From Riemann to Poincaré, topological methods pervaded the development of geometry, algebra and arithmetic. Homology and homotopy, sheaf theory and spectral sequences, derived and model categories have shaped a wealth of invariants applied to a wide variety of problems, crowned by Grothendieck’s refoundation of algebraic geometry. In this line, Voevodsky’s motivic theory has burst out in the nineties led by the proof of the Bloch-Kato conjecture. After Morel and Voevodsky’s foundation of motivic homotopy, the theory has strongly evolved around its close connection with algebraic topology, abutting to important foundational results such as cohomological orientation theory (motivic cohomology, algebraic cobordism), and the discovery of the quadratic nature of the A1-homotopical invariants: Morel’s computations of stable homotopy sheaves of spheres, Panin and Walter’s generalized orientations leading to Levine’s quadratic enumerative geometry. The aim of the project is to extend and apply A1-homotopical methods in three main complementary directions, a unifying motto being the role of characteristic classes, especially that of the fundamental class of the diagonal: - The study of non necessarily proper algebraic varieties through A1-homotopical methods, with the aim of establishing a theory of "A1-homotopy at infinity". A cornerstone would be the understanding of the fundamental class of the diagonal of open algebraic varieties. Our first target is to develop methods of computations from different perspectives, which could also give a new approach for computing these fundamental classes. We also hope to develop this line of thoughts in order to initiate an A1-homotopical study of singularities, knots and links following Mumford, Milnor, and others. A long-term motivation is to build unstable A1-homotopical invariants at infinity, and formulate an A1-homotopical analogue of the Poincaré conjecture. - Secondly, we want to apply the quadratic invariants of A1-homotopy theory to arithmetic problems. A first direction, of an arithmetic nature, is the development of a quadratic Riemann-Roch formula, based on the quadratic invariants of A1-homotopy such as Hermitian K-theory, and Chow-Witt groups. Further, we intend to develop the recent notion of formal ternary laws, an A1-homotopical analogue of formal group laws. Given the central role of the latter, we think this study is promising for the future of stable A1-homotopy. We also propose a more exploratory route, to uncover the possible role of quadratic invariants of A1-homotopy in Beilinson's conjectures on special values of L-functions. - Our last theme propose to develop new decomposition theorems for relative motives. Indeed, it can be formulated in terms of decomposition of the fundamental class of the diagonal, this time in the proper case. Some of our recent results give new cases where the relative Chow-Künneth conjecture can be proved. We hope to be able to extend this result to more arithmetical cases, by working over number fields or even number rings. We also plan to develop the theory of relative Nori motives and its links with Voevodsky's theory. It is finally possible to transport the methods and definitions of A1-homotopy at infinity to the category of Nori motives and to take advantage of this abelian theory.

Dispersive partial differential equations (PDEs) have important applications in different research areas such as hydrodynamics, optics, plasma physics and medical imaging. In this project these PDEs, including in higher dimensions and in periodic settings, will be studied with a unique innovative combination of analytical and numerical approaches and techniques from the theory of integrable systems, also applied to non-integrable PDEs. The goal is to use the predictive power of numerical techniques for breakthroughs on the analytical side, and analytical insight into the equations to generate innovative numerical schemes able to address challenges in applications numerically. An important question is to what extent the global behavior of solutions to integrable equations or systems holds true for the solutions of non-integrable versions of those equations, for instance Boussinesq systems appearing in the context of water waves, and this also for non-local equations. For integrable equations with global solutions, the long-time behavior is to be determined. Of particular interest are higher-dimensional situations, also in a periodic setting. The to be gained analytical insight and the to be developed numerical algorithms will be directly applicable to Electrical Impedance Tomography, a form of medical imagery also applicable in geology.

Handling large datasets has become a major challenge in fields such as applied mathematics, machine learning and statistics. However, many methods proposed in the literature do not take into account the fine structures (geometric or not) behind the underlying data. Such structures can often be modeled by graphs. Though many worldwide companies such as Google, Facebook or Twitter, have made their success extracting information where the signals live natively on a graph, a refined analysis of the underlying graph influence is still missing and most of the literature neglects, for simplicity, the underlying graph structure, or uses over simplistic linear estimators to overcome these issues. We advocate the use of robust non-linear regularizations to deal with inverse problems or classification tasks on such signals. Project GraVa is an endeavor to solve such concrete and difficult issues through the mathematical perspective of variational methods for graph signal processing. This stance raises several challenges: 1. Which estimators are good candidates for such tasks, and how to assess their performance? We will encompass recent contributions from communities of graph harmonic analysis, statistics and optimization, and develop new tools in nonlinear spectral graph theory which is only emerging. 2. How to design computationally tractable algorithms for these methods? We will rely on modern distributed and parallel optimization schemes. Our main ambition is to go beyond standard approaches by fully taking advantage of key graph properties (regularity, sparsity, etc.). 3. What is a good metric to compare graph signals and how to classify them? Measuring an L2-error is standard in signal processing but reflects a Gaussian assumption on the noise distribution. We intend to develop more robust and structure-dependent error metrics able to deal with inverse problems, segmentation and classification tasks. 4. How to tackle time-dependent signals? Dealing with static signals is a first step, but many networks requires considering time-evolving graphs structures. We plan to transfer time-series analysis to graph signals in order to deploy our achievements to real case scenarios. To help GraVa reach its objective to make an industrial impact, we intend to collaborate closely with Kwyk, a French startup specialized in e-learning solutions towards high school.

In complex geometry, one distinguishes three classes of varieties, according to the sign of the canonical bundle. Among those with trivial canonical bundle, the hyperKaehler varieties are the least understood, notably because examples are missing. However, subtle links were observed with some Fano varieties, whose geometry is more accessible. The goal of the project is to deepen those links. At the geometrical level, by studying Fano varieties of K3 type, from which we hope to construct hyperKaehler varieties as moduli spaces of cycles, or moduli spaces of objects in the derived category. At the categorical level: the derived category of a Fano variety may contain a subcategory similar to that of a hyperKaehler variety. At the level of algebraic cycles: the Chow rings of hyperKaehler varieties admit conjecturally some remarkable properties that we intend to understand in relation with the associated Fano varieties.

The aim of this project is the study of relativistic quantum systems in interaction within the framework of time-dependent partial differential equations (PDEs). We will derive and analyze effective equations in various asymptotic regimes, obtaining a simpler description of complex physical phenomena. Relativistic systems are described using the Dirac operator, a matrix-valued first order differential operator. Contrary to its nonrelativistic analogue, the Laplacian, the Dirac operator is associated with an indefinite energy functional. In the systems we consider, the interactions are modeled by coupling the Dirac equation to other evolution equations or via a nonlinearity in the Dirac equation. Both the indefiniteness of the energy and the presence of interactions seriously complicate the analysis regarding issues of well-posedness, long time behavior of solutions and accurate estimates in the parameters of the problem. The two main themes of the project are the analysis of coupled/nonlinear time-dependent Dirac equations and the dynamical study of the Dirac sea. In the first theme, several asymptotic regimes will be considered in order to derive simpler, effective equations. Various dynamical properties (well-posedness, stationary solutions and their stability) of these effective equations will be studied and the results will be used to enhance our understanding of to the original equations. In the second theme, an additional step in the modeling of relativistic quantum systems is made by taking into account the so-called Dirac sea. The analysis of the dynamical properties of the Dirac sea amounts to considering a quantum system with infinitely many particles (and infinite energy). This provides a description closer to relativistic quantum field theory, where the vacuum is a polarizable medium (the Dirac sea). From a mathematical point of view, this amounts to studying PDEs where the unknown is an operator rather than a wavefunction. In both themes, an emphasis is made on the derivation of effective, simpler equations which account for the complicated physical behavior of these systems. While the first theme can be connected to many existing works in related topics in PDEs, the second theme has not been investigated as much and thus is an exciting new direction of research. On the mathematically rigorous basis of the analysis of nonlinear PDEs and spectral theory, we aim to develop original methods to increase the theoretical knowledge of relativistic quantum systems and their asymptotic analysis. Through a precise partition of each theme into different tasks, we will answer questions about the behavior of relativistic systems relevant both from the mathematical and physical point of view. The investigator will be supported by a team of young collaborators with complementary viewpoints and a common background in relativistic quantum systems.