DUBTOP Fall 2025

DUBTOP 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 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.

Scheduled Talks

October 10     Jay Reiter

Deformations of stable ∞-categories

To a stable ∞-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 mapping  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 behind synthetic spectra. As applications, we will catch a glimpse of a few very strong algebraicity results--that is, results identifying stable ∞-categories of interest with the derived categories of certain abelian categories.

October 17     Albert Jinghui Yang

Motivic classifying spaces

The integral Hodge conjecture was famously disproved by Atiyah and Hirzebruch, who found a counterexample in the cohomology of the classifying space of a certain finite group. Following a question proposed by Totaro in his 2014 book, it is natural to ask whether analogous phenomena occur in the motivic setting. In this talk, I will study motivic classifying spaces in richer motivic cohomology theories, including algebraic K-theory and hermitian K-theory. This is joint work with Prerna Dhankhar, Rebecca Field, Arjun Nigam, and J.D. Quigley.

November 7     Daniel Rostamloo

Topological structures in derived algebraic geometry

In the past decade, several major advances in p-adic cohomology theory have ushered in a multitude of new questions and applications across homotopy theory, number theory, and algebraic geometry (extending even to birational geometry). At the heart of these developments is an elegant derived approach which has been essential for the discovery of new unifying theories. In this talk, I will give a survey of the language and objects involved in this beautiful story with a view toward the Hochschild-Kostant-Rosenberg theorem, a deep result connecting homotopy theory and de Rham cohomology.

November 14     Manyi Guo

Synthetic Toda brackets

In this talk I will explain how and why Toda brackets admit controllable bounds on their Adams filtration when considered in the synthetic setting. This control may seem subtle: one must keep track not only of the Adams filtrations of the maps, but also of the Adams filtrations of the requisite homotopies. The category of synthetic spectra provides exactly the right tool to organize this tracking. The synthetic version of Toda brackets also gives a refined understanding of “crossing differentials” as well as capturing the Adams filtration of the corresponding classical Toda brackets in the ordinary stable category. If time permits, I will also discuss examples from the work of Burklund--Hahn--Senger and of Lin--Wang--Xu.

November 21     Jackson Morris

(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.

December 5     Nelson Niu

(Title and abstract TBA)