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Title:
Polygons in Low Dimensional Topology
Speaker:
Abstract:
In his groundbreaking work, Jones constructed a polynomial invariant of knots in 1980’s. Jones polynomial is defined by a surgery formula relating the invariants of three knots. In another remarkable work in 1980’s, Casson introduced a powerful numerical invariant of integral homology 3-spheres. This invariant also satisfies a closely related surgery formula. Both Casson invariant and Jones polynomial were subsequently categorifed into instanton Floer homology and Khovanov homology. Both of these homology theories satisfy exact triangles which are categorifications of sugary formulas for Casson and Jones invariants. Nowadays, there are many homology theories in 3-manifold topology which satisfy some version of surgery exact triangle. In this talk, I will review such triangles in low dimensional topology. Then I will focus on the notion of ``surgery exact polygons'' which was recently discovered in the context of other Floer homology theories. I will also explain how this result provides evidence for a ``homological-mirror-symmetry-type'' conjecture relating Yang-Mills gauge theory and algebraic geometry.
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