Matteo Bechere
Nationality: Italian
Host: University of Konstanz
| Background: After obtaining his Scientific Matura, Matteo enrolled in the Bachelor’s program in Mathematics at the University of Cagliari, graduating cum laude in February 2023. He then continued his studies with the Master’s program in Pure Mathematics at the same university. During his Master’s studies, he participated in the University’s Excellence Programme and spent six months at the University of Cadiz as an Erasmus student. In July 2024, he received his Master’s degree cum laude, having written a thesis on the Moment Problem in the compact case. |
| Master thesis: The Moment Problem in the compact case: localization of the support. Abstract: The thesis deals with the following generalization of the classical full moment problem. Given a unital, commutative, real algebra A and a real valued linear functional L on A, does there exist a representation of L as an integral with respect to a Radon measure on the character space of A endowed with the weak topology? After a brief overview of the moment problem, from the classical formulation for polynomials in finitely many variables to the general formulation considered in this thesis, we present a generalized version of Riesz-Haviland’s celebrated theorem. This result establishes a beautiful duality between moment problems and Positivstellensätze also in the general setting here considered. Finally, we focus on the moment problem in the compact case, namely when the representing measure has compact support. A recent characterization of all linear functionals admitting a compactly supported representing measure is given in terms of a growth condition on the starting functional L, as well as the positivity of L on an Archimedean quadratic module of A, and also the continuity of L with respect to a seminorm defined on A. This result also includes the exact localization of the support of the representing measure for L. Note that in the previous literature, to the best of our knowledge, the support was only proved to be contained into some closed subset of the character space, but no explicit support description was given. The support, by virtue of the above characterization, is described in three equivalent ways: one in terms of the growth condition on L, one involving the smallest seminorm on A with respect to which L is continuous, and another concerning the largest Archimedean quadratic module of A on which L is positive. |
| Research interests: He has always been passionate about Measure Theory, and recently he has grown more and more interested in moment problems, their connection to Real Algebraic Geometry, and tensor decompositions. |
| Goals in TENORS: The name of the project he will take part in TENORS is “Tensor decompositions for sums of even powers of real polynomials”. The main goal is to find tensor decompositions of forms as sums of even powers and to exploit the Archimedean Positivstellensatz for sums of even powers modules for polynomial optimization. |
| Hobby: When he does not have to study mathematics, in his free time, he still studies mathematics. Sometimes he also likes to go for a walk, practice sports such as bodyweight training or padel, or play video games. |
