ERC FUNDED PROJECTS

In an era of rapid climate change there is a pressing need to understand whether and how organisms are able to adapt to novel environments. Such understanding is hampered by a major divide in the life sciences. Disciplines like systems biology or neurobiology make rapid progress in unravelling the mechanisms underlying the responses of organisms to their environment, but this knowledge is insufficiently integrated in eco-evolutionary theory. Current eco-evolutionary models focus on the response patterns themselves, largely neglecting the structures and mechanisms producing these patterns. Here I propose a new, mechanism-oriented framework that views the architecture of adaptation, rather than the resulting responses, as the primary target of natural selection. I am convinced that this change in perspective will yield fundamentally new insights, necessitating the re-evaluation of many seemingly well-established eco-evolutionary principles. My aim is to develop a comprehensive theory of the eco-evolutionary causes and consequences of the architecture underlying adaptive responses. In three parallel lines of investigation, I will study how architecture is shaped by selection, how evolved response strategies reflect the underlying architecture, and how these responses affect the eco-evolutionary dynamics and the capacity to adapt to novel conditions. All three lines have the potential of making ground-breaking contributions to eco-evolutionary theory, including: the specification of evolutionary tipping points; resolving the puzzle that real organisms evolve much faster than predicted by current theory; a new and general explanation for the evolutionary emergence of individual variation; and a framework for studying the evolution of learning and other general-purpose mechanisms. By making use of concepts from information theory and artificial intelligence, the project will also introduce various methodological innovations.

2 500 000 €

Start date: 2018-12-01, End date: 2023-11-30

The understanding of spatial, social and dynamical organization of large numbers of agents is presently a fundamental issue in modern science. ADORA focuses on problems motivated by biology because, more than anywhere else, access to precise and many data has opened the route to novel and complex biomathematical models. The problems we address are written in terms of nonlinear partial differential equations. The flux-limited Keller-Segel system, the integrate-and-fire Fokker-Planck equation, kinetic equations with internal state, nonlocal parabolic equations and constrained Hamilton-Jacobi equations are among examples of the equations under investigation. The role of mathematics is not only to understand the analytical structure of these new problems, but it is also to explain the qualitative behavior of solutions and to quantify their properties. The challenge arises here because these goals should be achieved through a hierarchy of scales. Indeed, the problems under consideration share the common feature that the large scale behavior cannot be understood precisely without access to a hierarchy of finer scales, down to the individual behavior and sometimes its molecular determinants. Major difficulties arise because the numerous scales present in these equations have to be discovered and singularities appear in the asymptotic process which yields deep compactness obstructions. Our vision is that the complexity inherent to models of biology can be enlightened by mathematical analysis and a classification of the possible asymptotic regimes. However an enormous effort is needed to uncover the equations intimate mathematical structures, and bring them at the level of conceptual understanding they deserve being given the applications motivating these questions which range from medical science or neuroscience to cell biology.

2 192 500 €

Start date: 2017-09-01, End date: 2023-02-28

"This proposal aims at strengthening the connections between more fundamentally oriented areas of mathematics like algebra, geometry, analysis, and topology, and the more applied oriented and more recently emerging disciplines of discrete mathematics, optimization, and algorithmics. The overall goal of the project is to obtain, with methods from fundamental mathematics, new effective tools to unravel the complexity of structures like graphs, networks, codes, knots, polynomials, and tensors, and to get a grip on such complex structures by new efficient characterizations, sharper bounds, and faster algorithms. In the last few years, there have been several new developments where methods from representation theory, invariant theory, algebraic geometry, measure theory, functional analysis, and topology found new applications in discrete mathematics and optimization, both theoretically and algorithmically. Among the typical application areas are networks, coding, routing, timetabling, statistical and quantum physics, and computer science. The project focuses in particular on: A. Understanding partition functions with invariant theory and algebraic geometry B. Graph limits, regularity, Hilbert spaces, and low rank approximation of polynomials C. Reducing complexity in optimization by exploiting symmetry with representation theory D. Reducing complexity in discrete optimization by homotopy and cohomology These research modules are interconnected by themes like symmetry, regularity, and complexity, and by common methods from algebra, analysis, geometry, and topology."

2 001 598 €

Start date: 2014-01-01, End date: 2018-12-31

The purpose of this project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980's, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings' theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact Kähler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.

1 809 345 €

Start date: 2015-09-01, End date: 2021-08-31