Bounds for real solutions to structured polynomial systems Understanding the real solutions to systems of polynomial equations is a difficult question with many applications. In particular, a non-trivial lower bound is an existence proof for solutions and non-trivial upper bounds give complexity bounds. The best-known upper bound is due to Khovanski and is unrealistically large. A recent breakthrough of Li, Rojas, and Wang suggests the possibility of more reasonable upper bounds. Lower bounds are a very recent phenomenon, having arisen in quantum cohomology and (separately) with the Wronski map in Schubert calculus. The results, though, are striking: "Most" rational curves of degree d interpolating 3d-1 real points in the plane are real, every rational function with only real critical points is real. In this talk, I will describe this background and then discuss recent work giving upper and lower bounds on the numbers of real solutions to some sparse polynomial systems. This is jout work with Soprunova (lower bounds) and with Bihan, Bertrand, and Rojas (upper bounds).