Affiliation:
Universität Düsseldorf
Date:
Wed, 20/02/2019 - 16:30 - 18:00
Motivic homotopy theory is a fusion of homotopy theory and algebraic
geometry. In analogy to the homotopy category of topological spaces,
obtained by making the unit interval contractible, one obtains a
homotopy category of schemes by making the affine line contractible.
This category has a rather topological flavour; one can for example talk
about suspensions, loop spaces and classifying spaces and represent
cohomology theories by spectra.
We will start with a picture of the homotopy theory of topological
spaces, recalling facts about homotopy (co)limits, fiber bundles and
cohomology theories.
We will then introduce motivic homotopy theory, emphasizing the
parallels to classical homotopy theory. We will use an abstract setup,
due to the speaker, which also encompasses complex and non-archimedian
analytic geometry and derived algebraic geometry, as well as many new
geometric settings. Starting from a cartesian closed, presentable
infinity category and a commutative group object therein, we will see a
representation theorem for line bundles, a Snaith type algebraic
K-theory spectrum, rational splittings into Adams eigenspaces and a
rational motivic Eilenberg-MacLane spectrum.
The results can be seen as descent statements for the corresponding
motivic objects to the field with one element, and actually deeper.
For much of the talk the only prerequisites are some basic category
theory and topology, and willingness to take the language of infinity
categories on trust.
