List of previous NRW topology meetings

Schedule:

Thu Nov 5, in room HGB 30:
13:00 Reception

13:15-14:05 Luca Pol - Separable algebras and the Galois theory of G-spectra

14:15-15:05 Tomás Mejia-Gomez - A-infinity structures in link and tangle invariants

15:05-15:45 coffee, tea, and snacks

15:45-16:35 Stefan Schwede - Equivariant K-theory as a global localization

16:45-17:35 Shay Ben-Moshe - The Redshift Bound from Quillen-Lichtenbaum

18:30 Dinner

Fri Nov 6, in room HMA 40:
9:30- 10:20 Jack Davies - Equivariant forms of affineness in derived geometry

10:20-11:00 Coffee break

11:00- 11:50 Catherine Ray - tba

12:00- 12:50 Anish Chedalavada - Finite presentation in tensor-triangular geometry

Abstracts

Luca Pol: Classifying separable commutative algebra objects can be viewed as a homotopical analogue of classifying finite étale extensions in algebra and geometry. In this talk, I will consider this question in the category of genuine G-spectra. I will explain how it is naturally connected to the Galois theory of stable homotopy categories developed by Akhil Mathew. After introducing the relevant notions, I will describe the separable commutative algebra objects in genuine G-spectra and determine the Galois group of the category. This is based on joint work with Niko Naumann and Maxime Ramzi. 

Tomás Mejia-Gomez: In the twenty-five years since the introduction of Khovanov homology, many refinements of the original construction have emerged: local and equivariant versions, stable homotopy types, matrix factorizations. Besides sometimes yielding stronger invariants of links, these higher structures bring a better structural understanding of Khovanov homology itself. After an overview of this philosophy, we turn to the A-infinity structures underlying Bar-Natan homology, an equivariant deformation of Khovanov homology. These emerge most naturally in the dg-category of matrix factorizations, where we prove that they assemble into a tangle invariant. Their power becomes most visible for 4-ended tangles: the derived indecomposables are classified by curves with local systems on the 4-punctured sphere, and structural results on Bar-Natan and Khovanov theories become statements about which curves that can appear. Time permitting, we will discuss the prospects for similar results in sl(n) Khovanov–Rozansky homology.

Stefan Schwede: A celebrated theorem of Snaith shows that complex K-theory can be obtained from the unreduced suspension spectrum of infinite complex projective space by inverting the Bott class. A contemporary reformulation of this result is that complex K-theory classifies orientations of the multiplicative group in higher algebra.

In want to explain a highly structured, globally-equivariant refinement of Snaith's localization result: a specific morphism of ultra-commutative global ring spectra from the unreduced suspension spectrum of the global classifying space of U(1) to the global K-theory spectrum is a localization away from the "pre-Bott classes", certain representation-graded U(n)-equivariant homotopy classes for n>0.

The localization property holds in different, but interrelated senses:

- as universal morphisms in the infty-categories of ultra-commutative ring spectra and of commutative global ring spectra;
- for every compact Lie group G, as universal examples in the infty-categories of ultra-commutative G-ring spectra and of commutative G-ring spectra;
- for every compact Lie group G, at the level of G-equivariant cohomology theories.

The construction of the pre-Bott classes exploits the global functoriality and power operations.

Shay Ben-Moshe: One of the main aspects of the redshift philosophy is that algebraic K-theory increases chromatic height by one (for suitable inputs). I will present a new proof of the upper bound part, originally due to Clausen-Mathew-Naumann-Noel. The argument proceeds by descent from the case of the Lubin-Tate spectrum, which holds by another aspect of redshift, the Quillen-Lichtenbaum property, proved by Hahn-Wilson.

Jack Davies: The derived category of quasi-coherent sheaves on a scheme (or a stack) X can be complicated for a multitude of different reasons. Perhaps it requires many generators or these generators might fail to be dualisable or compact. The simplest possible situation is when the structure sheaf itself is a compact generator of QCoh(X), in which case QCoh(X) is equivalent to the category of modules over the global sections of X in spectra. In this situation, we say that X is 0-affine. There is a surprising abundance of 0-affine stacks which are not literally affine, many born out of deep theorems in chromatic homotopy theory.

In this talk, I want to introduce a generalisation of 0-affineness called G-equivariant affineness for some compact Lie group G. A (derived/spectral) stack X is G-equivariantly affine if QCoh(X) is rigid and its generators are parametrised by modules over a G-ring spectrum. This gives rise to a lot more stacks X with well behaved QCoh(X), and also also connects the algebra of the G-ring spectrum in question with the local geometry of X. I'll talk about how this question naturally arises when studying QCoh(E) for an oriented elliptic curve E, give that and many related examples, and a sketch of the proof if time allows. This is all joint work with Sil Linskens and Leonard Tokic.

Anish Chedalavada: Given a stable, presentably symmetric monoidal ∞-category C, the telescope conjecture asks whether every smashing localization of C has compactly generated kernel. In this talk, I explore the extent to which this holds under certain "finite presentation" hypotheses on a given smashing localization. Using this, I will demonstrate a general descent result in TT-geometry: for example, I will sketch a computation of the Balmer spectrum of a general perfect stack in characteristic 0, using recent work of Hyslop.