# Permanent Members

## Tilman Bauer, Associate Professor, KTH.

My research is in stable and unstable homotopy theory, in particular elliptic cohomology, chromatic homotopy theory, completion and localization, and homotopy versions of Lie groups and their classifying spaces. I am currently working on a description of the algebraic structure on generalized cohomology of spaces. For singular cohomology, the theory of unstable modules and algebras over the Steenrod algebra has led to beautiful results and new theories such as the Sullivan conjecture, p-compact groups, and Lannes' T-functor theory. But little is known for other cohomology theories, in particular for the cohomology theories known as Morava K-theories, and the homological algebra arising from this. The aim is a better abstract understanding and computational access to Morava-K-theoretic localization of spaces and the construction and analysis of K(n)-compact groups.

## Alexander Berglund, Associate Professor, SU

My research revolves around interactions between algebraic topology and commutative algebra. In my PhD thesis, I used ideas from homotopy theory to solve problems in commutative algebra concerning the cohomology of Stanley-Reisner algebras and the classification of Golod rings. More recently, I have become interested in reversing the flow of ideas, using algebraic models to study homotopy theory. In particular, L-infinity algebra models for Q-local homotopy types and E-infinity algebra models for p-complete homotopy types. When working with the algebraic models, tools from homological algebra such as Koszul duality for operads, minimal models, and various notions of formality become available. I am developing such tools to study algebraic models for constructions such as mapping spaces, loop spaces and polyhedral products, with applications to string topology and toric topology. I am currently studying the cohomology of classifying spaces of homotopy automorphism groups or, equivalently, characteristic classes for fibrations with a specified fiber.

## Wojciech Chachólski, Professor, KTH

It is often the abundance of information and not the lack of it that makes it difficult to understand the objects we study. Homotopy theory is a simplifying tool. It extracts information by forgetting the irrelevant data. For example, if it is not the size but the shape that is important, then we might decide to identify intervals with points (forgetting the length). How would our world change after such identification? Homotopy theory has been developed to study this. Constructing objects out of a given one is a typical tool. In geometry traditionally spheres and discs have been the basic building blocks. Why restrict ourselves to just these spaces? We could use other objects such as projective spaces. Part of my research has been about understanding what happens if we do that. Of course it is not expected that all spaces can be constructed using a given one and that different spaces can be used to construct the same objects. A big triumph of homotopy theory has been to catalog all possible building blocks of our world. It has been a big surprise that this is even possible.