# Algebraic K-Theory and Redshift

## Fall school, Mainz, 18th - 22nd September 2023

About

The stable homotopy category, or the ∞-category of spectra, is a very complicated and still quite mysterious object. Chromatic homotopy theory provides a filtration on this object whose layers are more accessible. One rich source of spectra is the algebraic K-theory of rings or more generally ring spectra, and starting with work of Waldhausen in the early 80s mathematicians studied the interaction between algebraic K-theory and the chromatic filtration. As a milestone, several precise conjectures have been formulated by Rognes and Ausoni-Rognes around the year 2000. These are often referred to as "redshift conjectures". They predict, among other things, that algebraic K-theory increases the chromatic height of a commutative ring spectrum by 1. Recent years have seen a burst of new results in the direction of these conjectures, partly of theoretical, qualitative nature, partly of computational, quantitative nature.

This fall school will provide an introduction to this beautiful circle of ideas, leading from introductory courses on algebraic K-theory and chromatic homotopy theory up to some recent breakthrough results.

The school is aimed primarily towards graduate students and postdocs who are interested in working in one of the areas covered by the school. Some background in homotopy theory will be helpful for following the mini-courses.

Mini-courses

The school will consist of four mini-courses by:

- Christian Ausoni (Paris)
- Robert Burklund (Copenhagen)
- Thomas Nikolaus (Münster)
- Elizabeth Tatum (Stockholm)

Furthermore, there will be problem sessions accompanying the mini-courses.

Info

**Time:** 18th - 22nd September 2023 (Monday morning - Friday noon).

**Place:** Johannes Gutenberg Universität Mainz, Germany.

**Organisers:** Christian Dahlhausen, Lorenzo Mantovani, and Georg Tamme.

Registration & Financial Support

Registration will open in May 2023. Limited financial support for participants is available.

Acknowledgements

This event is funded by Johannes Gutenberg Universität Mainz,
by SFB/TR 45 "Periods, moduli spaces and arithmetic of algebraic varieties",
and by CRC326 - GAUS: "Geometry and Arithmetic of Uniformizing Structures".

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