Water is by far the most important and most studied fluid of all times, but the physics of water confined at the nanoscale remains largely mysterious, despite its major implication in numerous fields. In the project COWAT, we aim at measuring for the first time the properties of water confined at the nanoscale inside an individual, narrow (? 1.4 nm) carbon nanotube. We will make use of the exquisite sensitivity of the nanotube mechanical resonators, down to a single proton, to characterize both the hydrogen bond network and the transport properties of water confined at the nanoscale. We will be able to answer several open question in the nanofluidic community, among which the origin of fast water flow inside narrow carbon nanotube.

In 2020, the principal investigator of this project introduced a proof technique with applications in many areas related to combinatorics. This proof technique relies on a counting argument and belongs to a family of techniques such as the Lovasz Local Lemma and entropy compression. These techniques are instrumental in proving the existence of combinatorial objects under specific constraints. They have found numerous applications in combinatorics on words, graph theory, tilings, group theory, and many other related areas. The most basic version of the counting argument yields bounds identical to one of the versions of entropy compression. Moreover, the proofs are considerably simpler, relying on elementary combinatorial arguments such as inductions and bijections. This simplicity has enabled the PI and others to push this argument further to solve open problems that resisted other known techniques. The objective of the project is to study the counting argument and some related techniques to improve their applications. The success of this project will be measured by the new problems we can solve by utilizing these techniques.

The aim of our project is to bring together prominent mathematicians who work in areas related to geometric group theory, measurable group theory, probability and dynamics. More precisely, we have a certain number of themes, which already proved to be rich and fruitful and at the same time promising for future developments, and for which we expect new interactions to lead to important breakthroughs and new perpectives. Theme 1: Actions on metric spaces -- fixed points versus properness. The interactions between a group and the spaces on which it acts lie at the heart of geometric group theory. Concerning isometric actions on metric spaces, their study is structured between two essential and opposite poles : proper actions on the one hand and fixed point properties on the other. The most famous fixed point property, with countless applications, is certainly Kazhdan's Property (T), a definition of which can be formulated as follows: any isometric action on a Hilbert space has a fixed point. On the opposite side, the Haagerup property (existence of a proper isometric action on a Hilbert space) has drawn a lot of attention, notably due to its connection with the Baum-Connes conjecture. These properties can either be extended to more general Banach spaces, yielding strengthenings of Property (T); or specialized, for instance to actions on CAT(0) cube complexes. The relationships between these variants have been the object of many investigations. Despite several breakthrough results, many questions remain open. Theme 2: Topological and measured dynamics of group actions. Over the last decades, there has been a huge amount of work in generalising various aspects of dynamics and ergodic theory from one transformation to the action of a general, often countable, group. Though it might seem surprising at first sight, it appears that the action of a group G on its space of subgroups, denoted Sub(G), simply by conjugacy, plays an important role in dynamics of group actions. This is exemplified by the recent emergence of two notions that proved useful for a variety of questions : the notion of invariant random subgroup, in short IRS, which designates an invariant probability measure on Sub(G), and the notion of uniformly recurrent subgroup, in short URS, which is a minimal closed invariant subset of Sub(G). Further study is undoubtedly needed and constitutes two important objectives in this theme. The third one is the study of random walks on groups. Though this topic originated way earlier, it has been the object of spectacular recent progresses, which call for further developments. Theme 3: Coarse geometry of groups A central branch of geometric group theory is the study of finitely generated groups up to quasi-isometries. We plan to contribute to the classification problem, notably for the class of solvable groups, which remains largely open. But our goal is also to spread the study of coarse properties of groups in other directions. A natural extension of the question of whether two groups are quasi-isometric, is whether a group can ``embed" into another. The type of embeddings that have been mostly studied so far are quasi-isometric embeddings. For instance it is well-known that an amenable group does not quasi-isometrically embed into a hyperbolic group unless it is virtually cyclic. By contrast the more flexible notion of coarse embedding remains much more mysterious. It is however arguably more natural than quasi-isometric embeddings: for instance an injective morphism between finitely generated groups is a coarse embedding but is not quasi-isometric in general. In a different direction, we plan to investigate the relatively recent and quickly growing subject of ``quantitative" measure equivalence, which lies at the intersection of geometric group theory and measure group theory.

The SYNCOGEST project aims at modeling a part of the gestuality (mimics, postures and gestures) spontaneously deployed by speakers and their interlocutors during face-to-face communication, with a view to endowing embodied virtual conversational agents (ECAs) with a more natural and efficient attitude in interactional context. The aim is to propose a model for the automatic generation of appropriate gestures synchronized with speech in production (communicative gestures). This project is based on the articulation of complementary interdisciplinary approaches, bringing together specialists in artificial intelligence, language sciences and movement sciences. If the automatic generation of ECA gestures is of increasing interest, the complex relationship between gestures and speech is still largely to be determined; indeed, it implies identifying the types of gestures spontaneously performed by speakers and interlocutors (gestures related to verbal content or to the regulation of the interaction) and understanding their synchronization modes with speech (accentual, intonative, propositional criteria...). We therefore propose first to collect and annotate a dyadic reference corpus (gestures and speech), and then to work on a generation model based on deep learning adapted to this type of problem.

DynaCycloP (Dynamic combinatorial library of CycloPeptides) project lies in the field of chemical biology and aims at designing an innovative, economical and straightforward approach to build large libraries of conformationally constrained and biologically stable cyclopeptides for ligand screening. For this purpose, dynamic combinatorial chemistry (DCC) is used as a tool to graft amino acid side-chains on a well-ordered peptide scaffold, allowing the combination of ligand synthesis and screening in a single step. New analytical approaches and synthetic tools for the characterization of large dynamic combinatorial libraries (DCL) and identification of hits that will ultimately validate the developed strategy as a powerful tool for ligands discovery will be investigated. The proof of concept will be established using glycosaminoglycans (GAG) as a biological target, for the development of cell-targeting therapies.