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Infinite dimensional moment problem: open questions and applications
Infinite dimensional moment problems have a long history in diverse applied
areas dealing with the analysis of complex systems but progress is hindered by
the lack of a general understanding of the mathematical structure behind them.
Therefore, such problems have recently got great attention in real algebraic
geometry also because of their deep connection to the finite dimensional case.
In particular, our most recent collaboration with Murray Marshall and Mehdi
Ghasemi about the infinite dimensional moment problem on symmetric algebras of
locally convex spaces revealed intriguing questions and relations between real
algebraic geometry, functional and harmonic analysis. Motivated by this
promising interaction, the principal goal of this paper is to identify the main
current challenges in the theory of the infinite dimensional moment problem and
to highlight their impact in applied areas. The last advances achieved in this
emerging field and briefly reviewed throughout this paper led us to several
open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated
reference
Identifying Mixtures of Mixtures Using Bayesian Estimation
The use of a finite mixture of normal distributions in model-based clustering
allows to capture non-Gaussian data clusters. However, identifying the clusters
from the normal components is challenging and in general either achieved by
imposing constraints on the model or by using post-processing procedures.
Within the Bayesian framework we propose a different approach based on sparse
finite mixtures to achieve identifiability. We specify a hierarchical prior
where the hyperparameters are carefully selected such that they are reflective
of the cluster structure aimed at. In addition this prior allows to estimate
the model using standard MCMC sampling methods. In combination with a
post-processing approach which resolves the label switching issue and results
in an identified model, our approach allows to simultaneously (1) determine the
number of clusters, (2) flexibly approximate the cluster distributions in a
semi-parametric way using finite mixtures of normals and (3) identify
cluster-specific parameters and classify observations. The proposed approach is
illustrated in two simulation studies and on benchmark data sets.Comment: 49 page
Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
R-matrices are the solutions of the Yang-Baxter equation. At the origin of
the quantum group theory, they may be interpreted as intertwining operators.
Recent advances have been made independently in different directions.
Maulik-Okounkov have given a geometric approach to R-matrices with new tools in
symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a
conjecture on the categorification of cluster algebras by using R-matrices in a
crucial way. Eventually, a better understanding of the action of
transfer-matrices obtained from R-matrices led to the proof of several
conjectures about the corresponding quantum integrable systems.Comment: This is an English translation of the Bourbaki seminar 1129 (March
2017). The French version will appear in Ast\'erisqu
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