Real Polynomials: Counting and Stability

Understanding the real solutions of polynomials or systems of polynomial equations is at the heart of many problems both in the realm of pure mathematics and in the mathematical modeling of networks in systems biology, in computer aided geometric design, or in the control of mechanisms. While there exist a lot of celebrated results (for example, bounds exploiting the signs of the coefficients of a univariate polynomial or bounds taking into account the number of terms), even some basic questions are still unsolved. There has been a recent explosion of activity in this area, where different tools have started to be connected in new and unexpected ways, such as tools from the theory of matroids, from fewnomial theory, or from convex algebraic geometry. This workshop will bring together experts from close but different areas in real algebraic geometry to collaborate and will lead to further progress (both theoretical and practical) on a number of problems on the real solutions of polynomials. Specific consideration is given on the problem of counting and bounding the number of real roots as well as on questions of stability.