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SUMMARY:Genuine equivariant E_\\infty ring spectra\, normed algebras\, and
  lax limits - Tobias Lenz (Rheinische Friedrich-Wilhelms-Universität Bonn
 )
DTSTART:20250506T130000Z
DTEND:20250506T140000Z
UID:TALK230440@talks.cam.ac.uk
DESCRIPTION:Multiplicative structures on cohomology theories have been exp
 loited fruitfully throughout the history of algebraic topology\, ranging f
 rom the basic and classical argument that the Hopf maps are stably non-tri
 vial to the more recent (and very much non-basic) use of equivariant power
  operations in the solution of the Kervaire invariant one problem due to H
 ill--Hopkins--Ravenel. The latter crucially relies on a refined notion of 
 `genuine commutative algebras' in equivariant spectra\, containing more in
 formation (in the form of `twisted power operations') than just E_\\infty 
 algebras in the \\infty-category of spectra. While originally these object
 s were defined using well-behaved pointset models of spectra\, in more rec
 ent years an alternative \\infty-categorical approach has been suggested b
 y Bachmann-Hoyois and Nardin-Shah. In this talk I will report on joint wor
 k with Sil Linskens and Phil P&uuml\;tzst&uuml\;ck\, in which we construct
  an equivalence between the two suggested definitions of `genuine commutat
 ive G-ring spectra.' I will further explain how our methods yield an \\inf
 ty-categorical description of Schwede's `ultra-commutative global ring spe
 ctra' (again originally defined by a pointset model)\, and how this allows
  one to make precise the idea that an ultra-commutative global ring spectr
 um ought to be a compatible family of genuine commutative G-E_\\infty ring
  spectra for all (finite) groups G.
LOCATION:Seminar Room 2\, Newton Institute
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