# Motivic homotopies for fun and profit-Paul Arne Østvær

### Lecture 1. "Motivation"

Tuesday, July 8, 14 Uhr WSC-N-U-4.04

In this introductory talk we review the notions of homotopies in topology and algebra, and homotopy theory in categorical settings. Our main examples of interest are classical stable homotopy, equivariant stable homotopy, and motivic stable homotopy. We give examples of computations of homotopy groups in all three settings.

Lecture notes for Lecture 1

### Lecture 2. "Techniques"

Wednesday, July 9, 10 Uhr WSC-N-U-3.05

The three topics for this lecture are (1) The Hopf invariant one problem in topology solved by Adams, (2) Adams and Adams-Novikov spectral sequences in topology, and (3) Motivic Hopf maps.

Lecture notes for Lecture 2

### Lecture 3. "Examples of computations"

Friday, July 11 14 Uhr, WSC-S-U-3.03

We first give an outline of Voevodsky's proof of Milnor's conjecture in Galois cohomology with some details on motivic Eilenberg-MacLane spaces and the motivic Steenrod algebra. In the second part we discuss computations of motivic Adams and Adams-Novikov spectral sequences.

Lecture notes for Lecture 3

### Lecture 4. "Motivic slices"

Monday, July 14, 14 Uhr WSC-N-U-4.04

Just like for a pizza, there is a way of slicing the motivic stable homotopy category into smaller pieces which turn out to be motives. We explain this procedure and how one can hope to compute the homotopy groups of a motivic spectrum, using the so-called slice spectral sequence.

Lecture notes for Lecture 4

### Lecture 5. "Motivic stable stems"

Tues. July 15, 14 Uhr WSC-N-U-4.04

The homotopy of the motivic sphere spectrum is a universal motivic invariant. We explain current work with Rondigs and Spitzweck dealing with computations of these groups via the slice spectral sequence.

Lecture notes for Lecture 5