Central Extensions of Gerbes Amnon Yekutieli, BGU Abstract: The talk will begin with an explanation of the concept of extension of groupoids. This is a mild generalization of the familiar concept of extension of groups. Oddly this concept appears to be new; perhaps because it is not very interesting... Next I will explain what is a gerbe. This is a very complicated concept. My point of view is that a gerbe G, on a topological space X, is the geometric version of a nonempty connected groupoid -- much in the same way that a sheaf of groups on X is the geometric version of a group. I will try to avoid the horrid technicalities of 2-categories and stacks. The structure of nonabelian gerbes is known to be very complicated. Mainly one wants to know if a given gerbe G is trivial. The theory of nonabelian cohomology (Giraud, 1960's) was invented for this purpose. However this theory is too abstract to be useful. An extension of gerbes is gotten by geometrizing the concept of extension of groupoids. Now this is a very interesting concept. A special case is that of central extension of gerbes, and I will give some examples. Next I will state a couple of results about obstruction theory for central extensions of gerbes. This theory gives concrete criteria to determine if a given gerbe is trivial. A more elaborate result is about the structure of pronilpotent gerbes. If time permits I will explain how the last result is used in my work on twisted deformation quantization of algebraic varieties. The paper (same title as lecture) appeared in Advances Math. (2010)