1. The Moment-SOS Hierarchy and its applications
June 30 & July 1
1:00pm - 2:30pm
A3-3-301 of BIMSA
Roughly speaking, the Generalized Problem of Moments (GPM) is an infinite-dimensional linear optimization problem (i.e., an infinite dimensional linear program) on (possibly several) convex sets of measures whose supports are basic semi-algebraic sets. From a theoretical viewpoint, the GPM has developments and impact in various area of Mathematics like Real algebraic geometry, Fourier analysis, functional analysis, operator theory, probability and statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPM has a large number of important applications in various fields like optimization, probability, mathematical finance, optimal control, control and signal processing, chemistry, cristallography, tomography, quantum information & computing, etc.
In its full generality, the GPM is untractable numerically. However when its data are algebraic, then the situation is much nicer. Indeed, the Moment-SOS hierarchy is a systematic numerical scheme based on a sequence of (convex) semidefinite programs of increasing size whose associated monotone sequence
of optimal values converges to the optimal value of the GPM. Sometimes (e.g. in global optimization) finite convergence takes place and is generic.
In the talk, we will introduce the Moment-SOS hierarchy, and briefly describe several of its applications, notably in optimization, probability & statistics, optimal control and PDEs ....
2. The Christoffel Function: Some applications, Connections and Extensions
Even though the Christoffel function (CF) is well-known in approximation theory and orthogonal polynomials, it is only recently that some of its remarkable properties have been shown to be useful in some other applications, like data analysis and mining (e.g. for outlier detection and support inference), and approximation of possibly discontinuous functions with no Gibbs phenomenon. So in this talk we will briefly introduce the CF and describe how some of its main features can be exploited in some applications. Moreover we will also describe connections of the CF with seemingly unrelated fields, like positive polynomials, Pell’s equation and equilibrium measure of compact sets, and if time permits, we will introduce some variants with interesting additional properties.
主讲人介绍
Jean Bernard Lasserre is a leading mathematician and SIAM Fellow, affiliated with LAAS-CNRS, the Toulouse School of Economics (TSE), and the Institute of Mathematics at the University of Toulouse. His research spans optimization, probability, statistics, and dynamical systems, with particular emphasis on applications in machine learning, data analysis, inverse problems, and nonlinear partial differential equations. He is especially known for his pioneering contributions to the duality between moment problems and positive polynomials, as well as for advancing the use of the Christoffel function in data science and its connections to diverse mathematical fields.
He was an invited speaker at the International Congress of Mathematicians (ICM 2018) and has received numerous prestigious international awards, including the John von Neumann Theory Prize, the Lagrange Prize, the Khachiyan Prize, and the Grand Prize of the INRIA–French Academy of Sciences. He currently holds the Chair in “Polynomial Optimization” at ANITI, one of France’s national institutes for artificial intelligence. His recent international activities include leadership roles in the EU-funded POEMA and TENORS networks, as well as the DESCARTES project (CNRS@CREATE, Singapore).
