The last decades has witnessed a very fast and deep development in the field of evolution partial differential equations (and particularly the dispersive equations). These major advances allow some perspectives which appeared to be completely out of reach a few years ago and open the very exciting perspective of studying deep dynamical properties of solutions of Partial differential equations. On the other hand, specialists of dynamical systems successfully extended methods and ideas, developed for the study of finite dimensional models, to infinite dimensional ones. This led to spectacular results concerning long time behavior of solutions of some non-linear partial differential equations, especially in one space dimension. The tools developed in the PDE context to handle non-linear PDEs could also lead to major breakthroughs, when combined with some dynamical properties of the equations. Finally, the study of partial differential equations in the presence of randomness, a topic originating in ideas from statistical mechanics, has also recently seen spectacular results. The goal of this project is to blend together ideas from these three points of view, to gain some new insights on the behaviour of solutions of partial differential equations in some asymptotic regimes : -- Long time behaviour, -- Existence, scattering, -- Stabilities, instabilities of particular solutions -- Blow-up.

With the enormous amounts of data involved in training a machine learning system, as well as the trend to push artificial intelligence to mobile devices and embedded systems, taking into account computation and memory constraints is of primary interest from the get go when designing machine learning methods. A defining feature of modern approaches to artificial intelligence is that the different type of constraints coming from statistical efficiency (i.e. training data efficiency) and computational efficiency (i.e. memory and compute time efficiency), should be considered simultaneously, not separately. While thinking about efficient algorithms in the two above senses was present from the origins of the field of machine learning, joint consideration of these issues has accelerated significantly in the last decade.The overarching objective of this chair will be to combine expertise and tools from optimisation, statistics, and theoretical computer science to take into account structure hidden in the data, and design provably reliable, statistically efficient, and computationally efficient algorithms exploiting such structure. One key aspect that will be considered is that of adaptivity, i.e. automatic tuning of the statistical models or hyperparameters involved, as well as of the type and amount of computational acceleration (such as parallelization, low dimensional representation), driven by he data itself. For the teaching objectives, a central tenet is to expose masters’ students to relevant parts of the fields of mentioned above (optimisation, statistics, theoretical computer science) and to bring them in a position to understand, control and ultimately shape (and not only to use) the latest developments in the field of artificial intelligence. Interaction between the different fields will be emphasized, so that the student’s thinking and skillset is shaped by a principle of mutual cross-overs, as much as through specialized courses on technical and up-to-date topics.

Unsupervised Learning is one of the most fundamental problem of machine learning, and more generally, of artificial intelligence. In a broad sense, it amounts to learning some unobserved latent structure over data. This structure may be of interest per se, or may serve as an important stepping stone integrated in a complex data analysis pipe-line - since large amounts of unlabeled data are more common than costly labeled data. Arguably, one the cornerstones of unsupervised learning is clustering, where the aim is to recover a partition of the data into homogeneous groups. Beside vanilla clustering, unsupervised learning encompasses a large variety of related other problems such as hierarchical clustering, where the group structure is more complex and reveals both the backbone and fine-grain organization of the data, segmentation where the shape of the clusters is constrained by side information, or ranking or seriation problems where where no actual cluster structure exists, but where there is some implicit ordering between the data. All these problems have already found countless applications and interest in these methods is even strengthening due to the amount of available unlabelled data. We can for instance cite crowdsourcing - where individuals answer to a subset of questions, and where, depending on the context, one might want to e.g. cluster them depending on their field of expertise, rank them depending on their performances, or seriate them depending on their affinities. Such problems are extremely relevant for recommender systems - where individuals are users, and questions are items - and for social network analyses. The analysis of unsupervised learning procedures has a long history that takes its roots both in the computer science and in mathematical communities. In response to recent bridges between these two communities, groundbreaking advances have been made in the theoretical foundations of vanilla clustering. We believe that these recent advances hold the key for deep impacts on the broader field of unsupervised learning because of the pervasive nature of clustering. In this proposal, we first aim at propagating these recent ground-breaking advances in vanilla clustering to problems where the latent structure is either more complex or more constrained. We will consider problems of increasing latent structure complexity - starting from hierarchical clustering and heading toward ranking, seriation, and segmentation - and propose new algorithms that will build on each other, focusing on the interfaces between these problems. As a result, we expect to provide new methods that are valid under weaker assumptions in comparison to what is usually done - e.g. parametric assumptions - while being also able to adapt to the unknown intrinsic difficulty of the problem. Moreover, many modern unsupervised learning applications are essentially of an online nature - and sometimes decisions have to be made sequentially on top of that. For instance, consider a recommender systems that sequentially recommends items to users. In this context where sequential, active recommendations are made, it is important to leverage the latent structure underlying the individuals. While both the fields of unsupervised learning, and sequential, active learning, are thriving, research at the crossroad has been conducted mostly separately by each community - leading to procedures that can be improved. A second aim of this proposal will then be to bring together the fields of unsupervised learning and active learning, in order to propose new algorithms that are more efficient at leveraging sequentially the unknown latent structure. We will consider the same unsupervised learning problems as in the batch learning side of the proposal. We will focus on developing algorithms that fully take advantage of new advances in clustering, and of our own future work in batch learning.

This project will combine the expertise of many specialists in contact and symplectic topology. Our research objectives will concentrate around two specific themes: Lefschetz fibrations and open book decompositions on the one hand and persistent homology on the other hand. Lefschetz fibrations and open book decompositions are central tools to understand the structure of symplectic and contact manifolds. We plan to use these notions in order to derive new constraints on the topology of Lagrangian submanifolds. We also plan to gain a better understanding of their holomorphic curves invariants and of their properties, using the very recent theory of convex hypersurfaces. Exciting new results in C^0 symplectic topology were obtained using persistent homology. We plan to use it in order to extract richer information from homological invariants constructed using holomorphic curves, generating families or sheaves.

GeoMod is a Collaborative International Research project between France and Germany. Contemporary model theory studies abstract properties of mathematical structures from the point of view of first-order logic. It tries to isolate combinatorial properties of definable sets such as the existence of certain configurations, or of rank functions, and to use these properties to obtains structural consequences. These may be algebraic or geometric in nature, and can be applied to specific structures such as Berkovich geometry, difference-differential algebraic geometry, additive combinatorics or Erdos geometry. A good example of a combinatorial configuration implying algebraic structure is the group configuration theorem which asserts that certain combinatorial/dimension theoretic patterns are necessarily induced by the existence of a group, and that moreover the structure of the groups which might give rise to this configuration is highly restricted. This result, which itself generalizes the coordinatization theorems of geometric algebra was given its definitive form for stable theories by Hrushovski in his 1986 PhD thesis and hereafter became one of the most powerful tools in geometric stability theory, used to resolve open problems in classification theory, in the proof of the trichotomy theorem for Zariski geometries, and thereby the crucial component of the model theoretic solution of the function field Mordell-Lang and number field Manin-Mumford conjectures. More recently, the group configuration theorem and its avatars have taken center stage in applications to combinatorics, for example in the work of Bays and Breuillard on extensions of the Elekes-Szabó theorem. The model theoretic study of valued fields provides another example of the confluence of “pure” stability theory and “applied” algebraic model theory. Abraham Robinson identified ACVF, the theory of algebraically closed nontrivially valued fields, as the model companion of the theory of valued fields already in 1959, and for most of the next half century the theory maintained an “applied” character distinct from the stability theory of “pure” model theory. However, in order to describe quotients of definable sets by definable equivalence relations (imaginaries) in valued fields, Haskell, Hrushovski and Macpherson were led to the theory of stable domination and the pure and applied strands merged. The deep connections between these approaches to the theory of valued fields further manifested themselves in the Hrushovski-Loeser approach to nonarchimedian geometry, in which spaces of stably dominated types replaced Berkovich spaces. Our project is structured around these three themes: First we aim to strengthen the still fairly recent relations between model theory and combinatorics. Secondly, we aim to develop the model theory of valued fields, a subject which has traditionally been very strong both in France and in Germany, but using the sophisticated tools of geometric stability (or neostability). Finally, we will develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects.