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Concrete conditions for realizability of moment measures via quadratic modules

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If you have a question about this talk, please contact Mustapha Amrani.

Polynomial Optimisation

In this talk, we intend to give a brief introduction to the realizability problem presenting a new approach based on its deep connection to the moment theory. This is not only the key idea which allowed us to get interesting results about the full realizability problem, but it is also the base for a new research direction which links the realizability problem to polynomial optimization theory.

The realizability problem naturally arises from applications dealing with systems consisting of a huge number of components. The investigation of such systems is greatly facilitated if the attention is restricted to selected physical parameters (usually correlation functions) which encode the relevant structure of the system. The realizability problem exactly addresses the question whether a given candidate correlation function actually represents the correlation function of some random distribution.

We will present necessary and sufficient conditions for the realizability of an infinite sequence of moments given by Radon measures on a closed semi-algebraic subset of the space of distributions. Our approach is based on the interpretation of the realizability problem as an infinite dimensional moment problem and it exploits the quadratic module generated by the polynomials defining the semi-algebraic set in question. This result determines realizability conditions which can be more easily verified than the Haviland type conditions developed by A. Lenard. Moreover, it completely characterizes the support of the realizing process giving a solution of the full realizability problem for random measures.

In conclusion, we will introduce some open questions related to the extension of our result to the truncated case and we will sketch our idea to connect the realizability problem with polynomial optimization problems in order to get bounds for some material properties of great physical interest.

This talk is part of the Isaac Newton Institute Seminar Series series.

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