PhD Defense Kaif Hilman bin Muhammad Borhan Tan
Title:
Norms and periodicities in genuine equivariant hermitian K–theory
Abstract:
In this thesis, we explore various aspects of genuine G–equivariant K–theory on stable ∞–categories for finite groups G. The main theme is establishing equivariant multiplicative norms - in the sense of Hill–Hopkins–Ravenel - on these K–theories and our formalism of choice is that of the parametrised higher category theory by Barwick–Dotto–Glasman–Nardin–Shah. This thesis is divided into three parts, each one building up towards the next.
In Part I, we develop the theory of G–presentable and G–perfect–stable ∞– categories. This will serve as the technical underpinnings for our investigations on G–equivariant algebraic K–theory in Part II where we show that when G is a 2–group, algebraic K–theory refines to the structure of a ring G–spectrum equipped with the Hill–Hopkins–Ravenel norms. Along the way, we will obtain a “multiplicative Borelification principle” via a simple categorification–decategorification procedure which provides a huge source of examples of equivariant K–theory with norms. Finally, in Part III, we initiate the study of genuine equivariant hermitian K–theory by introducing the notion of G–Poincaré ∞–categories, generalising in the equivariant direction the recent advances made by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle. Among other things, we refine Borel equivariant Grothendieck–Witt theory to the structure of a normed ring G–spectrum when G is a 2–group, and we also obtain a new source of equivariantly periodic ring G–spectra in the form of equivariant L–theory.
Advisors:
Jesper Grodal, University of Copenhagen, Denmark
Markus Land, L.M.U. Munich, Germany
Assessment committe:
Søren Galatius (chair), University of Copenhagen, Denmark
John Greenlees, University of Warwick, United Kingdom
Thomas Nikolaus, University of Münster, Germany