Obvious natural morphisms of sheaves are unique: Theory and Applications of Categories, Vol. 29, 2014, No. 4, pp 48-99.
Abstract: We prove that a large class of natural transformations (consisting roughly of those constructed via composition from the “functorial” or “base change” transformations) between two functors of the form ···f∗g∗··· actually has only one element, and thus that any diagram of such maps necessarily commutes. We identify the precise axioms defining what we call a “geofibered category” that ensure that such a coherence theorem exists. Our results apply to all the usual sheaf-theoretic contexts of algebraic geometry. The analogous result that would include f! and g!-type functors remains unknown. (arXiv) (JMM 2012 talk)
Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian: Represent. Theory 16 (2012), 345-449.
Abstract: In this work we obtain a result simultaneously generalizing the results of Mirkovíc–Vilonen and Gaitsgory on the Satake equivalence. We consider the general notion of twisting by a gerbe and define the natural class of ``factorizable'' gerbes by which one can twist in the context of the Satake equivalence. These gerbes are almost entirely described by the quadratic forms on the weight lattice of a reductive group. We show that a suitable formalism exists such that the methods of Mirkovíc–Vilonen can be applied directly in this general context virtually without change and obtain a Satake equivalence for twisted perverse sheaves. In addition, we present new proofs of the properties of their structure as an abelian tensor category. This is the updated version of my thesis. (arXiv)
Notes on Beilinson's "How to glue perverse sheaves": Journal of Singularities, volume 1 (2010) pp. 94–115.
Abstract: The titular, foundational work of Beilinson not only gives a technique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These constructions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also defines a new, "maximal extension" functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems. (arXiv, errata)
My two talks at Dennis Gaitsgory's seminar in 2009-2010:
- The Hecke category (part I — Factorizable structure)
- The Hecke category (part II — Satake equivalence)
I gave a short talk at the 2012 Joint Meetings in Boston: Coherence of canonically-defined natural transformations in the derived category of l-adic sheaves. You can also find the TeX source and supporting package on my LaTeX page.
I spoke at the UC Davis algebra seminar on February 16th, 2012: The dual group for a twisted Satake equivalence, and quadratic forms from gerbes.