DUPTOP is the graduate student topology and homotopy theory seminar at UW. We take turns giving talks on topics of our choosing within a selected theme for the quarter. This quarter, the theme is Synthetic Spectra. To broaden appeal, we will welcome occasional talks on other unrelated topics if there are speakers interested in giving them!
We are meeting Fridays 1:30 - 2:30 in Thomson Hall (THO) 325.
Deformations of stable infinity categories
To a stable infinity category C equipped with a well-behaved homology functor to an abelian category A, Patchkoria--Pstrągowski associate a derived category which, in a precise sense, interpolates between objects in C and Adams spectral sequences in A abutting to them. In this talk, we will begin by showing how this derived category arises naturally when trying to categorify Adams resolutions of spectra and then take a close look at its construction which axiomatizes the deformation-theoretic ideas of synthetic spectra. As applications, we will catch a glimpse of a few very strong algebraicity results--that is, results identifying stable infinity categories of interest with the derived categories of certain abelian categories.
(A synthetic approach to detecting) v1-periodic families
The stable homotopy groups of spheres are hard to calculate. Chromatic homotopy theory offers an organizational
principle to sort the elements of these groups into periodic layers. Adams constructed the first infinite family of
v1-periodic elements in his study of the J-homomorphism; showing the nontriviality of these elements requires
use of what is known as the e-invariant. A more modern approach to show nontriviality involves the Adams and
Adams-Novikov spectral sequences, and an even more modem approach categorifies this into F2-synthetic
spectra. This talk will give a brief introduction to v1-periodicity before discussing the Carrick--Davies
(https://arxiv.org/pdf/2401.16508) method to detecting v1-periodicity via F2-synthetic spectra.