Joel Friedman - Sheaves on Graphs, L^2 Betti Numbers, and Applications.
Joel Friedman (University of British Columbia, Canada)
Sheaf theory and (co)homology, in the generality developed by
Grothendieck et al., seems to hold great promise for applications in
discrete mathematics. We shall describe sheaves on graphs and their
applications to (1) solving the Hanna Neumann Conjecture, (2) the girth
of graphs, and (3) understanding a generalization of the usual notion of
linear independence. It is not a priori clear that sheaf theory should
have any bearing on the above applications.
A fundamental tool is what we call the "maximum excess" of a sheaf; this
can be defined quite simply (as the maximum negative Euler
characteristic occurring over all subsheaves of a sheaf), without any
(co)homology theory. It is probably fundamental because it is
essentially an L^2 Betti number of the sheaf. In particular, Warren
Dicks has given much shoter version of application (1) using maximum
excess alone, strengthening and simplifying our methods using skew group
rings.
This talk assumes only basic linear algebra and graph theory. Part of
the material is joint with Alice Izsak and Lior Silberman.
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