Abstract: Hypergeometric
systems are a very interesting class of systems of PDEs that can be studied
using combinatorial (commutative) algebra. There are many open problems in
this area, and the one I will discuss concerns the dimension of the solution
space (the "holonomic rank") of such a system. All the available evidence
points at a very strong connection between rank fluctuations and the local
cohomology of the underlying toric ring. The precise relationship is a joint
conjecture with Ezra Miller. The talk will be self contained: I will assume
no hypergeometric or toric background, and among other things, I will show
how to combinatorially compute the local cohomology of a toric (or semigroup)
ring.
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