Francesco Mascarin
Nationality: Italian
Host: MPI
| Background: Francesco Maria is completing his Master’s degree in Mathematics at the University of Trieste, in northern Italy, where he also earned his Bachelor’s degree. After developing a solid foundation in general mathematics, he is now focusing on computational aspects of algebra and algebraic geometry, with a particular interest in algorithm design. His Master’s thesis explores homotopy continuation methods for solving polynomial equations, specifically aimed at numerically computing the degree of Gibbs varieties. His interest in numerical algebraic geometry led him to an internship at the Max Planck Institute for Mathematics in the Sciences (MPI MiS) in September and October 2024. During this time, he collaborated with the Numerical Algebraic Geometry research group on a project studying the Kuramoto model for coupled oscillators. Since the primary focus of research is on understanding equilibrium states – expressed as solutions to systems of algebraic equations he investigated a novel parameterization utilizing Lissajous varieties. This experience allowed him to work with researchers from diverse backgrounds, to participate in engaging talks and workshops, and to deepen his understanding of the field. |
| Master thesis: Homotopy Continuation Method for Computing the Degree of Gibbs Varieties |
| Abstract: This thesis focuses on using homotopy continuation methods to numerically compute the degree of Gibbs varieties. Homotopy continuation is a numerical method in algebraic geometry for solving polynomial systems. The thesis introduces this method, including the Euler-Newton predictor-corrector phase, endgames, and projective patches, while comparing it with symbolic approaches such as Grobner basis methods. Gibbs varieties, a non-commutative extension of toric varieties, play a determinantal role in semidefinite programming. The degree of a variety is defined as the number of intersection points between the variety and a generic linear space of complementary dimension, which reduces to solving a zero-dimensional polynomial system. This computation is carried out using homotopy continuation methods. |
| Research interests: His research interests span several areas of mathematics, including computational algebra, commutative algebra, nonlinear algebra, algebraic geometry, and numerical algebraic geometry. |
| Goals in TENORS: As a TENORS candidate, he aims to advance the understanding of Gibbs varieties by developing new theoretical results and efficient algorithms, particularly in the context of entropic regularization for semidefinite programming and quantum optimal transport. He also seeks to enhance his research and soft skills by collaborating with fellow researchers in the TENORS network and the broader scientific community. Through these interactions, he hopes to exchange ideas and enjoy meaningful moments of personal and professional growth. |
| Hobby: Francesco Maria enjoys exploring new places, meeting diverse people, and has a passion for nature and historical architecture. He has a keen interest in nutrition and hypertrophy training. In the future, he hopes to participate in bodybuilding training courses to further his knowledge and skills. |
