Research

I am an algebraic topologist. I refrain from trying to explain in detail what algebraic topology is. For some points of view on that the Essays about algebraic topology might be useful. My current reseach interest is stable homotopy theory, more specifically: structured ring spectra.

Since the mid 90s there are several strictly symmetric monoidal categories of spectra available which yield a model of the stable homotopy category. I am interested in multiplicative cohomology theories and particularly such cohomology theories that stem from so called commutative S-algebras; these are nothing but commutative monoids in the category of spectra.

Having Brave New Rings (aka structured ring spectra) one wants to understand their arithmetic properties. In joint work with Andy Baker we investigated Galois extensions of ring spectra in the sense of John Rognes, and Picard groups. We also try to understand Azumaya extensions of ring spectra and the corresponding notion of Brauer groups. This is related to work of Bertrand Toen, David Gepner, Niles Johnson and others.

I got involved in algebraic K-theory of ring spectra. There is a machinery that turns bimonoidal categories into ring spectra. If you plug in a ring, you get the corresponding Eilenberg-MacLane spectrum. A conjecture of Baas-Dundas-Rognes states that algebraic K-theory of such spectra can be described in one go via a suitable K-theory construction in a 2-categorical setting. In joint work with Baas, Dundas and Rognes we prove this conjecture. In particular this identifies K(ku) (the algebraic K-theory of connective complex topological K-theory) with the K-theory of 2-vector spaces. Ausoni and Rognes proved that K(ku) can be viewed as a version of elliptic cohomology.

In algebra it is most of the time obvious whether a ring is commutative or not. In topology, identifying a ring spectrum as a commutative one can be quite tricky. For instance, residue fields of commutative ring spectra are rarely commutative. Sometimes one has to use the old-fashioned machinery of so-called \(E_\infty\)(= homotopy everything) ring spectra to establish such structures. Being \(E_\infty\) is a property which is formulated in the language of operads. As a technical means these can be quite useful and part of my research is concerned with them. In a project with Muriel Livernet we study homology theories for so-called \(E_n\)-algebras (these are algebraic analogues of n-fold loop spaces). We provide an interpretation of such homology theories in terms of functor homology. we build some machinery for actual calculations of these homology groups. In a joint project with Brooke Shipley we show that there is a Quillen equivalence between differential graded \(E_\infty\)-algebras and commutative \(H\mathbb{Z}\)-algebra spectra.

Related work is on higher order (topological) Hochschild homology. Our WIT (women in topology) team, Irina, Inna, Kate, Ayelet and I, did some calculations of higher order THH of \(F_p\)-algebras. In work with Ayelet Lindenstrauss and Bjørn Dundas we show that higher THH of number rings detects ramification.

For finished projects and work in progress, have a look at my list of publications.

An excellent overview on the main players in stable homotopy theory (S,MU, formal groups) can be found in (the first 38 e-slides of) Neil Strickland's talk on the Abel Symposium 2007. For some insight about the characters behind the scene (BP mafia etc) have a look at Mike Hopkins' paper The mathematical work of Douglas C. Ravenel in HHA 10(3) 2008, 1--13.

Pigne d'Arolla 2008 Morton Brun, Holger Reich, John
Rognes, Birgit Richter, Allen Hatcher, Randy McCarthy, Christian
Ausoni, James Cruickshank 2001

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