7 feb 2019 - 15 feb 2019
Università di Verona
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Dear friends of mathematics,
here in Verona we are pleased to announce the following minicourse by Peter Arndt:
Introduction to abstract and motivic homotopy theory
by Peter Arndt (Heinrich-Heine-Universität Düsseldorf)
hosted at the Università di Verona
from Feb 7th to Feb 15th
comprising approx. 12 lectures
(full course description below)
We hereby invite all interested PhD and master's students to take part in the minicourse and get to know the wondrous world of abstract and motivic homotopy theory; please pass this invitation on to your students as you see fit. For most of the course the only prerequisites are some basic category theory and topology. In particular no knowledge of algebraic geometry is supposed until the final parts.
To those of you who have not yet had the pleasure to experience Peter Arndt in person, let me say that Peter is an excellent speaker who understands quite well how to give accessible talks. The setting of this minicourse will be informal, allowing lots of opportunities for discussion.
There is no participation fee; alas, while we gladly help with finding affordable housing, we can not reimburse any costs. Some further details will appear on http://profs.sci.univr.it/~angeleri/minicourses%202018.html (sorry for http) in due course; any organizational questions should be addressed to me, while any questions regarding the course contents can be addressed directly to Peter <Peter.Arndt(AT)uni-duesseldorf.de>. To those who require this, we can do our part so students can earn ECTS credit points for the course.
Verona is a beautiful city in northeast Italy with a rich artistic heritage. Two of Shakespeare's plays are set in Verona; in particular, you can visit the balcony where Romeo promised Juliet eternal love.
Cheers,
Ingo
PS: So as to provide more value to those participants with a long journey to Verona, in the same time frame there will be an introductory tutorial to Agda. Agda is one of the several major interactive theorem provers we can use to write machine-checked proofs. While Agda, and indeed any of the currently existing proof assistants, still needs a long way to go till it can interpret hasty mathematical notes handwritten in informal English, it is useful to working mathematicians even in its current form. The Agda tutorial will both explain how to do classical mathematics with it and how to use it to learn and do homotopy type theory (HoTT).
Full course description:
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. Apart from the ground-breaking proof of the Bloch–Kato conjecture the approach has led to a multitude of results, for example on quadratic forms, vector bundles over schemes, algebraic cycles, algebraic K-theory and computations of stable homotopy groups of spheres in topology.
The minicourse will start with an introduction to abstract homotopy theory: We will see how inverting a class of arrows in a category leads to an infinity category, and how one can work with this using several formalisms. The special case of topological spaces with the class of weak equivalences serves both as a guiding example and as a technical foundation; thus we will simultaneously review some classical homotopy theory of topological spaces.
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-archimedean 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, Adams operations, rational splittings and a rational motivic Eilenberg-MacLane spectrum. We will also present some more special results for schemes and analytic spaces.
For most of the course the only prerequisites are some basic category theory and topology. In particular no knowledge of algebraic geometry is supposed until the final parts.