## tt-geometry in Trondheim

• Trondheim, Norway

This is the project page for the project tt-geometry in Trondheim', supported by the Trond Mohn Foundation. For background reading on tensor-triangulated geometry, a good introduction can be found in the survey papers of Balmer:

A recent online lecture given by Drew Heard can be found here.

The image on the left is due to Balmer and Sanders, and shows a part of a computation of the tensor-triangulated geometry of equivariant homotopy theory.

### Members

Clover May, postdoc (August 2021 - ).

### Publications

Rational local systems and connected finite loop spaces. - Drew Heard. Glasgow Mathematical Journal (arXiv link)

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree $$G$$-spectra. More generally, we show that if $$K$$ is a closed subgroup of a compact Lie group $$G$$ such that the Weyl group $$W_GK$$ is connected, then a certain category of rational $$G$$-spectra at $$K$$' has an algebraic model. For example, when $$K$$ is the trivial group, this is just the category of rational cofree $$G$$-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

### Preprints

On conjectures of Hovey–Strickland and Chai - Tobias Barthel, Drew Heard, and Niko Naumann.

We prove the height two case of a conjecture of Hovey and Strickland that provides a $$K(n)$$-local analogue of the Hopkins–Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross &ndash Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava $$E$$-theory is coherent, and that every finitely generated Morava module can be realized by a $$K(n)$$-local spectrum as long as $$2p-2>n^2+n$$. Finally, we deduce consequences of our results for descent of Balmer spectra.