Seminar in Real and Complex Geometry

Thursday, 12.01.2017, 16:30-17:30, Schreiber building, room 209

Mikhail Borovoi, Tel Aviv University

Reeder 's puzzle and orbits in real homogeneous spaces


In the talk I shall describe a puzzle for children. We have a pile of stones and a graph D with n vertices. At most one stone may be placed on a vertex, so a vertex has one of two states: stoned or unstoned. We move by selecting a vertex v having an *odd* number of stoned neighbors, and then change the state of v. Given an initial configuration of stones on the graph D, we try to reduce the total number of stones as much as possible. How to determine this minimal number of stones from the initial configuration? This puzzle, introduced by Mark Reeder in 2005, is related to the Galois cohomology set H^1(R,G), where G is a simply connected, simple, compact algebraic group over the field R of real numbers. The graph D is the Dynkin diagram of G. We solve a generalized version of the puzzle. Our solution of generalized Reeder's puzzle gives a method to compute the number of connected components of (G/H)(R), where G is a simply connected semisimple R-group, H is a simply connected semisimple R-subgroup of G, and G/H is the homogeneous space of G by H, which is an algebraic variety over R. This will be a colloquium-style talk. No preliminary knowledge of algebraic groups, Dynkin diagrams or Galois cohomology will be assumed. The talk is based on a joint work with Zachi Evenor (TAU).