Systems of multivariate polynomial equations are ubiquitous throughout mathematics and neighboring scientific and engineering fields such as kinematics, computer vision, power flow systems, chemical reaction networks, and systems biology. For such polynomial systems arising in applications, typically only real solutions are of interest including synthesizing a mechanism to perform given tasks, three-dimensional image reconstruction in computer vision, and equilibria of a dynamical system. Of course, if a polynomial system has only finitely many solutions over the complex numbers, then one approach is to compute all of the complex solutions and retain only the subset of real solutions. Although such an approach has been used effectively on problems of modest size, typically only a very small fraction of the complex solutions are real. Thus, a key research area is to develop numerical approaches for computing and manipulating only the real solution set to a system of polynomial equations yielding the subject area of real numerical algebraic geometry.
The field of complex numbers is algebraically closed and the fundamental theorem of algebra posits the existence of solutions. However, such theory no longer holds over the real numbers. Nonetheless, there is a growing body of research and software tools
for numerically computing only real solutions. Additional techniques using random exploration have been combined with methods to validate that all real solutions have been computed. Random exploration with topological data analysis and machine learning can learn information about the real solution set such as the number of connected components. Such techniques can be employed to build maps of robot configuration spaces as an enabler for robot path planning. During this MRC, participants will learn about methods in real numerical algebraic geometry, use software implementations, and explore various applications of computing real solutions to systems of polynomial equations.
To be successful in this MRC, participants should have some background in numerical linear algebra and Newton's method together with some familiarity with numerical methods for solving ordinary differential equations (e.g., Euler and Runge-Kutta methods). For example, these topics would have been covered in an introductory course in numerical analysis. Additionally, each participant should have at least either a basic knowledge of algebraic geometry over the complex numbers or knowledge in an application area of interest for this MRC such as kinematics, computer vision, or chemical reaction networks.
Applications will be accepted on MathPrograms.org from Friday, December 13, 2024 through Saturday, February 15, 2024 (11:59 p.m. EST).
