Moment Problems and Applications - Minisymposium at DMV2015
in Memoriam Murray Angus Marshall 24.3.1940 - 1.5.2015

Schedule

Titles and Abstracts

• Sergio Albeverio :
  Title. Some moment problems in one to infinite dimensions.
  Abstract.
  We present some problems of the theory of the moment problem as related to infinite dimensional analysis, the theory of stochastic process and quantum (field) theory. Relations with spectral theory and some integrable systems are also discussed.


• Sabine Burgdorf :
  Title. The operator theoretic moment problem.
  Abstract.
  The moment problem has a direct application in polynomial optimization, where one wants to optimize the value a polynomial can attain over a given set. Several interesting problems in polynomial optimization turn out to be hard, but a suitable method to approximate these problems is the so-called Lasserre relaxation, i.e. one replaces positive polynomials by sums of squares. This results in a semidefinite program (SDP) which can be dualized via conic duality, resulting in an SDP where one optimizes over linear functionals. In this step the moment problem plays a crucial role: If the optimizing linear functional turns out to be a moment function, i.e. its moments are the moments of a positive measure, then the relaxation is actually exact and one obtains the optimal value of the original polynomial optimization problem.
  The classical moment problem for linear functionals on polynomials in commuting variables has a long and fruitful history, whereas the investigation of the moment problem in non-commuting variables is relatively new. The latter is closely related to operator theory and one can end up in possibly infinite-dimensional spaces. In this talk we will introduce the non-commutative equivalent(s) of the moment problem and show how the theory can be applied to polynomial optimization problems arising in quantum theory.


• Maria Infusino in memory of Murray Marshall:
  Title. A continuous moment problem for locally convex spaces.
  Abstract.
  This talk was supposed to be given by Murray A. Marshall who suddenly passed away on the 1st of May 2015 and is given in his memory.
  It is explained how a locally convex (lc) topology \(\tau\) on a real vector space \(V\) extends naturally to a locally multiplicatively convex (lmc) topology \(\overline{\tau}\) on the symmetric algebra \(S(V)\). This allows application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of \(\overline{\tau}\)-continuous linear functionals \(L: S(V)\rightarrow \mathbb{R}\) satisfying \(L(\sum S(V)^{2d}) \subseteq [0,\infty)\) (more generally, of \(\overline{\tau}\)-continuous linear functionals \(L: S(V)\rightarrow \mathbb{R}\) satisfying \(L(M) \subseteq [0,\infty)\) for some \(2d\)-power module \(M\) of \(S(V)\)) as integrals with respect to uniquely determined Radon measures \(\mu\) supported by special sorts of closed balls in the dual space of \(V\). The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and Šifrin. It is more general because \(V\) can be any locally convex topological space (not just a separable nuclear space), the result holds for arbitrary \(2d\)-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that \(L : S(V) \rightarrow \mathbb{R}\) is \(\overline{\tau}\)-continuous (not just that \(L\) is continuous on the homogeneous parts of degree \(k\) of \(S(V)\), for each \(k\ge 0\)).
  This is a joint work with Mehdi Ghasemi, Salma Kuhlmann and Murray Marshall.


• David Kimsey :
  Title. Multidimensional moment problems, the subnormal completion problem and cubature rules.
  Abstract.
  Given a positive integer \(t\), a set \(K \subseteq \mathbb{R}^d\) and a real multisequence \(s = \{ s_{\gamma_1, \ldots, \gamma_d} \}_{0 \leq \gamma_1+\ldots+\gamma_d \leq m}\) we will formulate new moment matrix conditions for \(s\) have a \(K\)-representing measure \(\sigma= \sum_{q=1}^t \varrho_q \delta_{w_q}\) with \(t\) atoms, i.e., $$s_{\gamma_1, \ldots, \gamma_d} = \int_{\mathbb{R}^d} x_1^{\gamma_1} \cdots x_d^{\gamma_d} d\sigma(x_1, \ldots, x_d) \quad {\rm for} \quad 0 \leq \gamma_1+\ldots + \gamma_d \leq m$$ and $$w_1, \ldots, w_t \in K.$$ Using these conditions, we will establish new minimal inside cubature rules for planar measures in \(\mathbb{R}^2\) and also pose a solution to the subnormal completion problem in \(d\) variables, i.e., given a collection of positive numbers \(\mathcal{C} = \{ \alpha_{(\gamma}^{(1)}, \ldots, \alpha_{\gamma}^{(d)}) \}_{0 \leq |\gamma| \leq m}\) we wish to determine whether or not \(\mathcal{C}\) gives rise to a \(d\)-variable subnormal weighted shift operator whose initial weights are given by \(\mathcal{C}\). We will also highlight recent results for a full moment problem in a countably infinite number of variables and briefly discuss an application to stochastic processes.
  This talk is partially based on joint work with Daniel Alpay and Palle Jorgensen.


• Ognyan Kounchev :
  Title. Multidimensional moment problem on the sphere and application to cubature formulas on the sphere.
  Abstract.
  A recent breakthrough was the discovery that the spherical polynomials have an Almansi type representation. Hence, one may follow the framework of the multidimensional moment problem developed in previous works for the euclidean space: arxiv.0509380 and arxiv.0802.0023 We formulate the full moment problem on the sphere \(S^n\), and define also the truncated moment problem. We provide a solution for the truncated pseudo-positive moment problem. As a by-product we discover new cubature formulas on the sphere. The remarkable thing is that the new cubature method satisfies the classical criterion for weak* convergence of Osgood, Vitali, Lebesgue, Polya, and Banach.
  This is a joint work with Hermann Render.


• Christian Kuehn :
  Title. Moment Closure - A Brief Review.
  Abstract.
  Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one moment, a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure. In this talk, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.


• Eugene Lytvynov : CANCELLED!
  Title. A moment problem for random discrete measures.
  Abstract.
  Let \(X\) be a locally compact Polish space. A random measure on \(X\) is a probability measure on the space of all (nonnegative) Radon measures on \(X\). Denote by \(\mathbb K(X)\) the cone of all Radon measures \(\eta\) on \(X\) which are of the form \(\eta=\sum_{i}s_i\delta_{x_i}\), where, for each \(i\), \(s_i>0\) and \(\delta_{x_i}\) is the Dirac measure at \(x_i\in X\). A random discrete measure on \(X\) is a probability measure on \(\mathbb K(X)\). The main result of the talk states a necessary and sufficient condition (conditional upon a mild a priori bound) when a random measure \(\mu\) is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure \(\mu\). Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterisation via moments is given when a random measure is a point process.
  This is a joint work with Yuri Kondratiev and Tobias Kuna.


• Frank Vallentin :
  Title. New upper bounds for the density of translative packings of superspheres.
  Abstract.
  In this talk I will present new upper bounds for the maximum density of translative packings of superspheres in three dimensions (unit balls for the \(l^p\)-norm). This will give some strong indications that the lattice packings experimentally found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proof of our new upper bounds is computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.
  This is joint work with Maria Dostert, Cristobál Guzmán, and Fernando Mário de Oliveira Filho.


• Victor Vinnikov :
  Title. Multievolution scattering systems and interpolation problems on the polydisc.
  Abstract.
  In dimension one, moment problems are closely related to classical interpolation problems for bounded analytic functions of the unit disc (or on the upper half plane). The analogues of these interpolation problems in higher dimension were considered largely intractable till the groundbreaking work of Agler in the late 1980s -- early 1990s who discovered the (correct) relationship with multivariable operator theory (von Neumann inequality) and (as it became apparent only quite recently) certain sums of squares decompositions. I will review some of these developments, with a particular emphasis on the joint work with Joe Ball and Cora Sadosky that both provides a more constructive approach and shows a relation to the work of Geronimo and Woerdeman on the two-dimensional trigonometric moment problem.