorphisms over finite categories in which
all endomorphisms are invertible. This result can be seen as a
generalization of the additivity of traces in monoidal categories
with a compatible triangulation.
 Comments. In the meantime the same result has
been obtained independently
here.
Traces, homotopy theory, and motivic Galois groups
The LefschetzVerdier trace formula and a generalization of a
theorem of Fujiwara
after Y. Varshavsky
 @ucla:

 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:

 SS15:
 TA for "Algebraic
Geometry 2"
 FS14:
 TA for "Algebraic Geometry 1"
 SS14:
 I was on "sabbatical".
 FS13:
 TA for "Algebra"
 SS13:
 TA for "Analysis 2"
 FS12:
 TA for "Analysis" (for future teachers)
 SS12:
 TA for "Grundlagen der Mathematik" (for future teachers)
 FS11:
 TA for "Analysis" (for future teachers)