This project focuses on various aspects of branching processes in fixed, variable or random environments, whether they are single-type or multitype. We propose to identify the limit of Bienaymé-Galton-Watson trees conditioned by their total population through their coding by multi-indexed and matrix-valued random walks. Then we will study the problem of the extinction of a part of the population for continuous multitype branching processes. We will construct the continuous analogue of multitype Bienaymé-Galton-Watson trees. These continuous random trees will then be obtained in the stable case as scaling limits of the renormalized discrete trees. These continuous random trees will be associated with continuous multi-type branching processes. We will also study discrete-time multitype branching processes in random environments to obtain asymptotic properties of the corresponding population size and survival probability; in particular, the problems of large deviations and asymptotic normalization will be considered. To this end, we will first deepen the study of the products of random matrices, in particular through the study of the multidimensional processes corresponding to the linear action of these products of matrices. We will be particularly interested in the cases where these processes are conditioned to remain in a cone of the Euclidean space. We will then establish limit theorems (invariance principle, local limit theorem, ...) for these conditioned processes. We will finally focus on the fundamental branching martingale associated to these Bienaymé-Galton-Watson trees, defined from the corresponding products of random matrices.

The aim of this project is to define and study new lagrangian objects, in the framework of derived symplectic geometry, and deduce results in traditional representation theory or string theory via topological field theories. We split this project in two axe which can be treated independently, but are motivated by common questions arising from physics. The first axis has several tasks particularly suitable for a postdoctoral researcher, while the second will be carried out in close collaboration with Damien Calaque and Julien Grivaux. Symplectic geometry is a natural setting for the hamiltonian formulation of classical mechanics, as most phase spaces appear to be symplectic. The cotangent bundle of a manifold is an example of a symplectic manifold, and actually, symplectic manifolds do not have local invariants: it follows from a theorem of Darboux that every symplectic manifold is locally the cotangent to R^n. Lagrangian submanifolds play a crucial role in symplectic geometry. Generalizing Darboux’s theorem, Weinstein proved that in the neighborhood of a lagrangian submanifold L, every symplectic manifold is a neighborhood of the zero section of T*L. Thus lagrangian submanifolds can naturally be interpreted as generalized configurations of a classical mechanical system. These submanifolds pop up everywhere: graphs of closed 1-forms, graphs of symplectomorphisms (an example of which is the time t=1 flow of a hamiltonian vector field), conormal bundles, zero loci of moment maps, etc. The need to deal with singular spaces becomes clear, which is the purpose of derived geometry. A leitmotiv of derived geometry is to replace geometric perturbations with homological perturbations in order to compute fiber products. Homological perturbations can be made functorial (in a higher categorical sense), and make sense in the algebro-geometric context, resolving pathological behaviours of many moduli spaces appearing in classical physics (spaces of solutions of equations of motion - as opposed to quantum physics). The first Axis of this project consists in studying new cohomological Hall algebras (COHAs) built on lagrangian subvarieties constructed in the framework of derived geometry by Bozec, Calaque and Scherotzke. These COHAs form the quintessence of geometric representation theory, consisting in studying quantum groups by geometric means. It led to the definition of algebraic structures on cohomology groups of lagrangian subvarieties, yielding the recent resolution of crucial conjectures. The second axis aims at solving a conjecture of Moore and Tachikawa regarding topological field theories (TFTs). The TFT they’re aiming at is a functor from the category of oriented cobordisms X of dimension 2 to a category of symplectic holomorphic hamiltonian varieties. In physics, classical field theories typically associate to a manifold a space of fields, where the precise notion of space required might be some derived stack. In many examples the space of fields is a mapping space Map(X,T) for some fixed target T, endowed with a symplectic structure which plays an important role in quantizing the classical theory. This involves the so-called AKSZ construction which has been promoted to the context of derived geometry, where constructions require a small number of assumption.

Derived algebraic geometry goes back to intersection theory and particularly to the famous Serre's intersection formula introduced in the 50'. This formula express an intersection number as an alternating sum of dimensions of the higher Tor's of the structure sheaves of two algebraic sub-varieties. In the early 90', Kontsevich has pushed the story one step further by introducing the notion of quasi-manifolds and virtual fundamental classes in his treatment of the moduli space of stable maps for then purpose of enumerative geometry of curves. The theory of quasi-manifolds and virtual classes have evolved independently in the late 90'. On the one side Kapranov and Ciocan-Fontanine introduced the notion of dg-schemes as a formalization of the notion of quasi-manifold. On the other side virtual classes has been defined in great generality by Behrend and Fantechi based on the notion of obstruction theories. Both of these notions have a serious drawback: the lack of functoriality, which in practice implies un-necessary technical complications as well as un- reachable constructions. This has led several authors to develop new foundations for the whole subject. The modern foundations of what is now called "derived algebraic geometry" have been developed during the last decade by Toën-Vezzosi (HAG1, HAG2), and later on by Lurie. They introduced the notion of derived Artin n-stacks, a far reaching generalization of the notion of algebraic stacks in the sense of Artin, higher algebraic stacks in the sense of Simpson, and of dg-schemes in the sense of Kapranov and Ciocan-Fontanine. These foundations are based on techniques from homotopical algebra and higher category theory making the subject extremely flexible and therefore extremely rich in examples. Derived algebraic geometry has been developing fast during the last few years, with the works of various mathematicians: Pantev, Toën, Vaquié, Vezzosi, Lurie, Francis, Gaitsgory, Rozenblyum, Preygel, Ben-Zvi, Nadler, Brav, Bussi, Joyce, Costello, Ginot, Calaque, Bhatt, Schuerg. Thanks to these recent developments, derived algebraic geometry is today a central subject, with very solid foundations as well as rich interactions with various subjects in mathematics, as for instance: singularity theory (matrix factorizations), symplectic geometry (shifted or derived symplectic structures, moduli of objects in Fukaya categories), quantization by deformation (formality and higher formality theorems via derived algebraic geometry), moduli theory (moduli space of objects in a Calabi-Yau 3-fold), enumerative geometry (Donaldson-Thomas and Gromov-Witten invariants), \etc The main objective of the present proposal is to bring together mathematicians with international recognition whose research domains are related to categorification's problems. We divide them in two main themes : Gromov-Witten theory and quantization. In Gromov-Witten theory, we plan to refund the subject using derived algebraic geometry to find hidden functorialty properties and also to define Gromov-Witten invariants in non-archimedian geometry. On the quantization side, we plan to categorify Donaldson-Thomas invariants and study deformation quantization.

A fundamental goal of Algebraic Geometry is to classify algebraic varieties up to isomorphism. This is extremely hard, already for surfaces, and open in general. It has become clear that we can only hope for a classification up to birational maps, that is, isomorphisms between dense open sets. Understanding birational maps is therefore a key step towards the classification of algebraic varieties. For one of the largest families of algebraic varieties, so-called Mori fibre spaces, any birational map between any two of them is composed of special birational maps called Sarkisov links. For surfaces over nice fields, Sarkisov links are well-understood, but little is known about them in dimension three or higher, over any field. The understanding of Sarkisov links will mean an enormous advance in the study of birational maps and a substantial leap towards a classification of a large family of algebraic varieties. The very ambitious aim of this project is to describe all Sarkisov links completely in any dimension and in several non-classical settings in terms of base-locus, contracted hypersurfaces and induced rational map on the bases of the implicated Mori fibre spaces. If achieved, it will revolutionize the study of birational maps and provide new exciting tools to determine classes of algebraic varieties in several settings. In dimension three and higher, already the classification of Sarkisov links over the field of complex numbers is extremely ambitious. Another very difficult task is to classify Sarkisov links over a field of positive characteristic, as the geometry of algebraic varieties over such fields is even more challenging than it is over the field of complex numbers. The Minimal Model program, a major active research area in Biratonal Geometry, has made tremendous advances in the last decades. Recently developed ideas and techniques allow the attack on birational maps between algebraic varieties by studying Sarkisov links.

Real algebraic geometry focuses on zero sets of polynomials with real coefficients (algebraic sets) and on sets where such polynomials are of constant sign (semialgebraic sets). Although real algebraic geometry shares common notions with complex algebraic geometry, real algebraic varieties behave quite differently from the complex ones (for instance, the real projective space is an affine real algebraic variety) and Grothendieck's scheme theory is less suited to study real algebraic varieties as Hilbert's Nullstellensatz fails over the real numbers. Therefore real algebraic geometry has followed a rather different path from the one of complex algebraic geometry: it relies less on commutative algebra and more on analytic tools. This branch of mathematics is at the intersection of several areas such as algebraic geometry, commutative algebra, analytic geometry, differential topology, and model theory. More recently, effective real algebraic geometry has grown rapidly with the development of algorithmic methods, in connection with more applied problems coming from robotics or computer-aided design. The NewMIRAGE project is divided into two axes, each one having its own research problems. Axis 1. A major problem in real algebraic geometry consists in defining a natural ring of functions on a real algebraic variety. This ring should have good algebraic properties and be rigid enough to encode information about the variety. The first axis of the NewMIRAGE project aims to focus on two classes of functions, namely continuous rational functions and regulous functions. Axis 2. The Milnor fibre is a fundamental source of invariants to study singular analytic hypersurface in the complex case. Recently, a common generalisation of both the topological and motivic Milnor fibres has been introduced by J.-B. Campesato, G. Fichou and A. Parusinski. The second axis of the NewMIRAGE project consists in adapting and studying this framework to the real setting.