Moments and Positive Polynomials
We are interested in the very rich theory of moments and its dual facet, that of positive polynomials. We have been a pioneer in the development of what is called the MomentSOS hierarchy (see also the Lasserre hierarchy and SumofSquare Optimization), and notably for global optimization. This approach is very natural and the resulting hierarchy of semidefinite relaxations perfectly matches both sides of the same theory, namely the theory of moments and its dual theory of positive polynomials. As a byproduct one also obtains KarushKuhnTucker global optimality conditions, in which the multipliers are nonnegative polynomials instead of scalars as in the usual (local) optimality conditions. In fact the MomentSOS hierarchy also applies to solving the Generalized Moment Problem (GMP) (with polynomial data) whose list of important applications is endless. In fact, global polynomial optimization is the simplest instance. To appreciate the impact of this methodology in and outside optimization, see for instance:
 Optimization over polynomials: Selected Topics, M. Laurent, Proceedings of ICM 2014 Congress, Seoul, 2014.
 On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields, TomLukas Kriel and M. Schweighofer, Foundations of Computational Mathematics 19, 1223–1263 (2019)
and especially for its impact in Combinatorial Optimization and Theoretical Computer Science, see for instance:
 SumofSquares proofs and the quest toward optimal algorithms, B. Barak and D. Steurer, Proceedings of ICM 2014 Congress, Seoul, 2014.
 Candidate Lasserre Integrality Gap For Unique Games, S. Khot and D. Mohskovitz, Electronic Colloquium on Computational Complexity, Report No. 142 (2014).
 High dimensional estimation via SumofSquares proofs, D. Steurer, Proceedings of ICM 2018 Congress, Rio de janeiro
 The MomentSOS Hierarchy, J.B. Lasserre, Proceedings of ICM 2018 Congress, Rio de Janeiro
In particular this methodology has been used:
 for Motion Planning in Robotics. See the algorithm NUROA ( Numerical Roadmap Algorithm) and the associated paper:
NUROA: A Numerical Roadmap Algorithm, Reza Iraji, Hamidreza Chitsaz. Proceedings IEEE CDC Conference, Los Angeles, December 20214. arXiv:1403.5384
 For solving the Optimal Power Flow Problem, an important problem for the optimal management of energy networks. See e.g. the recent papers:

Advanced optimization methods for power systems, P. Panciatici, M.C. Campi, S. Garratti, S.H. Low, D.K. Molzhan, A.X. Sun and L. Wehenkel. Proceedings 18th Power Systems Computation Conference, Wroclaw, Poland, August 2014.

SparsityExploiting MomentBased Relaxations of the Optimal Power Flow Problem, D.K. Molzahn and I.A. Hiskens, IEEE Transactions on Power Systems, 2015. arXiv:1404.5071

Global Optimization for power dispatch problems based on theory of moments, J. Tian, H. Wei and J. Tan, Electrical Power and Energy Systems 71, 2015, pp. 184194.

Lasserre hierarchy for largescale polynomial optimization in real and complex variabales, D. Molzahn, C. Josz, SIAM J. Optim. 28 (2018), pp. 10171048

Deciding robust feasibility and infeasibility using a set containment approach: An application to stationary passive gas network operations, D. Haussmann, F. Liers, M. Stingl, J.C. Vera, SIAM J. Optim. 28 (2018), pp. 24892517

MomentSOS relaxation of the medium term hydrothermal dispatch problem, F. Cicconet, K.C. Almeida, Int. J. Elec. Power & Energy Systems 104, pp. 124133, 2019
 In Computer Vision and Pattern Recognition :
 Convex Relaxations for Consensus and NonMinimal Problems in 3D Vision, T. Probst, D. Dani Paudel, A. Chhatkuli, L. Van Gool, IEEE Int. Conf. Comp. Vision (ICCV), 2019, pp. 1023310242.
 Certifiably optimal and robust camera pose estimation from points and lines, Lei Sun, Zhongliang Deng, IEEE Access, 2020.
 In perfect shape: Certifiably optimal 3D shap reconstruction from 2D landmarks, Heng Yang, L. Carlone, Proceedings of teh IEEE/CVR conference on Computer Vision and Pattern Recognition, (CVPR), pp. 621630, 2020
 In Computer Graphics: Hexahedral mesh repair via SumofSquares relaxations. Z. Marschner, D. Palmer, P. Zhang, J Solomon, Eurographics Symposium on Geometry Processing 39, No 5, pp. 133147, 2020
 In IoT Signal Processing for radar applications: Localization with 1Bit passive radars in Narrow Band IoTapplications using multivariate polynomial optimization, Saeid Sedighi, Kumar Vijay Mishra, Bhavani Shankar M. R., Björn Ottersten (2021), IEEE Transactions on Signal Processing 69, pp. 25252540.
 In Signal Processing for SuperResolution, extending seminal work of Candès & FernandezGranda: Exact solutions to Super Resolution on semialgebraic domains in higher dimensions, De Castro Y, Gamboa F, Henrion D., Lasserre J.B. (2017) , IEEE Trans. Info. Theory 63, pp. 621630, and also:

Low Complexity Beamspace Super Resolution for DOA Estimation of Linear Array, Pan Jie, Fu Jiang, Sensors 20, No 8, 2020.

Sparse signal reconstruction for nonlinear models via piecewise rational optimization, A. Marmin, M. Castella, JC. Pequet, L. Duval, Signal Processing 179, 2021
 In Statistics for Optimal Design: Approximate Optimal Designs for Multivariate Polynomial Regression, De Castro Y, Gamboa F, Henrion D., Hess R., Lasserre J.B. (2019) , Annals of Statistics 47, pp. 125157.
 In Machine Learning for Certification of Robustness for Neural Networks: Lipschitz Constant Estimator of Neural Networks via Sparse Polynomial Optimization, F. Latorre, P. Rolland, V. Cevher, ICLR, December 2020.
 In Physics for bounding groundstate energy of interacting particule systems: Moment methods in energy minimization: New bounds for Riesz minimal energy problems, D. de Laat, Trans. Amer. Math. Soc. 373, pp. 14071453, 2020
 In Quantum Information for several problems in entanglement theory: Complete hierarchies of efficient approximations to problems in entanglement theory, Jens Eisert, Philipp Hyllus, Otfried Gühne, and Marcos Curty, Physical Review A 70, 062317, 2004, and Ruling out static latent homophily in citation networks, P. Wittek, S. Darany, l G. Nelhans, Scientometrics 110(2), pp. 765777, 2016, Semideviceindependent selftesting of unsharp measurments, N. Miklin, J. Borkala, M. Pawlowski, Physical Review Research 2, pp. 033014, 2020. Correlations constrained by compsite measurement, Lukasz Czekaj, Ana Bel´en Sainz, John H. Selby, an Michal Horodecki, PArXiv:2009.04994, 2020.
 in traffic networks for bounding travel time: Momentbased travel time reliability assessment with Lasserre's relaxation, Xiangfeng Ji, Xuegang (Jeff) Ban, Jian Zhang, Bin Ran, Transportmetrica B: Transport Dynamics 7(1), pp. 401422, 2019
 In models for control of chemo and immunotherapy: Robust Optimal Controlbased Design of Combined ChemoandImmunotherapy Delivery Profile, K. Moussa, M. Fiacchini, M. Alamir, IFAC Papers On Line 5226 (2019), pp. 76–81
This approach also permits to obtain (numerically) bounds on measures that satisfy prespecified moment conditions. As a consequence we can apply these techniques to provide upper and lower bounds in several problems of performance evaluation. For instance, for option pricing models in financial mathematics based on diffusions (such as Black and Scholes, OrnsteinUlhenbeck, CoxIngersoll interest models) we may obtain good upper and lower bound on e.g. Asian and Barrier options. The method is generic to diffusion type models. It can be also applied (and provides a new methodology) :
 For solving weak formulations of Optimal Control problems and of a class of nonlinear hyperbolic PDEs (e.g. the Burgers equation).
 For solving some problems in computational geometry and probability (e.g. Lebesgue and/or Gaussian volume of semialgebraic sets, Lebesgue decomposition of a measure).
 For modelling chanceconstraints (or probabilistic constraints) and distributionally robust chanceconstraints in control or optimization under uncertainty.
 For SuperResolution in Signal Processing, i.e., to recover a sparse signal (an atomic signed measure) from a few moments.
 For Optimal Design in Statistics,
to cite a few of its applications, some described in the book :
The MomentSOS Hierarchy:
Lectures in Lectures in Probability, Statistics, Computational Geometry, Control and Nonlinear PDEs
by D. Henrion, M. Korda and J.B. Lasserre, World Scientific Publishing, Singapore, 2020.
More generally, we are also interested in all aspects of positivity of polynomials (and even semialgebraic functions) on basic semialgebraic sets, including certificates of positivity with tractable characterization (to be amenable to practical computation).