Tag - Topology

James Cameron: Homological residue fields for tensor triangulated categories and cooperations

15th February, 2022

Many tensor triangulated categories admit 'residue field functors' that control their large-scale structure. The derived category of a ring is controlled by the residue fields of the ring, the structure of the stable homotopy category is controlled by the Morava K-theories, and in modular representation theory there are the pi-points. Unfortunately, it is not known if every tensor triangulated category has a notion of tensor triangulated residue fields. Homological residue fields were introduced by Balmer, Krause, and Stevenson as an abelian avatar of the putative tensor triangulated residue fields. They exist in complete generality, but they are hard to understand and compute with in general. I will discuss how to connect homological residue fields with the tensor triangulated residue fields that exist in examples. I will show that for the derived category of a ring, homological residue fields are closely related to usual residue fields, and in stable homotopy theory they are closely related to Morava K-theories. In fact, the homological residue fields have even more structure, and can be identified with comodules for a Tor coalgebra which in the case of the stable homotopy category is the coalgebra of coooperations for a Morava K-theory. I will introduce homological residue fields, give some examples, and mention some open problems. This is joint work with Paul Balmer and with Greg Stevenson. 

Kai Cieliebak: Poincaré duality for loop spaces

7th February, 2022

This talk is about ongoing joint work with Nancy Hingston and Alexandru Oancea. I will explain how various puzzles in string topology get resolved in terms of symplectic geometry: Loop space homology and cohomology are merged into a larger space, Rabinowitz Floer homology, which carries a product and coproduct extending those from string topology and satisfies Poincaré duality.

Niall Taggart: Homological localizations of orthogonal calculus

1st February, 2022

Orthogonal calculus is a version of functor calculus that sits at the interface between geometry and homotopy theory; the calculus takes as input functors defined on Euclidean spaces and outputs a Taylor tower of functors reminiscent of a Taylor series of functions from differential calculus. The interplay between the geometric nature of the functors and the homotopical constructions produces a calculus in which computations are incredibly complex. These complexities ultimately result in orthogonal calculus being an underexplored variant of functor calculus.

On the other hand, homological localizations are ubiquitous in homotopy theory. They are employed to split ‘integral' information into ‘prime' pieces, typically simplifying both computation and theory.

In this talk, I will describe a 'local' version of orthogonal calculus for homological localizations, and survey several immediate applications.

Thibaut Mazuir: Higher algebra of A∞-algebras in Morse theory

28th January, 2022

In this short talk, I will introduce the notion of n-morphisms between two A∞-algebras. These higher morphisms are such that 0-morphisms correspond to standard A∞-morphisms and 1-morphisms correspond to A∞-homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra. Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A∞-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.

Doug Ravenel: String cobordism at the prime 3

18th January, 2022

String cobordism refers to the Thom spectrum for the 7-connected cover of BO, the classifying space for real vector bundles. I will describe progress toward a description of its 3-primary homotopy type in joint work with Vitaly Lorman and Carl McTague. It supports a map to tmf (the spectrum associated with topological modular forms) which is surjective in homotopy groups.

Jérôme Scherer: Floyd’s manifold is a conjugation space

7th December, 2021

This is joint work with Wolfgang Pitsch. We illustrate how equivariant stable homotopy methods can help us recognize the structure of a conjugation space, as introduced by Hausmann, Holm, and Puppe. We first explain their definition and present a characterization in terms of purity (obtained in previous joint work with Nicolas Ricka). We then perform equivariantly Floyd's construction from the 1970s of a pair of 5- and 10-dimensional manifolds with four cells, relying on Lück and Uribe’s work on equivariant bundles. The 10-dimensional one is a conjugation space.

Fosco Loregian: Towards a formal category theory of derivators

23rd November, 2021

Derivator theory, initiated by Grothendieck and Heller in the '90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (∞,1)-category theory, and its applications to algebraic geometry/topology.

For a 2-category theorist, a (pre)derivator is a familiar object - (a suitably co/complete) prestack on the category cat of small categories - and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.

After briefly outlining the essentials of "formal category theory'' (2-categories can be used to organize the theory of "categories with structure" just as category theory organizes the theory of "sets with structure"), I will report on a conjecture regarding the possibility to provide a "yoneda structure" or a "proarrow equipment" to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and myself, this allows to speak about "locally presentable" and "accessible" objects (showing that Adamek-Rosický and Renaudin's definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).

Arik Wilbert: Real Springer fibers and odd arc algebras

9th November, 2021

Arc algebras were introduced by Khovanov in a successful attempt to lift the quantum 𝔰𝔩2 Reshetikhin-Turaev invariant for tangles to a homological invariant. When restricted to knots and links, Khovanov’s homology theory categorifies the Jones polynomial. Ozsváth-Rasmussen-Szabó discovered a different categorification of the Jones polynomial called odd Khovanov homology. Recently, Naisse-Putyra were able to extend odd Khovanov homology to tangles using so-called odd arc algebras which were originally constructed by Naisse-Vaz. The goal of this talk is to discuss a geometric approach to understanding odd arc algebras and odd Khovanov homology using Springer fibers over the real numbers.