I am currently a Ph.d. student at the university of Copenhagen. My scientific advisor is Ryszard Nest. My CV contains additional information.
I am affiliated with the Non-commutative Geometry Group, and with the Centre for Symmetry and Deformation.
These days I am mainly interested in geometric and measurable group theory, and von Neumann algebras. My favorite invariants are the L^2-Betti numbers of a group(oid). The first L^2-Betti number has many known applications and interpretations - for some recent-ish work see e.g. the paper "Group cocycles and the ring of affiliated operators" by J. Peterson and A. Thom. Not so much of this kind of work involving higher L^2-Betti numbers instead of just the first one seems to exist. I'd like to change that.
Lueck's amenability conjecture states that a group G is amenable if and only if the inclusion of the group ring in the group von Neumann algebra has a certain flatness property (it is "dimension flat"). Lueck proved the "only if" part in his work on L^2-Betti numbers of countable discrete groups, and furthermore pointed out that if G contains a copy of the free group on two generators then this inclusion of rings cannot be dimension flat. The idea for this paper was to use Gaboriau-Lyons' measure-theoretic converse to the von Neumann problem to produce, for any given non-amenable G, a finite cyclic group A such that the wreath product of A and G is not "dimension flat" hence proving Lueck's conjecture "up to taking wreath products." There seem to be some difficulties in tracking down the exact group algebra inside the von Neumann algebra, but we do end up with two partial results: One is a characterization of amenability of G in terms of "dimension flatness" of groupoid algebras naturally associated with G. The other finds subalgebras that are not dimension flat and have countable linear dimension.
(preprint, 2011) L^2-Betti Numbers of Locally Compact Groups
One motivation for this work is to unify definitions of the L^2-Betti numbers for (countable) discrete groups and (countable) locally finite graphs, the latter introduced by D. Gaboriau in this paper. This is done by giving a general definition of L^2-Betti numbers for locally compact unimodular groups in terms of the continuous cohomology, analogously to Lueck's definition for discrete groups. A key feature of L^2-Betti numbers compared to just the classical Betti numbers is that they scale proportionally when passing to finite index subgroups. We also explore extensions of this phenomenon in the setting of passing to finite covolume subgroups, focusing on the case where the subgroup is countable discrete, i.e. a lattice. At the time of writing the results we get are not quite as good as one could have wished for and some questions remain.
(to appear in IEE Transactions on Information Theory) (with F. Topsøe) Computation of universal objects for distributions over co-trees
In this paper, we compute in closed form the universal code (a.k.a. universal predictor) for the model consisting of distributions respecting the order structure given by an arbitrary (finite) co-tree.
We also develop an algorithm to compute it exactly, in polynomial time.
(preprint) Composability in a certain family of entropies
I give a proof that in a somewhat general family of entropies, the Tsallis entropies are characterized as the ones enjoying the property of composability.
This was apparently known to the physicists, though I never did manage to understand how. In any case, the proof relies on no physical assumptions.
There is also an errata