Topology seminars: 2018-19

Greenlees-Shipley developed a Cellularisation Principle for Quillen adjunctions in order to attack the problem of constructing algebraic models for rational G-spectra. One example of this was the classification of free rational G-spectra as torsion modules over the cohomology ring H*(BG) (for G connected). This has some disadvantages; namely that it is not monoidal and that torsion modules supports only an injective model structure.

I will explain a related method called the Left Localisation Principle, and how this can be used to construct a monoidal algebraic model for cofree G-spectra. This will require a tour through the different kinds of completions available in homotopy theory. This is joint work with Luca Pol.

There is a construction of Freyd which associates, to any ring R, the free abelian category on R. That abelian category may be realised as the category of finitely presented functors on finitely presented R-modules.

It has an alternative interpretation as the category of (model-theoretic) imaginaries for the category of R-modules. In fact, this extends to additive categories much more general than module categories, in particular to finitely accessible categories with products and to compactly generated triangulated categories.

I will describe this and give some examples of its applications.

I will describe an extremely easy construction with formal group laws, and a slightly more subtle argument to show that it can be done in a coordinate-free way with formal groups. I will then describe connections with a range of other phenomena in stable homotopy theory, although I still have many more questions than answers about these.

In particular, this should illuminate the relationship between the Lambda algebra and the Dyer-Lashof algebra at the prime 2, and possibly suggest better ways to think about related things at odd primes.

The Morava K-theory of symmetric groups is well understood if we quotient out by transfers, but somewhat mysterious if we do not pass to that quotient; there are some suggestions that dilation will again be a key ingredient in resolving this.

The real topological Hochschild and cyclic homology (THR, TCR) are invariants for rings with anti-involution which approximate the real algebraic K-theory.

In this talk we will introduce these objects and report about recent computations. In particular we will discuss components of THR and TCR and some recent and ongoing computations for finite fields. This is all joint with E. Dotto and K. Moi.

Prismatic homotopy theory is the study of stable monoidal homotopy theories through their Balmer spectrum. In this talk, I will discuss how one can use localised p-complete data at each Balmer prime in an adelic fashion to reconstruct the homotopy theory in question.

There are two such models, one is done by moving to categories of modules, which, for example, recovers the algebraic models for G-equivariant cohomology theories. The other, newer model, works purely at the categorical level and requires the theory of weighted homotopy limits. This is joint work with J.P.C Greenlees.

Half a century ago, Barry Mazur and David Mumford suggested a remarkable dictionary between prime numbers and knots. I will explain how the story of exodromy permits one to make this dictionary precise, and I will describe some applications.

I will explain how the category of discrete monoids models the homotopy category of connected spaces. This correspondence is based on derived localisation of associative algebras and could be viewed as an algebraisation result, somewhat similar to rational homotopy theory (although not as structured).

Closely related to this circle of ideas is a generalisation of Adams’s cobar construction to general nonsimply connected spaces due to recent works of Rivera-Zeinalian and Hess-Tonks. This is joint work with J. Chuang and J. Holstein.

In the 90s, Kuperberg defined a scalar invariant of 3-manifolds from each finite-dimensional involutory Hopf algebra over a field. The construction is based on the presentation of 3-manifolds by Heegaard diagrams and involves tensor products of the structure tensors of the Hopf algebra. These tensor products are then contracted using integrals of the Hopf algebra to obtain the scalar invariant.

We generalise this construction by contracting the tensor products with other morphisms. Examples of such morphisms are derived from involutory Hopf algebras in symmetric monoidal categories. This is a joint work with R. Kashaev.

In joint work with John Rognes, we have computed the 2-local homotopy of tmf, with full details. We first compute the cohomology of A(2) by a method of general interest. Grobner bases play a key role in allowing us to give a useful description it. I will briefly describe this.

We then show that all the Adams spectral sequence differentials follow from general properties together with three key relations in the homotopy of spheres. We then compute the hidden extensions and the relations in homotopy using the cofibers of 2, eta and nu. This allows us to give a clear and memorable description of tmf_*.

I will end with a brief description of the duality present in tmf_* coming from the Anderson duality for tmf.

Knots and their groups are a traditional topic of geometric topology. In this talk I will explain how aspects of the subject can be approached using ideas from Quillen’s homotopical algebra, rephrasing old results and leading to new ones.

Let G be a finite group. The coefficients of G-equivariant cohomology theories naturally form a type of structure called a Mackey functor, which incorporates data coming from each subgroup of G.

When the cohomology theory is a G-ring commutative spectrum – meaning that is has an equivariant multiplication – interesting new structures arise. In particular, work of Brun and of Strickland shows that the zeroth homotopy groups have norm maps which yield the structure of a Tambara functor.

In this talk, I discuss joint work with Vigleik Angeltveit on the algebraic structure induced by norm maps on the higher homotopy groups, which we call a graded Tambara functor.

The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I'll show how it arises in the context of category theory using categories enriched over the extended real numbers. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (eg in Galois theory or algebraic geometry) arise from a relation between sets. I will assume no knowledge of the Legendre-Fenchel transform and no knowledge of enriched categories.