Francesco Maria Mascarin (DC6) is about to complete his secondment at CWI Amsterdam, under the supervision of Monique Laurent, after almost one year of collaboration.
During this secondment, he started a new research project with his supervisors, Monique Laurent and Simon Telen, which emerged spontaneously from discussions about previous joint work with Simon Telen and its possible applications to semidefinite programming. This collaboration led to a preprint in May, The algebraic boundary of graph elliptopes (https://arxiv.org/abs/2605.02484).
Elliptopes are sets of correlation matrices, namely positive semidefinite matrices with ones on the diagonal. Graph elliptopes are convex semialgebraic sets obtained by projecting the complete elliptope onto the variables corresponding to the edges of a graph. A natural question in algebraic geometry is to determine the algebraic boundary of convex semialgebraic sets: the smallest algebraic variety containing their Euclidean boundary. The algebraic boundary captures important structural properties of a convex body and provides a natural generalization of the notion of hyperplane arrangements for polytopes.
In this project, the algebraic boundary of graph elliptopes was studied, leading to several interesting results, including a description of the algebraic boundary of graph elliptopes for certain graph classes, such as cycle-completable graphs. As a byproduct, the project led to the proof of a conjecture of Sturmfels and Uhler concerning the degree of homogeneous cycle polynomials, as well as to the characterization of graph elliptopes that are spectrahedra. A spectrahedron is the feasible set of a semidefinite program, and the complete elliptope is, by construction, a spectrahedron. Graph elliptopes, on the other hand, are projections of spectrahedra, also known as spectrahedral shadows. It was shown that graph elliptopes are themselves spectrahedra if and only if the underlying graph is chordal.
This project was particularly relevant because it brought together two areas that the candidate encountered at different stages of his PhD: algebraic geometry, which he explored during the first months of his PhD at MPI Leipzig, and semidefinite programming, which he further developed during his stay at CWI Amsterdam.
During his stay in Amsterdam, the candidate also had the opportunity to attend numerous workshops and conferences, both inside and outside the Netherlands. Many interesting discussions arose after talks given throughout Europe about the projects developed so far. In particular, the candidate attended several workshops organized within the TENORS network, in Konstanz,Tromsø, and Trento, as well as events outside the network in Copenhagen, Trieste, Edinburgh, Narvik, and Durham, and further research meetings in the Netherlands, including in Lunteren, Tilburg, and Eindhoven. These trips provided opportunities to establish many important connections and led to meaningful research problems and new questions, leaving the candidate with several promising directions to investigate in the future.
Apart from research, the candidate also spent a magnificent time enjoying the city of Amsterdam and participating in the social events organized by CWI, such as the Christmas and Summer parties. Through these events, he had the opportunity to meet numerous people, ranging from new colleagues to prominent figures in mathematics and computer science, such as Guido van
Rossum, who created the famous programming language Python while working at CWI.
Following the completion of his second project, the candidate started a follow-up project with Monique Laurent, with the ambitious goal of finding an algebraic-geometric characterization of cycle-completable graphs by studying the algebraic boundaries of their corresponding elliptopes. In parallel, the candidate is working on a solo project concerning Lissajous varieties, algebraic varieties that play an important role in the description of the algebraic boundary of graph elliptopes and that also arise in dynamical systems, connecting this work with his earlier project with Simon Telen. Moreover, he is in active discussions with Hongjin Wu, a postdoctoral researcher at CWI, who is working on global synchronization of the Kuramoto model. This topic
is closely related to the candidate’s previous work and may lead to further research collaborations.
After almost a year of research at CWI, the candidate is now ready to leave Amsterdam for his next secondment at HSBC, first in Singapore and then in London, to further enrich his research experience.