Title: Cloak and dagger: Is effectivity really stability in disguise?
Speaker: Brent Doran (Oxford University)
Abstract: The quest to understand the commonalities and differences between topology and algebraic geometry is central to many mathematical developments. Here we consider a number of seemingly unrelated but well-known unsolved problems (in geometry, number theory, representation theory): for instance, dating back to Newton's time, what is the minimal degree curve passing through n chosen points in the plane with multiplicities? Relatedly, a common, but famously difficult, broad question is to determine when a ``homology" class is represented by a (symplectic, algebraic, etc.) geometric object, that is to say, when is a class effective?
We recast these problems using a canonical construction that crucially exploits an oft-overlooked difference between topological and algebraic structures. The resulting "uncloaked" problems can then all be attacked with the same weapon. At least up to scaling, this is a form of stability analysis using both additive and multiplicative groups. Beautiful combinatorial structures, like scuffed polytopes and modifications of Okounkov bodies arise naturally. The results are suggestive that effectivity may reduce to stability much more broadly still, and that many wall-crossings in geometry may admit interpretation as a change in this form of stability.
If there is time, we may briefly mention links with quantum entanglement and entropies, and question whether physics is sensitive to some of these differences between topology and algebraic geometry.
Some of this work is joint with Frances Kirwan.
Seminar URL: https://research.math.osu.edu/agseminar/
