The main focus of the present proposal is the study of representations and cohomology of EI-categories - that is, categories all of whose endomorphisms are isomorphisms. The motivations for this proposal have their roots in long standing conjectures in modular representation theory as well as open problems in the homotopy theory of p-local groups, a certain type of p-complete topological spaces whose homotopy theory has been developed in work by Broto, Levi and Oliver.
The principal investigator has recently shown that one of the most well-known conjectures in modular representation theory, Alperin's weight conjecture, (which is originally a numerical equality involving the number of simple modules of a p-block of a finite group), has a structural reformulation in terms of the cohomology of a certain functor on a suitable EI-category.
One of the directions the principal investigator proposes to explore is that there should be similar structural results - exploiting work of G. R. Robinson - regarding Dade's conjectures (which are, very roughly, refinements of Alperin's weight conjecture taking into account more subtle invariants of characters). The problem is to find the ``right" EI-category.
Two important open problems in block theory can be formulated in a completely general way as ``gluing problems" of 2-cocycles of certain functors on appropriate categories: the question whether one can associate a classifying space with each p-block and the question whether the K"ulshammer-Puig 2-cocycles on automorphism groups of centric subgroups of a block are the restrictions of a 2-cocycle defined on the fusion system of the block. Another recent result of the principal investigator is that there is a spectral sequence relating the cohomology of a functor on a regular EI-category to the cohomology of functors on the poset of isomorphism classes of objects of that category. This spectral sequence should be useful for addressing the two mentioned gluing problems; this is another direction the principal investigator proposes to take.
One way to look at a - not even necessarily mathematical - object is to consider all maps of that object to itself which preserve its structure (the group of its "symmetries") and deduce from those maps properties of the considered object - this is, in a very simplified way, the general principle of representation theory.
