Algebraic K-theory and Chromatic Homotopy Theory

The most universal kind of linear algebra is based not on abelian groups, but on homotopy-theoretic objects known as spectra. According to chromatic homotopy theory, one can systematically organize spectra into periodic families.  On the other hand, a natural source of spectra is provided by algebraic K-theory, a highly refined cohomological invariant of rings (or schemes, etc).  This leads to the subject of this course: the interaction of the chromatic theory with algebraic K-theory.  The story begins with classical theorems of Thomason, Mitchell, and Hesselholt-Madsen.  Bold generalizations of these theorems were conjectured by Rognes and Ausoni-Rognes, under the umbrella term of « redshift ».  Several of these conjectures are now theorems due to recent work of many people.  Remarkably, this work has applications to « pure » chromatic homotopy theory: Burklund-Hahn-Levy-Schlank used it to settle (in the negative) the « telescope conjecture », the last of Ravenel’s conjectures.
Lecture 1: Introduction to chromatic homotopy theory.Lecture 2: Descent and « soft redshift ».
 Lecture 3: « Hard redshift », a.k.a. the Lichtenbaum-Quillen property.
 Lecture 4: The telescope conjecture.