Nonamenable products are not treeable

LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct productX×Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product has infinitely many connected components almost surely. Research partially supported by NSF grant DMS-9803597.     

Asymptotic behavior and symmetry of condensate solutions in electroweak theory

We study condensate solutions of a nonlinear elliptic equation in ℝ², which models a W-boson with a cosmic string background. The existence of condensate solutions and an energy identity are discussed, based on which the refined asymptotic behavior of condensate solutions is established by studying the corresponding evolution dynamical system. Applying the “shrinking-sphere” method, we also prove the symmetry under inversions of condensate solutions for some special cases.     

A Short Synthesis of (+)-Brefeldin C through Enantioselective Radical Hydroalkynylation

A very concise total synthesis of (+)-brefeldin C starting from 2-furanylcyclopentene is described. This approach is based on an unprecedented enantioselective radical hydroalkynylation process to introduce the two cyclopentane stereocenters in a single step. The use of a furan substituent allows a high trans diastereoselectivity to be achieved during the radical process and it contains the four carbon atoms C1–C4 of the natural product in an oxidation state closely related to the one of the target molecule. The eight-step synthesis requires six product purifications and it provides (+)-brefeldin C in 18 % overall yield.