### Concrete conditions for realizability of moment functions via
quadratic modules

Infusino, M *(University of Reading)*

Thursday 18 July 2013, 15:00-15:30

Seminar Room 1, Newton Institute

#### Abstract

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 generalized functions 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 Radon measures.
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#### Presentation

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