Seminars in Geometry/Topology

Topology Seminar Thursday 11th December, 13:00-14:00, 656 Simastuen, Sentralbygg 2

Marius Verner Bach Nielsen (NTNU): Filtered ring spectra, and a new proof of Lurie’s faithfully flat descent.

Abstract: In this talk I will introduce the filtered spectra and the higher algebra of filtered ring spectra. Along the way I will give a gentle introduction of filtered spectra as a replacement for spectral sequences and argue how one might recover many classical spectral sequences from this setup. After this, I will introduce the faithfully flat topology on filtered ring spectra and prove a version of faithfully flat descent for filtered ring spectra. This has as a surprising corollary, namely that E_\infty-ring spectra satisfy descent for the pi_*-flat topology.

If time permits, I will discuss how this relates to modern objects of interest like Pstragowski’s synthetic spectra and the even filtration of Hahn, Raksit and Wilson.

Topology Seminar Wednesday 29th October, 10:15-11:00, 656 Simastuen, Sentralbygg 2

Séverin Philip (Stockholm): The fields of semi-stability

Abstract: A fundamental theorem of Grothendieck asserts that any abelian variety over a number field acquires semi-stable reduction after a finite extension. I will present the notion of semi-stable reduction and how field extensions interact with it starting from the case of elliptic curves before moving to abelian varieties. I will present the semi-stable reduction theorem of Grothendieck and an effective version. I will then introduce the finite monodromy groups, groups that represent the local obstruction to semi-stable reduction and recent characterization results of those.

Topology Seminar Wednesday 29th October, 11:15-12:00, 656 Simastuen, Sentralbygg 2

William Hornslien (Grénoble): An explicit construction of a rank 2 vector bundle on projective 3-space

Abstract: Atiyah and Rees proved that the Chern classes and a mod 2 invariant, named the alpha invariant, classifies all rank 2 topological vector bundles on P3. They also showed that a construction by Horrocks provides algebraic representatives for all topological rank 2 bundles. However, Horrocks’ construction is non-explicit. The goal of this talk is to construct an algebraic rank 2 bundle on P3 with trivial Chern classes and non-trivial alpha invariant. We will use methods from motivic homotopy theory to construct an explicit algebraic description of the bundle. Along the way we will also see an interesting trick with bigraded homotopy groups.

This project is joint work with Jean Fasel.

Topology Seminar Wednesday 28th May, 11:00-12:00, 656 Simaestuen, Sentralbygg 2

Walker Stern (TU München): A Braided Monoidal Hecke Category

Abstract: Hecke algebras occupy a key position in low-dimensional topology and representation theory, as they govern a broad class of quantum knot invariants. Of particular use, a simulatenous categorification of several Hecke algebras carries a braided monoidal structure induced by a kind of convolution. In this talk,I will present work in progress with Jonte Gödicke, Quoc Ho, and Yang Hu, in which we show that this braided monoidal structure arises naturally as a linearization a far richer structure: lax E_2 algebra in an $(\infty,2)$-category of spans. This lax E_2 algebra can be accessed relatively simply, using the combinatorics of flags and a classification of algebras in spans. Moreover, various linearizations retrieve not only the Hecke algebras, but also the braided monoidal $(\infty,2)$-category of Soergel bimodules recently constructed using obstruction-theoretic techniques.

Topology Seminar Monday 10th February, 13:15-14:15, 656 Simaestuen, Sentralbygg 2

Thomas Blom (MPI Bonn): Cofree fibrations and endomorphism objects

Abstract: Cocartesian fibrations are one of the most important tools in higher category. They can be thought of as functors between ∞-categories that admit a notion of "transport" between the fibers, and they feature in Lurie's famous straightening-unstraightening equivalence.

A useful construction that has many applications is to "freely adjoin cocartesian lifts" to a functor. Not so well-known is that there also exists a dual construction: one can "cofreely adjoin cocartesian lifts" to a functor.

I will describe this dual construction and give various applications. In particular, I will show how it allows one to simplify and generalize Lurie's Day convolution construction and his construction of endomorphism objects in ∞-categories.