2:30 pm Wednesday, August 9, 2017
Thesis Defense: GHK mirror symmetry and the Knutson-Tao hive cone by
Timothy Magee [mail] in RLM 9.166
In 1998, Allen Knutson and Terry Tao introduced a rational polyhedral cone with some amazing combinatorial properties in their proof of the saturation conjecture. The "Knutson-Tao hive cone" encodes the number of copies of a given irreducible representation of appearing in the tensor product of two others-- so it tells us how to rewrite a tensor product as a direct sum. Choosing the representations of interest slices the cone to give a bounded polytope, and counting the integral points in this bounded polytope gives the number we're looking for. This cone (and plenty of others with the same wonderful combinatorial properties) can actually be obtained by completely general mirror symmetry considerations, without any representation theory at all. We'll get much more than the cone too. In this setting, integral points are elements of a canonical basis, and the combinatorial data is just the cardinality this basis. Moreover, this is a construction that in theory applies whenever you have a space equipped with the right sort of volume form, so the Knutson-Tao hive cone is part of a very broad framework when viewed in this way. I'll give an overview of how this all works. This talk will be based on my thesis, which in turn is based on the log Calabi-Yau mirror symmetry program being developed by my advisor Sean Keel and his collaborators (Mark Gross, Paul Hacking, Bernd Siebert, ...). Submitted by
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