Our very own Rodion Déev will speak in the geometry seminar on Monday 8th June at 2pm in the Salle des Profs (9th floor of building NO). Rodion’s title is “Holomorphic families of knots” and his abstract is below. Hope to see you there!
It has long been known that geometric structures on a manifold give rise to different geometric structures, sometimes more familiar, on their space of knots (that is, space of all smooth immersions of a circle up to reparametrisation). For example, a volume form on a three-dimensional manifold yield an infinite-dimensional symplectic structure, and a conformal class of a Riemannian metric an infinite-dimensional complex manifold. However, these spaces are infinite-dimensional and not exactly convenient to deal with. We study them from the point of view of their finite-dimensional complex submanifolds. A classical example is given by a theorem of Hitchin who showed that the space of geodesics in a three-dimensional space form is a complex surface. We prove a converse to this theorem, namely, that existence of a compact complex surface X subject to a couple of natural conditions inside a space of knots Knots(M, [g]) yields (M, [g]) to be a three-sphere or RP^3 with the conformal class of a round metric, and X to be the surface of geodesics in some round metric in this class. We also show that dropping of either condition allows counterexamples to this statement. This is a joint work with Vasily Rogov.