Date:	Tue, November 3, 2015
Time:	14:15
Place:	Research I Seminar Room

Abstract: In dynamical systems the following questions appears to be of a great importance. When is a topological branched self-covering of the 2-sphere equivalent to a rational map? William Thurston first answered this question more than 30 years ago (in 1982), by giving a "negative" characterization: a branched cover is NOT rational if and only if there IS a combinatorial obstruction. Since then, mathematicians did quite a lot of research trying to deeply understand the result and techniques, generalize them to different families of maps and make it to an algorithm. The latter seems to be difficult to achieve as the combinatorial obstruction is given by an invariant multi-curve satisfying certain condition. Thus, Thurston's characterization is hard to apply in practice, since it involves checking infinitely many multi-curves. Recently, Dylan Thurston (the son of William Thurston) gave a "positive" combinatorial characterization for a somewhat restricted class of maps: a branched cover CAN be realized by a rational map if and only if there IS a certain combinatorial object (namely, an elastic graph with a particular "self-embedding" property). The result links three subjects - dynamics of rational maps on the Riemann sphere, - embeddings of conformal surfaces, - elastic graphs, through the study of so-called "stretch factors". We are going to have a series of lectures based on D. Thurston's papers (partly joined with J. Kahn and K. Pilgrim) with the aim to understand the concepts and techniques which lead to the proof of the new characterization of rational maps. I will start the first talk of that series with formulating the main result and sketching a couple of examples of its applications. Next, I will outline the general scheme of the proof and explain which concepts and preliminary results come to play at what time. The details should be explained in the further talks.