Structure of Real Algebraic Varieties via Monodromy and Topology

The structure and behavior of real algebraic varieties can be insightful for real world applications where computations over the complex numbers are not meaningful. First, I will present on a new definition of monodromy action over R which encodes tiered characteristics regarding real solutions. Examples will be given to show the benefits of this definition over a naive extension of the monodromy group (over C). In addition, an application in kinematics will be discussed to highlight the computational method and impact on calibration. Next, I will discuss a method for computing topological information such as Euler characteristic, genus, Betti numbers, and the generators of the fundamental group for smooth, compact, and orientable real algebraic surfaces. This underlying approach is via a cell decomposition computed using numerical algebraic geometry from the software Bertini_Real and then creating a simplicial complex modelling the real algebraic surface. The software Javaplex is then used to compute the desired topological information. Several examples will be discussed to demonstrate the approach.