Algebraic Geometry Seminar

Interference is the major bottleneck in wireless communications. We study the so-called interference channel model, which has multiple point-to-point communication links, all mutually interfering. Restricting attention to the natural class of linear strategies, the question is how many dimensions can be used at each transmitter while still enabling each receiver to decode. The problem reduces to the inherently geometric one of understanding a set of incidence relations among subspaces. For the three-user interference channel, we describe an explicit optimal construction. For K>3 users we obtain a partial generalization, but several clean mathematical problems remain unsolved, which I will describe. The talk is based on joint work with Dustin Cartwright and David Tse.

Coamoebas of reduced A-discriminants arise when studying the convergence of Mellin-Barnes integrals for the solutions to the associated A-hypergeometric system. Nilsson and Passare described these coamoebas, in dimension 2, as topological chains in the 2-torus T^2 with piecewise-linear boundary. This boundary, with opposite orientation, is the boundary of a natural centrally symmetric zonotope in T^2, and they showed that the union of these two chains is a cycle equal to vol(A)\cdot[T^2], i.e., it covers T^2 vol(A)-many times. Their proof could not be generalized to higher dimensions, and it gave no intuition about the multiplicity. In this talk, which is joint work with Passare, we give a new, simpler, and elementary proof of these facts which identifies the multiplicity from the pushforward of a homology cycle in a torus T^A to T^2. The ingredients of this proof generalize to all dimensions, giving hope for a complete understanding of A-discriminant coamoebas.
Abstract