17/07/2012, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Marco Robalo, Université Montpellier 2
Noncommutative Motives: A universal characterization of the motivic
stable homotopy theory of schemes.
In this talk I will explain a new approach to the theory of
noncommutative motives based in the construction of a motivic
stable homotopy theory for the noncommutative schemes of
Kontsevich.
In the first part of the talk, we review the original motivic
homotopy theory of schemes as constructed by
Voevodsky-Morel-Jardine and explain its recent universal
characterization. The fundamental step in this characterization is
to understand the nature of the construction of symmetric spectrum
objects in a model category M. I will try to sketch the idea in
detail.
In the second part we explain the construction of a new motivic
theory for the noncommutative schemes which mimics the classical
one for schemes. Every scheme gives rise to a noncommutative one
and because of the universal property described in the first part,
this assignment gives birth to a canonical monoidal comparison map
between the commutative and the new noncommutative motivic
theories. This work is ongoing and it is part of my PhD thesis
under the direction of B. Toën in the Université de
Montpellier.
The talk will require (at all times) the language of
-categories, for which I will provide a small introduction
in the beginning.
References
- M. Kontsevich,
Noncommutative motives
- J. Lurie,
Higher Algebra
- M. Robalo,
Non-commutative motives I: A Universal Characterization of the
Motivic Stable Homotopy Theory of Schemes
- G. Tabuada, A guided tour
through the Garden of Noncommutative Motives
