Following Berger's holonomy classification and Atiyah and Donaldson's achievements in Yang-Mills theory, differential geometers have studied the interactions between variational and algebraic perspectives. Our project combines these traditions in the study of special geometric structures, such as extremal Kähler/Sasaki, special holonomy, generalized geometry and the interplay of all these concepts in Ströminger systems. In practice, these problems belong to gauge theory : a space of connections, a curvature equation to solve, a group of symmetries to control. In each case, the expected outcome is a correspondence between a special geometry and an algebraic condition, as provided by Kobayashi-Hitchin-D-U-Y, which allows to describe the local structure of the moduli space in terms of stability. In addition to constructing new families of examples, our objective is to understand their global topology, deformations and algebraic obstructions.

The goal of the proposal is to develop New Applications of Quantum Invariants to 3- and 4-dimensional Topology. Its main objectives will be divided into three parts: First, applications of quantum representations to the theory of mapping class groups of surfaces, where we propose to study the faithfulness of two families of linear representations of mapping class groups: the non semi-simple quantum representations and the Heisenberg homological representations of configuration spaces on surfaces. Both are serious candidates for being faithful linear representations of mapping class groups, which would solve one of the biggest open problem on mapping class groups. We will also study Ivanov's question about finite index subgroups of mapping class groups of surfaces having finite abelianization on the family of kernels of semi-simple quantum representations modulo some ideals. Our second axis is to build up methods to relate quantum invariants to twisted homology on configuration spaces, hopefully providing new insights on the geometric content of those invariants. We expect such a program to give applications to the various conjectures relating quantum and classical invariants, and to detection problems (e.g. do the colored Jones polynomials detect the unknot ?). Finally, the third aspect addresses new 4-dimensional applications of quantum invariants. We plan to use the theory of trisections to extract new invariants of 4-manifolds or of knotted surfaces in S^4 from the established quantum invariants in dimension 3: colored Jones polynomials, WRT invariants... We will focus particularly on one interesting lead that results from the recent proof of Witten's finiteness conjecture for skein modules: we plan to investigate how to define a 3+1 TQFT whose values on 3-manifolds are the Kauffman bracket skein modules. In particular, we hope to find effective methods for computing dimensions of skein modules, giving a constructive proof of the finiteness conjecture.

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.

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.

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.