Tag - Topology

Hans-Werner Henn: On the Brown Comenetz dual of the K(2)-local sphere at the prime 2  

21st June, 2022

The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n)-local version In of I. For a type n-complex X, this is given by the Pontryagin dual of the 0-th homotopy group of the K(n)-localization of X. By work of Hopkins and Gross, the homotopy type of the spectra In for a prime p is determined by its Morava module if p is sufficiently large. For small primes, the result of Hopkins and Gross determines In modulo an "error term". For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almost 30 years. For n>2 very little is known if the prime is small. For n=2 every prime bigger than 3 is sufficiently large. The case p=3 has been settled in joint work with Paul Goerss. This talk is a report on work in progress with Paul Goerss on the case p=2. The "error term" is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 29. We use chromatic splitting in order to get information on the error term.     

Sergei Gukov: Surprising VOA Structures from Quantum Topology

6th June, 2022

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold.

Jesper Grodal: A guided tour to the Picard group of the stable module category

24th May, 2022

The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.

Jordan Williamson: Duality and definability in triangulated categories

17th May, 2022

In the category of modules over a ring, purity may be viewed as a weakening of splitting - a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. However, in general, it can be hard to check whether a class is closed under purity operations. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and illustrate this with a range of examples, often from the tensor-triangular perspective. I will also discuss an application to the study of definable subcategories of triangulated categories.

Martin Frankland: On good morphisms of exact triangles

3rd May, 2022

When studying the Adams spectral sequence in triangulated categories, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles. The talk will include examples from stable module categories of group algebras. 

Neil Strickland: Questions around the Chromatic Splitting Conjecture

19th April, 2022

The chromatic splitting conjecture (CSC) is an important open problem in stable homotopy theory. Although Beaudry has shown that the strongest version fails when n=p=2, one can still hope that the conjecture is valid at large primes, or up to a filtration that is nearly split in some appropriate sense.

The CSC gives a description of Ln-1LK(n)(S), but from that one can deduce similar descriptions of many other spectra, including those of the form LK(n1)LK(n2)...LK(nr)(S) with n1 < ... < nr.  The spectrum Ln(S) can be expressed as the homotopy inverse limit of a diagram of spectra of that form, and one can ask whether that, and various similar phenomena, are consistent with the CSC. We will explain a conjecture about how all this fits together, which involves some interesting algebra and combinatorics.  We will also explain how Morava K-theory Euler characteristics can be used to do some basic consistency checks.

Camila Sehnem: Equilibrium on Toeplitz extensions of higher-dimensional non-commutative tori

1st April, 2022

The C∗-algebra generated by the left-regular representation of ℕn twisted by a 2-cocycle is a Toeplitz extension of an n-dimensional non-commutative torus, on which each vector r ∈ [0,∞)n determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge and M. Laca, in which we show that the equilibrium states of the resulting C∗-dynamical system are parametrized by tracial states of the non-commutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of r. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalize the phase transition on the Toeplitz non-commutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.

Beren Sanders: Stratification in tensor triangular geometry with applications to spectral Mackey functors

22nd February, 2022

In a recent paper, joint with Tobias Barthel and Drew Heard, we develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. In this talk, I will provide an introduction to the problem of classifying thick and localizing tensor-ideals via theories of support, describe in broad strokes some of the highlights of our theory (which builds on the work of Balmer-Favi, Stevenson, and Benson-Iyengar-Krause) and, time-permitting, discuss our applications in equivariant homotopy theory. The starting point for these equivariant applications is a recent computation (joint with Irakli Patchkoria and Christian Wimmer) of the Balmer spectrum of the category of derived Mackey functors. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors, and harness our geometric theory of stratification to classify the localizing tensor-ideals of both categories.