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 Events
Nathaniel Bottman, Assistant Professor
University of Southern California
Mon 16 Dec 2019, 2:00pm
Department Colloquium
MATX 1100
What analysis, combinatorics, and quilted spheres can tell us about symplectic geometry
MATX 1100
Mon 16 Dec 2019, 2:00pm-3:00pm

Abstract

Over the past three decades, symplectic geometers have constructed powerful curve-counting invariants of symplectic manifolds. The chief example is the Fukaya category, which revealed a deep connection with algebraic geometry via Kontsevich's Homological Mirror Symmetry conjecture. In this talk, I will describe my program to relate the Fukaya categories of different symplectic manifolds. The key objects are "witch balls" (coupled systems of PDEs whose domain is the Riemann sphere decorated with circles and points), as well as the configuration spaces of these domains, which are posets called "2-associahedra". I will describe applications to symplectic geometry, algebraic geometry, and topology.
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Nathaniel Bottman
University of Southern California
Tue 17 Dec 2019, 2:00pm
MATH 204
Symplectic Geometry Seminar: Functoriality for the Fukaya category and a compactified moduli space of pointed vertical lines in C^2
MATH 204
Tue 17 Dec 2019, 2:00pm-3:00pm

Details

Abstract: A Lagrangian correspondence between symplectic manifolds induces a functor between their respective Fukaya categories. I will begin by introducing this construction, along with a family of abstract polytopes called 2-associahedra (introduced in math/1709.00119), which control the coherences among this collection of functors. Next, I will describe new joint work with Alexei Oblomkov (math/1910.02037), in which we construct a compactification of the moduli space of configurations of pointed vertical lines in C^2 modulo affine transformations (x,y) -> (ax+b,ay+c). These spaces are proper complex varieties with toric lci singularities, which are equipped with forgetful maps to \overline{M}_{0,r}. Our work yields a smooth structure on the 2-associahedra, thus completing one of the last remaining steps toward a complete functoriality structure for the Fukaya category.
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