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Junior Geometry
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Jayden Wang, PMA, Floor 12: Equivariant cohomology and (conditional) oriented matroid
Tuesday, September 06, 2022, 03:45pm - 04:45pm
According to Taoist wisdom, "Truths, if expressible by language, are not general truths". This has an interesting manifestation in math: theorems, if relying on geometry, are not general theorems. Therefore, I am going to talk about non-geometry. Given a collection of linear hyperplanes in \R^d, one can construct two spaces M_1 and M_3. M_1 is the complement of the union of all hyperplanes, while M_3 is the complement of R^{3d} of the union of corresponding subspaces of codimension 3. The cohomology of M_1, H^*(M_1) is a boring ring with an interesting filtration. Gel'fand and Varchenko gave presentations for H^*(M_1) and later on de Longueville & Schultz gave presentations for H^*(M_3). It happens to be the case that H^*(M_3) is exactly the associated graded algebra of H^*(M_1). Moseley realized that the reason for this connection lies in the fact that both rings are specializations of the torus-equivariant cohomology of M_3. Oriented matroids are combinatorial abstractions of real hyperplane arrangements. This is a vast generalization since almost all matroids don't come from hyperplane arrangements. All the cohomology rings mentioned above only depend on the underlying oriented matroids, so in principle, the above phenomenon should still hold even if the space M_1 and M_3 don't exist. This is proved in full generality in a joint work with Dorpalen-Barry and Proudfoot. Bodies die and vanish, but souls don't. In a similar way, cohomology exists without a space, for geometry dies and vanishes, but combinatorics doesn't.
Location: PMA, Floor 12

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