The study of fine geometric properties of sparse random graphs and of random maps has been a very active field for decades. More recently, a whole branch of the theory has been devoted to the study of random processes living on these objects. In particular random walks, percolation models, Ising and Potts model, or interacting particle systems like the contact process have been considered with motivations ranging from theoretical physics to biological sciences. The project ProGraM gathers young mathematicians as well as a graduate student and a postgraduate student that are experts in probability theory, combinatorics and theoretical physics. It aims at understanding the behavior of processes coming from statistical mechanics on local limits of random graphs and random maps, and how such processes can shed some light about the geometry of the underlying objects through four axes. 1- Random maps coupled with matter. We will study the geometric properties of random maps coupled with an Ising model. Their asymptotic properties are conjectured to be identical to those of the Brownian map when the Ising model is not critical. On the other hand, taking a critical Ising model should bring these models out of this universality class. 2- Exploration algorithms. The breadth and depth first search algorithms on a connected graph can be seen as random walks whose trace form a spanning tree of the graph. On a large number of classical models of random graphs, we want to show that the profile of this spanning tree converges towards a deterministic limit. The study of these limits and variants of these algorithms will then allow us to obtain information on long paths in the graph. 3- Adjacency spectrum. The study of the spectral properties of large random graphs is in full swing. We want to contribute via the information on long paths that we will get with the previous point. We also intend to study the influence of vertices of abnormally large degrees on the presence of outliers in the spectrum of a graph. 4- Interacting particles. The behavior of interacting particle systems modeling the propagation of an infection or a rumor on a graph is highly dependent on the geometric properties of the graph. In particular, the contact process is very sensitive to the distribution of vertices of high degrees in the graph if these degrees are unbounded. We will explore this link further by using an original percolation process introduced by two members of the project.

AMERISKA: Analysis of Multivariate Extremes and RISKs Assessment. An international research network on food and environmental risks assessment. The project AMERISKA aims at encouraging interactions of people of different backgrounds and from different countries to assess risks of different kinds. In particular, the project focuses on the assessment of risks in the contexts of food, hydrology and climatology which are major issues of concern for society. • Food risks assessments: The use of chemical products and the degradation of the natural environment are responsible for the presence of contaminants in food usually accumulated by human body. The degree of toxicity of the products and the consequences for human health require better stochastic modeling of the accumulation process. • Environmental risks assessments: modeling accurately the spatio-temporal structures of some atmospheric extreme variables like heavy rainfall, storms, still remains a statistical challenge. This is due to the complex multivariate structure within and between rare events. Inferring in space and time will be at the core of this environmental applications. To assess the aforementioned risks, a careful use of statistics is required. One major difficulty is that extremal events are often not well modeled due to the lack of observations. It is a major challenge for applied mathematicians to understand heavy tail phenomena observed. In particular, it appears that the importance of the events requires a global point of view, involving many researchers form different disciplines. It seems possible to use more information on extremal events because of the emergence of bigger mass of data. A major problem is to deal with the interaction (dependence) of extremes which necessarily leads to a multivariate context. Suitable mathematical tools have been developed only recently: multivariate regular variation processes are suitable new concepts for dealing with extremes and dependence in space and time. The literature on stochastic models for spatio-temporal extremal phenomena is still rather sparse. Statistical inference on these phenomena has just started and convincing applications are still missing. It is necessary to bundle the working forces of various researchers to face the challenges. AMERISKA will be a project where applied mathematicians concerns on issues of extremal risks will met; they will discuss the problems mentioned and and collaborate on their solution. One of the goals is the organization of a semester on risks that could be partly funded by Labex MME-DII. The research will be coordinated and led by 4 principal investigators: Olivier Wintenberger, Patrice Bertail, Philippe Naveau and Thomas Mikosch. They will manage a team of 8 experts from different countries, 12 french professors with strong experiments, 9 young french researchers and 2 PhD students, all working in the domain of quantitative risk analysis.

Dynamic resources allocation concerns the setting where an 'agent' sequentially makes choices in a set of possible actions based on the current context, the different choices leading to different stochastic rewards. The goal is to design and analyze computationally efficient dynamic rules of decision, called 'policies', for optimizing the future choices based on past observations. The key issue is to find the right trade-off between exploitation and exploration, i.e., the right balance between staying with the option that gave highest rewards in the past and exploring new options that might give even higher rewards in the future. Originally motivated by clinical trials, suchs models now appear in several industrial domains too, as modern technologies create many opportunities for new applications. The study of 'bandit' problems (the word refers to the paradigmatic situation of a gambler facing a row of slot-machines and deciding which one to choose in order to maximize his/her rewards) dates back to the early 1930s and the seminal works of Thompson. It has engendered a prolific literature, notably from the machine learning community. This literature addresses a wide range of issues, theoretical and computational, through developments rooted in probability theory and optimization. The statistical community also contributed under under the denomination of 'sequential inference', with a focus on asymptotic results. Semiparametric models based statistics ('semiparametrics' for short) has been a thriving research field for the last thirteen years or so. The still growing interest in semiparametrics is explained not only by its intrinsic fascinating theoretical complexity. Important theoretical advances backed by algorithmic advances and the avaibility of massive computational resources have enabled its application to a variety of real-life scientific problems, each characterized by its specific context (prior knowledge, questions at stake). As semiparametrics keeps proving its excellent scientific utility, the class of theoretical questions raised by it, both general and specific, steadily grows; so does the class of algorithmic challenges posed by its implementation. Recently, in the machine learning literature, new algorithms have been proposed, improving on the former methods. Because these new algorithms incorporate more involved inference procedures, theoretical semiparametrics appears to be a bottleneck to further progress. Symmetrically, the class of general bandit problems for which efficient algorithms are available steadily grows, and deserves more consideration from the statistical community, and especially the biostatistical community involved in clinical trials. The present proposal, called SPADRO, aims at providing new methods and new analyses for dynamic resources allocation problems by cross-fertilizing the latest breakthroughs in machine learning and semiparametrics.