In recent years, together with several collaborators I have initiated a new direction of research, focussing on the interaction between probabilistic methods and tools from infinite-dimensional variational calculus – allowing one to analytically and explicitly study the fluctuations of random objects having the form of non-linear transformations of general random fields. The aim of this project is to consolidate these results, as well as to significantly expand them in several new directions, in order to attack a number of challenging open questions in the field of stochastic geometry – with particular emphasis on the geometric structure on the level and excursion sets of Gaussian and non-Gaussian random fields defined on homogeneous spaces and on the high-dimensional behaviour of geometric objects arising in the theory of random graphs and combinatorial optimisation. We also plan to study some open problems connected to real polarization problems in Banach space theory. From a foundational standpoint, our principal aims will be the following: (i) To extend the existing theory of entropic limit theorems on a Gaussian space (or in the more general framework of Markov semigroups) in order to include random variables with infinite polynomial complexity; (ii) to prove novel functional and concentration inequalities based on the use of variational techniques; (iii) to systematically attack the problem of obtaining explicit Berry-Esseen bounds in limit theorems involving elements of the second andhigher Wiener chaoses of a given Gaussian field; (iv) to characterise the set of adherent points (in the sense of weak convergence) associated with the class of probability distributions within a fixed sum of Wiener chaoses; (v) to prove novel concentration and super-concentration estimates on spaces of random point configurations, with special emphasis on improved Poincaré inequalities; (vi) to eventually solve the long-standing ‘Gaussian Product Conjecture’ and the ‘Real Polarisation Problem’.From a geometric perspective, we will focus on the following tasks: (a) to quantitatively study second order fluctuations of geometric quantities associated with sums of random waves, that is, of random fields that are solutions to the Schrödinger equation associated with a given Riemaniann manifold – with special emphasis on cancellation phenomena and the dichotomy between non-Gaussian and Gaussian fluctuations on compact manifolds and on Euclidean domains – such geometric quantities include nodal lines, nodal intersections, critical points, Euler-Poincaré characteristics and occupation densities; (b) to establish a number of universality statements for the objects at point (a), thus bypassing the customary Gaussianity assumptions that are ubiquitous in the literature, (c) to develop a new approach towards second order results on configuration spaces, based on a generalised notion of stabilization, with specific emphasis on combinatorial optimisation problems, and via the use of carré-du-champ operators ; (d) to develop a new panoply of results, characterising the superconcentration of geometric objects associated with random point configurations. Achieving the main tasks of our project will require to successfully combine tools from different areas of Mathematics, namely: stochastic analysis, functional analysis, geometry, combinatorics and arithmetics. For this, we aim at hiring two scientific collaborators that will work in our group for a period of 2 1/2 and 2 years, respectively, that should be ideally proficient in two or more of the above areas. Part of the budget will be devoted to the consolidation and improvement of the PI’s international collaborations, as well as to the organisation of relevant scientific events, in order to enhance the dissemination and visibility of the obtained results.