Unstable motivic homotopy theory

The purpose of the school is to bring together people interested in learning about recent progress in unstable motivic homotopy theory, whether they are graduate students, young researchers or established scientists. The school will consist of four courses leading up to the proof of the motivic Freudenthal suspension theorem and some of its applications.

Organized by: Tom Bachmann (JGU Mainz) | Manuel Blickle (JGU Mainz) | Klaus Mattis (JGU Mainz)

Courses

Aravind Asok (USC Dornsife): The motivic Freudenthal suspension theorem.

Abstract: In lecture one I will introduce the notion of weak cellular classes in various motivic categories (both stable and unstable) and study its basic properties. In lecture 2 I will discuss nilpotent motivic spaces and afterwards I will introduce the refined Whitehead and Postnikov towers. These are towers where the layers can be made to inherit cellularity properties possessed by the initial space. Then in lecture 3 I will discuss the Freudenthal property for motivic spaces (a weak cellular estimate on the fiber of the unit of the P^1-loop-suspension adjunction). Using the behavior of this property under fiber sequences, I will explain how to reduce proving the motivic Freudenthal suspension theorem to the case of zero-spaces of P^1-spectra. Finally in lecture 4 I will discuss some applications of the motivic Freudenthal suspension theorem, in particular the resolution of Murthy's conjecture on splitting of corank 1 projective modules on smooth affine varieties over an algebraically closed field having characteristic 0.  

Tom Bachmann (JGU Mainz): The Rost-Schmid complex of a strongly A¹-invariant sheaf (after F. Morel).

Abstract: The aim of my lectures is to prove an important foundational result in motivic homotopy theory over perfect fields, due to Fabien Morel: any strongly A^1-invariant sheaf of abelian groups is strictly A^1-invariant. In the first lecture I will first explain the terminology and outline some of the consequences of this result. Afterwards I will discuss some standard topics, including Milnor-Witt K-theory and Gabber's presentation lemma. The second lecture focuses on contractions of strongly A^1-invariant sheaves of abelian groups, their relations to local cohomology, and the construction of transfers on them. The third lecture explores these topics further and more systematically, by studying certain Gersten complexes. Finally in the fourth lecture I will define the Rost-Schmid complex and sketch the proof of the main theorem.

Mike Hopkins (Harvard): Homotopy theoretic aspects of the motivic Freudenthal theorem.

Abstract: The proof of the motivic Freudenthal suspension theorem proceeds by first establishing it for symmetric powers, using a geometric argument, from that case deducing it for the zero-spaces of P^1-spectra, and from there proceeding to the general case.  These lectures will discuss the passage from symmetric powers to the zero-spaces of P^1-spectra.  I will focus on a more homotopy theoretic approach than appears in the paper.

Marc Levine (Duisburg-Essen): The homotopy coniveau tower.

Abstract: The homotopy coniveau tower is based on Bloch’s construction of his cycle complex defining the higher Chow groups, and its generalization to the Bloch-Lichtenbaum construction of a spectral sequence starting from the higher Chow groups and converging to higher algebraic K-theory. In the first lecture, I will recall and generalize these constructions, and state how they relate to Voevodsky's slice tower. In the second lecture I will discuss important properties of the homotopy coniveau tower. In the third lecture I will justify the statements about the relationship with the slice filtration. In the final lecture I will use the homotopy coniveau tower to compute slices of some motivic spectra of interest.

Registration and financial support

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