Computational Methods in Real Algebraic Geometry and Applications

At its most basic algebraic geometry studies the solution sets of systems of polynomial equations. Over the last several decades, beginning with the development of Gröbner basis, a wide variety of computational methods have arose which allow us to study these solutions quite effectively, particularly over the complex numbers. Many problems in science and engineering can be reduced to the problem of solving systems of polynomial equations, making it natural to apply these computational tools to study such applied problems. However, often the natural setting for such applied problems is not the complex numbers, but is rather the real numbers or some subset of the real numbers (e.g. the positive real numbers). Unlike the complex case when working over the real numbers there are relatively few computational tools available and many of these are extremely computationally expensive or only give partial answers to he questions of interest to scientists and engineers. Guided by the interests of the student this project would explore current symbolic, numeric, and combinatorial methods for studying real solutions to polynomial systems and how these can be employed to solve applied problems. The project could also explore how existing methods could be improved or adapted for better performance on applications of interest. Applications considered could range from systems biology and chemical reaction networks, to problems in robotics, or to problems coming from branches of pure mathematics such as complex analysis or algebraic geometry, depending on the interests of the student.