This project focuses on various aspects of branching processes in fixed, variable or random environments, whether they are single-type or multitype. We propose to identify the limit of Bienaymé-Galton-Watson trees conditioned by their total population through their coding by multi-indexed and matrix-valued random walks. Then we will study the problem of the extinction of a part of the population for continuous multitype branching processes. We will construct the continuous analogue of multitype Bienaymé-Galton-Watson trees. These continuous random trees will then be obtained in the stable case as scaling limits of the renormalized discrete trees. These continuous random trees will be associated with continuous multi-type branching processes. We will also study discrete-time multitype branching processes in random environments to obtain asymptotic properties of the corresponding population size and survival probability; in particular, the problems of large deviations and asymptotic normalization will be considered. To this end, we will first deepen the study of the products of random matrices, in particular through the study of the multidimensional processes corresponding to the linear action of these products of matrices. We will be particularly interested in the cases where these processes are conditioned to remain in a cone of the Euclidean space. We will then establish limit theorems (invariance principle, local limit theorem, ...) for these conditioned processes. We will finally focus on the fundamental branching martingale associated to these Bienaymé-Galton-Watson trees, defined from the corresponding products of random matrices.

This collaborative research project aims to bring together researchers from various areas - namely, geometry and topology, minimal surface theory and geometric analysis, and computational geometry and algorithms - to work on a precise theme around min-max constructions and waist estimates. These past few years, min-max techniques led to groundbreaking advances in the field of geometry, including the resolution of the Willmore conjecture and the Yau conjecture, using an approach developed by Marques and Neves. These breakthroughs combine analytical techniques from the Almgren-Pitts theory ensuring the existence of minimal hypersurfaces through min-max arguments and recent min-max estimates of Gromov and Guth based on topological considerations. Relying on different branches of geometry, analysis and topology, the resolution of these conjectures opened a new chapter of differential geometry by promoting the min-max theory and its applications, as Perelman's resolution of Poincaré's conjecture opened a new chapter with the Ricci flow. The central theme of this proposal is to study the geometry and topology of geometrical objects through Morse theory min-max processes on the space of cycles for various functionals measuring the size of the cycles. A special focus will be given to the implementation of new geometric constructions effective enough to lead to the development of algorithms in computational geometry. In the description of the project, we set forth three largely overlapping themes about minimal surface theory, quantitative homotopy theory, and combinatorial and non-combinatorial topology: (1) Minimal surface theory, (a) Index estimates, topology and classification, (b) Discrete surfaces in 3-manifolds and applications; (2) Quantitative homotopy theory, (a) Sweepout estimates in Riemannian geometry, (b) Pants decomposition, (c) Arborescent sweepouts, embedded graphs and algorithms; (3) Combinatorial and non-combinatorial topology, (a) Leray's acyclic cover theorem and the Kalai-Meshulam projection theorem, (b) From the selection lemma to waist theorems and back. This project calls for close collaboration between researchers from three scientific communities since progress in one area can often be adapted to solve problems in another area as illustrated by some of the questions raised in this proposal. By joining the complementary expertise of our consortium, our goal is to develop this min-max approach from different perspectives, making our project extremely coherent around this fast-growing topic.

This project is focused on three related topics in the field of higher-dimensional complex dynamics : bifurcations, wandering domains, and parabolic dynamics. Each of these topics has gone through recent breakthroughs by members of the project. Bifurcation theory in higher dimensions has been developped only recently. We intend to address e.g. the following questions: are bifurcation loci homegeneous, or are there stronger bifurcation sub-loci ? Can we extend the existing theory to wider classes of dynamical systems ? The first examples of wandering domains for two important classes of maps have been recently constructed by the scientific coordinator and two other members of the project. We plan to better describe the associated possible dynamics, and their repartition in parameter spaces. Finally, parabolic implosion techniques have been used crucially in one construction of wandering domains. We plan to contribute to the further development of these techniques.

This project focuses on the effect of stochastic perturbations on oscillatory phenomena in dynamical systems. Oscillations are present in a vast number of systems in physics, biology and chemistry. Noise acting on these systems may drastically modify the oscillation patterns, or, in the case of excitable systems, create oscillations that were absent in the unperturbed system. Systems displaying oscillations are by essence irreversible, and therefore the mathematical theory of their stochastic perturbations is still in its infancy. Recently a number of new mathematical techniques have emerged that promise substantial progress in the description of non-reversible systems. The aim of this project is to develop these methods further and to combine them in order to obtain effective tools for the study of oscillations in stochastic systems. These tools will be applied to the description of oscillations in dusty plasmas, and to several models originating in mathematical biology.

This project aims to study and further develop combinatorial objects appearing in the representation theory of Coxeter groups, Lie algebras or their generalizations (complex reflections groups, Kac-Moody algebras), and simultaneously, to use them for investigating discrete probabilistic models and their connections with problems in mathematical physics. There are numerous interactions between models of these types based on the combinatorics of partitions (conditioning random walks, percolation problems, Tasep, card shuffling, cut-off phenomenon). The aim this project is to develop these interactions by using new results and objects that were introduced recently in representation theory (crystal graphs, shifted Schur functions, Hall-Littlewood and Macdonald polynomials, basic sets, generalizations of the RSK-procedure etc.). One of the original features of this project is to propose a unified approach to these different themes.