Seminar Algebra and Geometry: Peter Feller (ETHZ)

Complex plane curves, their intersection with round spheres, and knot concordance

We start by recalling that a smooth algebraic curve of degree d in CP² is a genus (d-1)(d-2)/2 surface (read `smooth 2-manifold'). The `Thom Conjecture', proven by Kronheimer and Mrowka, asserts that such algebraic curves have a surprising minimizing property. We derive consequences of the Thom Conjecture for transversal intersections of algebraic curves with round spheres, describe the knots one finds as such intersections following Rudolph, and give precise instances of the sentiment that these intersections constitute very special elements in the so-called smooth concordance group. In contrast, in the topological category, we prove that all knots are topological concordant to such an intersection. Based on joint work with Maciej Borodzik. No knowledge about knot theory and concordance theory---the study of 1-manifolds in the 3-dimensional sphere and surfaces in 4-dimensional ball bounding them---will be assumed.

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