We propose to develop new approaches to solve unconventional mathematical questions inspired by the evolutionary dynamics of structured populations. The evolutionary dynamics of phenotypically structured populations are governed by stochastic and deterministic processes. These processes describe individual or collective dynamics and typically have various temporal regimes. We are interested in a class of models based on multi-scale integro-differential equations, describing large populations, with possible stochastic components. We will focus on the study of the impact of spatial heterogeneity, interaction of species or mixing of the gene pool on the Darwinian evolution of species. Such phenomena can be modeled by nonlocal parabolic Lotka-Volterra type equations or by nonstandard kinetic equations depending on whether or not mixing of the gene pool is taken into account. In the study of these unconventional equations several original features arise which require the development of new ideas and tools.

Compact Kähler manifolds are transcendental generalizations of complex projective manifolds, bearing a special metric structure. Inside the large class of compact Kähler manifolds with non-positive curvature, many aspects of such manifolds (topological, metric, geometric) are now reasonably well understood. However, the study of their moduli spaces or the search for minimal models in birational geometry have made it imperative to extend our understanding to non-compact or singular varieties. In that direction, we propose to investigate a wide array of questions using both analytic and algebraic methods, among which: - The uniformization problem - Analyze the behavior of canonical metrics at the boundary and construct explicit examples - Study the geometry of special subvarieties of moduli spaces or quotients of bounded symmetric domains - The Iitaka conjecture in the non-positive curvature case - Study hyperbolicity of base spaces of singular Calabi-Yau fibrations

Our project is devoted to the mathematical analysis of systems of nonlinear partial differential equations of the reaction-diffusion type which serve as the canonical model of the evolution of the density of several populations under the combined effects of reaction and diffusion. Such systems have a rich structure and often exhibit stable propagating solutions, or traveling waves. They have received much interest since the early 30's and nevertheless still present an active field of research. Furthermore, they typically arise in a wide range of applications from both physical and natural sciences. Therefore, the analytical study of such systems is often essential in the understanding of spreading phenomena, e.g. flame combustion, epidemics, ecological or bacterial invasions, spreading of neural activity. The typical questions that have been addressed by mathematicians and of crucial importance in applications can be formulated as follows. Suppose that at initially one is given an initial density of n species, what can be said about the asymptotic behavior of each density as time goes to infinity? Does each species spread and colonize the whole environment, and if yes, can one quantify this spreading? What is the final geometric configuration of the densities, if it exists? For scalar equations having a comparison principle or for monotone systems, answers to the above questions are relatively standard. However, without further assumptions on the form of the reaction terms, the analysis of propagation phenomena for systems is highly challenging. It is also very interesting from both a mathematical and modeling perspective since more involved and realistic behaviors are expected to happen in systems. Our main objective is precisely to develop techniques for systems that lack such comparison structures in order to provide insights on how interactions among species (including competition and prey-predator relationships) influence each other's spreading dynamics, and may even lead to the possibility of diffusion driven instabilities. Below, we have identified four outstanding problems which constitute the main directions of the Indyana project: - analysis of stage-invasion fronts and locking mechanisms; - analysis of spreading dynamics in heterogeneous and cross-diffusion systems; - analysis of spiraling dynamics in reaction-diffusion systems; - threshold of propagation in reaction-diffusion systems with applications. These are well identified problems in the reaction-diffusion community and present a very active field of research at which our group is at the leading edge. These projects are largely interconnected.

This is a project of fundamental research in mathematics, more specifically at the intersection of homotopy theory and algebraic geometry. Its main objective is to better understand the geometry of various types of moduli spaces in terms of Lie algebraic structures. The last two decades have seen great advances in the use of homotopy theoretic methods in geometry, driven by the development of derived geometry and higher category theory. Notably, it has lead to the introduction in recent years of a variety of generalisations of the notion of a Lie algebra. Through Koszul duality, these new types of Lie algebras provide important tools to study the infinitesimal and deformation theoretic properties of moduli spaces. The goal of this project is to expand this perspective in two directions: to algebraic geometry in positive characteristic, by further developing the theory of partition Lie algebras and partition Lie algebroids, and to complex geometry, by establishing a new model for derived complex analytic spaces in terms of curved refinements of complex Lie algebras.

Gaussian processes provide Bayesian priors over functions and crucial uncertainty quantification tools. They are widely employed in several fields of science and technology, among which geostatistics, computer experiments and machine learning. The GAP project will tackle two major bottlenecks regarding Gaussian processes. (1) Computational developments of Gaussian processes are at a more advanced stage than their corresponding theoretical developments. (2) Gaussian processes are not sufficiently exploited, outside of standard statistics and machine learning tasks, which is a loss of opportunity. These two limitations will be addressed by a team with expertise in mathematical and computational statistics, applied mathematics and machine learning. By cross-fertilising mathematical techniques from asymptotic statistics, probability and functional analysis, as well as methodological and computational developments, the four following axes will be carried-out. Axes (1) and (2) below are more theoretical while Axes (3) and (4) also aim for real data applications, for example in computer experiments, biology, medicine and finance. (1) Posterior concentration rates will be obtained for various deep Gaussian process models. This will provide the first mathematical guarantees for a rich class of Bayesian models, that notably solves the stationarity issues of standard Gaussian process models. (2) Error bounds will be obtained on approximation procedures for large data sets, more precisely inducing points and variational inference. These bounds will tackle theoretically unexplored settings: exact observations and classification. (3) Constrained Gaussian process models will be extended to high dimension, thanks to additive Gaussian processes and variable selection. This will yield the first high-dimensional Gaussian process models guaranteeing to satisfy the constraints on the whole input space. (4) Two novel applications of Gaussian processes will be provided. First, theoretically grounded source separation procedures will be obtained, under fixed-domain asymptotics. Second, Gaussian processes will enable to perform valid statistical inference post selection of regions. On this latter application, a graphical user interface for practitioners will be implemented.