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I'm an algebraic
topologist.
My
main field of interest are the interactions between group theory and
homotopy theory. In particular, I work on fusion systems. I
also
work on homological algebra of categories of functors.
Current projects:
- (with
A. Libman)
We work on the (p-localized) Burnside ring of fusion systems and
study its
algebraic properties. For example, we have proven that this
p-localized Burnside ring has a unit. We have also proven a version of
Segal's conjecture for p-local finite groups.
- (with
A.
Glesser, N. Mazza, S. Park and R. Stancu) We work on generalizations of
classical group theoretical results to the fusion system setting. The
following are examples of results which have been already
generalized:
Alperin's fusion theorem, Frobenius' theorem on normal p-complements,
Glauberman and Thompson's p-nilpotency criterion and Glauberman's
ZJ-theorem.
- Concerning homological
algebra I study functors
which have vanishing higher limits. I have found conditions on functors
from a poset to abelian groups such that the direct (inverse) higher
limits vanish. I call these conditions pseudo-projectivity
(pesudo-injectivity). Recently Jesper Grodal came up with
new ideas on how to generalize these results to (extended) Reedy
categories using model category theory.
- I study
posets whose integral
cohomology is given by a chain complex analogue to that of simplicial
complexes, i.e., posets for which one subdivision can be removed when
computing their cohomology. I have already found conditions on a poset
such that its cohomology can be computed via such a simplified
chain complex. This applies to Quillen's complex of a finite group G, i.e., to the poset of the non-trivial elementary abelian p-subgroups of G.
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