ERC FUNDED PROJECTS

It is a basic textbook notion that the plasma membranes of virtually all organisms display an asymmetric lipid distribution between inner and outer leaflets far removed from thermodynamic equilibrium. As a fundamental biological principle, lipid asymmetry has been linked to numerous cellular processes. However, a clear mechanistic justification for the continued existence of lipid asymmetry throughout evolution has yet to be established. We propose here that lipid asymmetry serves as a store of potential energy that is used to fuel energy-intense membrane remodelling and signalling events for instance during membrane fusion and fission. This implies that rapid, local changes of trans-membrane lipid distribution rather than a continuously maintained out-of-equilibrium situation are crucial for cellular function. Consequently, new methods for quantifying the kinetics of lipid trans-bilayer movement are required, as traditional approaches are mostly suited for analysing quasi-steady-state conditions. Addressing this need, we will develop and employ novel photochemical lipid probes and lipid biosensors to quantify localized trans-bilayer lipid movement. We will use these tools for identifying yet unknown protein components of the lipid asymmetry regulating machinery and analyse their function with regard to membrane dynamics and signalling in cell motility. Focussing on cell motility enables targeted chemical and genetic perturbations while monitoring lipid dynamics on timescales and in membrane structures that are well suited for light microscopy. Ultimately, we aim to reconstitute lipid asymmetry as a driving force for membrane remodelling in vitro. We expect that our work will break new ground in explaining one of the least understood features of the plasma membrane and pave the way for a new, dynamic membrane model. Since the plasma membrane serves as the major signalling hub, this will have impact in almost every area of the life sciences.

1 500 000 €

Start date: 2018-01-01, End date: 2022-12-31

The present proposal is concerned with the analysis of the Einstein equations of general relativity, a non-linear system of geometric partial differential equations describing phenomena from the bending of light to the dynamics of black holes. The theory has recently been confirmed in a spectacular fashion with the detection of gravitational waves. The main objective of the proposal is to consolidate my research group based at Imperial College by developing novel mathematical techniques that will fundamentally advance our understanding of the Einstein equations. Here the proposal builds on mathematical progress in the last decade resulting from achievements in the fields of partial differential equations, differential geometry, microlocal analysis and theoretical physics. The Black Hole Stability Problem A major open problem in general relativity is to prove the non-linear stability of the Kerr family of black hole solutions. Recent advances in the problem of linear stability made by the PI and collaborators open the door to finally address a complete resolution of the stability problem. In this proposal we will describe what non-linear techniques will need to be developed in addition to achieve this goal. A successful resolution of this program would conclude an almost 50-year-old problem. The Analysis of asymptotically anti-de Sitter (aAdS) spacetimes We propose to prove the stability of pure AdS if so-called dissipative boundary conditions are imposed at the boundary. This result would align with the well-known stability results for the other maximally-symmetric solutions of the Einstein equations, Minkowski space and de Sitter space. As a second -- related -- theme we propose to formulate and prove a unique continuation principle for the full non-linear Einstein equations on aAdS spacetimes. This goal will be achieved by advancing techniques that have recently been developed by the PI and collaborators for non-linear wave equations on aAdS spacetimes.

1 999 755 €

Start date: 2018-11-01, End date: 2023-10-31

Resolution of singularities is one of classical, central and difficult areas of algebraic geometry, with a centennial history of intensive research and contributions of such great names as Zariski, Hironaka and Abhyankar. Nowadays, desingularization of schemes of characteristic zero is very well understood, while semistable reduction of morphisms and desingularization in positive characteristic are still waiting for major breakthroughs. In addition to the classical techniques with their triumph in characteristic zero, modern resolution of singularities includes de Jong's method of alterations, toroidal methods, formal analytic and non-archimedean methods, etc. The aim of the proposed research is to study nearly all directions in resolution of singularities and semistable reduction, as well as the wild ramification phenomena, which are probably the main obstacle to transfer methods from characteristic zero to positive characteristic. The methods of algebraic and non-archimedean geometries are intertwined in the proposal, though algebraic geometry is somewhat dominating, especially due to the new stack-theoretic techniques. It seems very probable that increasing the symbiosis between birational and non-archimedean geometries will be one of by-products of this research.

1 365 600 €

Start date: 2018-05-01, End date: 2023-04-30

According to string theory, coherent sheaves on three-dimensional Calabi-Yau spaces encode fundamental properties of the universe. On the other hand, they have a purely mathematical definition. We will develop and use the new field of categorified Donaldson-Thomas (DT) theory, which counts these objects. Via the powerful perspective of noncommutative algebraic geometry, this theory has found application in recent years in a wide variety of contexts, far from classical algebraic geometry. Categorification has proved tremendously powerful across mathematics, for example the entire subject of algebraic topology was started by the categorification of Betti numbers. The categorification of DT theory leads to the replacement of the numbers of DT theory by vector spaces, of which these numbers are the dimensions. In the area of categorified DT theory we have been able to prove fundamental conjectures upgrading the famous wall crossing formula and integrality conjecture in noncommutative algebraic geometry. The first three projects involve applications of the resulting new subject: 1. Complete the categorification of quantum cluster algebras, proving the strong positivity conjecture. 2. Use cohomological DT theory to prove the outstanding conjectures in the nonabelian Hodge theory of Riemann surfaces, and the subject of Higgs bundles. 3. Prove the comparison conjecture, realising the study of Yangian quantum groups and the geometric representation theory around them as a special case of DT theory. The final objective involves coming full circle, and applying our recent advances in noncommutative DT theory to the original theory that united string theory with algebraic geometry: 4. Develop a generalised theory of categorified DT theory extending our results in noncommutative DT theory, proving the integrality conjecture for categories of coherent sheaves on Calabi-Yau 3-folds.

1 239 435 €

Start date: 2017-11-01, End date: 2022-10-31

During an organism’s development it must integrate internal and external information. An example in plants, whose development stretches across their lifetime, is the coordination between environmental stimuli and endogenous cues on regulating the key hormone gibberellin (GA). The present challenge is to understand how these diverse signals influence GA levels and how GA signalling leads to diverse GA responses. This challenge is deepened by a fundamental problem in hormone research: the specific responses directed by a given hormone often depend on the cell-type, timing, and amount of hormone accumulation, but hormone concentrations are most often assessed at the organism or tissue level. Our approach, based on a novel optogenetic biosensor, GA Perception Sensor 1 (GPS1), brings the goal of high-resolution quantification of GA in vivo within reach. In plants expressing GPS1, we observe gradients of GA in elongating root and shoot tissues. We now aim to understand how a series of independently tunable enzymatic and transport activities combine to articulate the GA gradients that we observe. We further aim to discover the mechanisms by which endogenous and environmental signals regulate these GA enzymes and transporters. Finally, we aim to understand how one of these signals, light, regulates GA patterns to influence dynamic cell growth and organ behavior. Our overarching goal is a systems level understanding of the signal integration upstream and growth programming downstream of GA. The groundbreaking aspect of this proposal is our focus at the cellular level, and we are uniquely positioned to carry out our multidisciplinary aims involving biosensor engineering, innovative imaging, and multiscale modelling. We anticipate that the discoveries stemming from this project will provide the detailed understanding necessary to make strategic interventions into GA dynamic patterning in crop plants for specific improvements in growth, development, and environmental responses.

1 499 616 €

Start date: 2018-01-01, End date: 2022-12-31

"This proposal connects three areas that are considered distant from each other: computational complexity, algebraic geometry, and numerics. In the last decade, it became clear that the fundamental questions of computational complexity (P vs NP) should be studied in algebraic settings, linking them to problems in algebraic geometry. Recent progress on this challenging and very difficult questions led to surprising progress in computational invariant theory, which we want to explore thoroughly. We expect this to lead to solutions of computational problems in invariant theory that currently are considered infeasible. The complexity of Hilbert's null cone (the set of ""singular objects'') appears of paramount importance here. These investigations will also shed new light on the foundational questions of algebraic complexity theory. As an essential new ingredient to achieve this, we will tackle the arising algebraic computational problems by means of approximate numeric computations, taking into account the concept of numerical condition. A related goal of the proposal is to develop a theory of efficient and numerically stable algorithms in algebraic geometry that reflects the properties of structured systems of polynomial equations, possibly with singularities. While there are various heuristics, a satisfactory theory so far only exists for unstructured systems over the complex numbers (recent solution of Smale's 17th problem), which seriously limits its range of applications. In this framework, the quality of numerical algorithms is gauged by a probabilistic analysis that shows small average (or smoothed) running time. One of the main challenges here consists of a probabilistic study of random structured polynomial systems. We will also develop and analyze numerical algorithms for finding or describing the set of real solutions, e.g., in terms of their homology. "

2 297 163 €

Start date: 2019-01-01, End date: 2023-12-31

Entropy (admissible) weak solutions are widely considered to be the standard solution framework for hyperbolic systems of conservation laws and incompressible Euler equations. However, the lack of global existence results in several space dimensions, the recent demonstration of non-uniqueness of these solutions and computations showing the lack of convergence of state of the art numerical methods to them, have reinforced the need to seek alternative solution paradigms. Although one can show that numerical approximations of these nonlinear PDEs converge to measure-valued solutions i.e Young measures, these solutions are not unique and we need to constrain them further. Statistical solutions i.e, time-parametrized probability measures on spaces of integrable functions, are a promising framework in this regard as they can be characterized as a measure-valued solution that also contains information about all possible multi-point spatial correlations. So far, well-posedness of statistical solutions has been shown only in the case of scalar conservation laws. The main aim of the proposed project is to analyze statistical solutions of systems of conservation laws and incompressible Euler equations and to design efficient numerical approximations for them. We aim to prove global existence of statistical solutions in several space dimensions, by showing convergence of these numerical approximations, and to identify suitable additional admissibility criteria for statistical solutions that can ensure uniqueness. We will use these numerical methods to compute statistical quantities of interest and relate them to existing theories (and observations) for unstable and turbulent fluid flows. Successful completion of this project aims to establish statistical solutions as the appropriate solution paradigm for inviscid fluid flows, even for deterministic initial data, and will pave the way for applications to astrophysics, climate science and uncertainty quantification.

1 959 323 €

Start date: 2018-08-01, End date: 2023-07-31

One of the main challenges in condensed matter physics is to understand strongly correlated quantum systems. Our purpose is to approach this issue from the point of view of rigorous mathematical analysis. The goals are twofold: develop a mathematical framework applicable to physically relevant scenarii, take inspiration from the physics to introduce new topics in mathematics. The scope of the proposal thus goes from physically oriented questions (theoretical description and modelization of physical systems) to analytical ones (rigorous derivation and analysis of reduced models) in several cases where strong correlations play the key role. In a first part, we aim at developing mathematical methods of general applicability to go beyond mean-field theory in different contexts. Our long-term goal is to forge new tools to attack important open problems in the field. Particular emphasis will be put on the structural properties of large quantum states as a general tool. A second part is concerned with so-called fractional quantum Hall states, host of the fractional quantum Hall effect. Despite the appealing structure of their built-in correlations, their mathematical study is in its infancy. They however constitute an excellent testing ground to develop ideas of possible wider applicability. In particular, we introduce and study a new class of many-body variational problems. In the third part we discuss so-called anyons, exotic quasi-particles thought to emerge as excitations of highly-correlated quantum systems. Their modelization gives rise to rather unusual, strongly interacting, many-body Hamiltonians with a topological content. Mathematical analysis will help us shed light on those, clarifying the characteristic properties that could ultimately be experimentally tested.

1 056 664 €

Start date: 2018-01-01, End date: 2022-12-31

Statistical physics is a theory allowing the derivation of the statistical behavior of macroscopic systems from the description of the interactions of their microscopic constituents. For more than a century, lattice models (i.e. random systems defined on lattices) have been introduced as discrete models describing the phase transition for a large variety of phenomena, ranging from ferroelectrics to lattice gas. In the last decades, our understanding of percolation and the Ising model, two classical exam- ples of lattice models, progressed greatly. Nonetheless, major questions remain open on these two models. The goal of this project is to break new grounds in the understanding of phase transition in statistical physics by using and aggregating in a pioneering way multiple techniques from proba- bility, combinatorics, analysis and integrable systems. In this project, we will focus on three main goals: Objective A Provide a solid mathematical framework for the study of universality for Bernoulli percolation and the Ising model in two dimensions. Objective B Advance in the understanding of the critical behavior of Bernoulli percolation and the Ising model in dimensions larger or equal to 3. Objective C Greatly improve the understanding of planar lattice models obtained by general- izations of percolation and the Ising model, through the design of an innovative mathematical theory of phase transition dedicated to graphical representations of classical lattice models, such as Fortuin-Kasteleyn percolation, Ashkin-Teller models and Loop models. Most of the questions that we propose to tackle are notoriously difficult open problems. We believe that breakthroughs in these fundamental questions would reshape significantly our math- ematical understanding of phase transition.

1 499 912 €

Start date: 2018-09-01, End date: 2023-08-31

The polarity of the single cell from which many organisms develop determines the polarity of the body axis. However, the polarity of these single cells is often inherited. For example, zygote polarity is inherited from the polarized egg cell of Arabidopsis thaliana. By contrast, polarity is not pre-set in the spore cell that forms the Marchantia polymorpha (Marchantia) plant. An environmental cue – directional light – polarises the spore cell which, in turn, directs the formation of the first (apical-basal) axis and the fates of the two daughter cells formed when the spore cell divides. Using Marchantia, we will discover how cell polarity is established de novo in the developing spore cell and how this, in turn, directs the specification of the first major axis in the plant. The proposed research is feasible because of the unique characteristics of the Marchantia system: 1. Isolated single apolar cells become polarized allowing us to exploit the real-time imaging with experimental manipulation of polarising cues at each stage of development. 2. Haploid genetics can be exploited to carry out genetic screens of unprecedented depth and we can identify mutant genes using a fully annotated genome sequence. 3. Gene expression can be measured with high temporal resolution during polarization. We propose to: 1. Describe the cellular and morphogenetic events that occur as the spore cell polarizes, divides asymmetrically to form cells at either end of the apical-basal axis. 2. Define the mechanism underpinning the de novo establishment of polarity using a combination of forward and reverse genetics and determine if this mechanism is conserved among land plants. 3. Determine the role of auxin in transmitting spore cell polarity to the cells at both ends of the apical-basal axis. This will describe, for the first time, the molecular mechanism controlling the de novo polarization of a single cell that develops into a plant.

2 499 224 €

Start date: 2018-10-01, End date: 2023-09-30

Female germ cells, oocytes, are highly specialised cells. They ensure the continuity of species by providing the female genome and mitochondria along with most of the nutrients and housekeeping machinery the early embryo needs after fertilisation. Oocytes are remarkable in their ability to survive for long periods of time, up to 50 years in humans, and retain the ability to give rise to a young organism while other cells age and die. Surprisingly little is known about oocyte dormancy. A key feature of dormant oocytes of virtually all vertebrates is the presence of a Balbiani body, which is a non-membrane bound compartment that contains most of the organelles in dormant oocytes and disappears as the oocyte matures. The goal of this proposal is to combine genetic and biochemical perturbations with imaging and the state of the art proteomics techniques to reveal the mechanisms dormant oocytes employ to remain viable. My previous research has shown that the Balbiani body forms an amyloid-like cage around organelles that could be protective. This has led me to identify the large number of unanswered questions about the cell biology of a dormant oocyte. In this proposal, we will study three of these questions: 1) What is the metabolic nature of organelles in dormant oocytes? 2) How does the Balbiani body disassemble and release the complement of organelles when oocytes start to mature? 3) What is the structure and function of the Balbiani body in mammals? We will use oocytes from two vertebrate species, frogs and mice, which are complementary for their ease of handling and relationship to human physiology. By studying the Balbiani body, this proposal will provide fundamental insights into organisation and function of organelles in oocytes and the regulation of physiological amyloid-like structures. More generally, the proposed experiments open up new avenues into the mechanisms that protect organelles from ageing and how oocytes stay dormant for many decades.

1 381 286 €

Start date: 2018-03-01, End date: 2023-02-28

A central theme of extremal combinatorics is the interplay and relationship between the parameters of combinatorial objects. The first and most immediate question which arises in this context is that of the (i) existence of objects with a given set of parameters. Once this has been answered, the next step is to seek for (ii) the number of such objects - i.e. to ask for a counting result. This is of central importance in the context of many combinatorial questions arising in statistical physics. A very effective approach here is to seek asymptotic results - rather than exact formulas. This asymptotic approach sometimes makes it possible to go even further and ultimately uncover the (iii) typical structure of the objects in such a given class. In this project, we will consider the above perspective with a focus on inter-related topics involving combinatorial designs, decompositions, Latin squares as well as matchings in graphs and hypergraphs. The project themes have close connections e.g. to statistical physics, probability, algebra and theoretical computer science. A common feature of the structures considered in this proposal is that the constraints describing them are of a "global nature". This makes their study extremely challenging. However, recently initiated methods have opened up completely new avenues, bringing questions within reach that were considered inaccessible until now. (In fact, one of the objectives involves the study of algebraic structures which had been conjectured not even to exist.) The aim of the project is the development of general tools and approaches which make the asymptotic study of such structures far more accessible. These tools will be mostly of a probabilistic nature. Indeed, the probabilistic perspective has already been the driving force behind recent advances which underpin the proposal. But it seems that overall, this development is still in its early stages - a situation we aim to address in the current project.

1 797 111 €

Start date: 2019-01-01, End date: 2023-12-31

A fundamental problem in the study of dynamical systems is to ascertain whether the effect of a perturbation on an integrable Hamiltonian system accumulates over time and leads to a large effect (instability) or it averages out (stability). Instabilities in nearly integrable systems, usually called Arnold diffusion, take place along resonances and by means of a framework of partially hyperbolic invariant objects and their homoclinic and heteroclinic connections. The goal of this project is to develop new techniques, relying on the role of invariant manifolds in the global dynamics, to prove the existence of physically relevant instabilities and homoclinic phenomena in several problems in celestial mechanics and Hamiltonian Partial Differential Equations. The N body problem models the interaction of N puntual masses under gravitational force. Astronomers have deeply analyzed the role of resonances in this model. Nevertheless, mathematical results showing instabilities along them are rather scarce. I plan to develop a new theory to analyze the transversal intersection between invariant manifolds along mean motion and secular resonances to prove the existence of Arnold diffusion. I will also apply this theory to construct oscillatory motions. Several Partial Differential Equations such as the nonlinear Schrödinger, the Klein-Gordon and the wave equations can be seen as infinite dimensional Hamiltonian systems. Using dynamical systems techniques and understanding the role of invariant manifolds in these Hamiltonian PDEs, I will study two type of solutions: transfer of energy solutions, namely solutions that push energy to arbitrarily high modes as time evolves by drifting along resonances; and breathers, spatially localized and periodic in time solutions, which in a proper setting can be seen as homoclinic orbits to a stationary solution.

1 100 348 €

Start date: 2018-01-01, End date: 2022-12-31

Singular solutions to variational problems and to partial differential equations are naturally ubiquitous in many contexts, and among these minimal surfaces theory and free boundary problems are two prominent examples both for their analytical content and their physical interest. A crucial aspect in this regard is the co-dimension of the objects under consideration: indeed, many of the analytical and geometric principles which are valid for minimal hypersurfaces or regular points of the free boundary do not apply to higher co-dimension surfaces or singular free boundary points. The aim of this project is to investigate some of the most compelling questions about the singularities of two classical problems in the geometric calculus of variations in higher co-dimension: I. Mass-minimizing integer rectifiable currents, i.e. solutions to the Plateau problem of finding the surfaces of least area, attacking specific conjectures about the structure of the singular set, most prominently the boundedness of its measure. II. The thin obstacle problem, consisting in minimizing the Dirichlet energy (or a variant of it) among functions constrained above an obstacle that is assigned on a lower dimensional space, with the purpose of answering some of the main open questions on the singular free boundary points. The main unifying theme of the project is the central role played by geometric measure theory, which underlines various common aspects of these two problems and makes them suited to be treated in an unified framework. Although these are classical questions with a long tradition, our knowledge about them is still limited and their investigation is among the most challenging issues in regularity theory. This is the central focus of the project, with the final goal to develop suitable analytical techniques that provides valuable insights on the mathematics at the basis of higher co-dimension singularities, eventually fruitful in other geometric and analytical settings.

1 341 250 €

Start date: 2018-02-01, End date: 2023-01-31

Moduli spaces are spaces which describe all mathematical objects of some type. This proposal concerns the study of certain moduli spaces via techniques from homotopy theory, from several different points of view. The main moduli spaces in which we are interested are moduli spaces of manifolds, or equivalently classifying spaces of diffeomorphism groups of manifolds. We are also interested in spaces of positive scalar curvature metrics on smooth manifolds, which we study by relating them to moduli spaces of smooth manifolds. The study of moduli spaces of manifolds via homotopy theory has seen a great deal of development in the last 20 years, the breakthrough result being Madsen and Weiss' calculation of the stable homology of moduli spaces of surfaces. More recently, Galatius and I have established analogous results for manifolds of higher dimension. A main goal of this proposal is to study the homology of moduli spaces from a multiplicative point of view. This leads to higher-order forms of the phenomenon of homological stability in which the failure of ordinary homological stability is itself stable. Remarkably, our methods developed to handle moduli spaces of manifolds are sufficiently general to yield deep new results when applied to other moduli spaces in algebra and topology, such as moduli spaces of modules (equivalently, classifying spaces of general linear groups) or moduli spaces of graphs (equivalently, classifying spaces of automorphism groups of free groups). In each case our methods give new information about their homology outside of the traditional stable range. Other goals of this proposal are to form new connections between spaces of Riemannian metrics of positive scalar curvature and infinite loop spaces, and to investigate the structure of tautological subrings of the cohomology of moduli spaces of manifolds, especially in relation to the tautological rings of moduli spaces of Riemann surfaces studied in algebraic geometry.

974 526 €

Start date: 2018-10-01, End date: 2023-09-30

Cells face a phosphate challenge. Growth requires a minimal concentration of this limiting resource because intracellular phosphate (Pi) is a compound of nucleic acids and modifies most cellular proteins. At the same time, cytosolic Pi may not rise much, because elevated cytosolic Pi can stall metabolism. It reduces the free energy that nucleotide triphosphate hydrolysis can provide to drive energetically unfavorable reactions. I will undertake a pioneering study to elucidate how cells strike this critical balance. We will identify a novel pathway for intracellular phosphate reception and signaling (INPHORS) and explore the role of acidocalcisomes in it. These studies may identify a key function of these very poorly understood organelles, provide one reason for their evolutionary conservation and elucidate a novel homeostatic system of critical importance for cellular metabolism. We recently provided first hints that a dedicated pathway for sensing and signaling intracellular Pi might exist, which regulates multiple systems for import, export and acidocalcisomal storage of Pi, such that cytosolic Pi homeostasis is guaranteed 1. Yeast cells will serve as an powerful model system for exploring this pathway and its physiological relevance. Yeast Pi transport and storage proteins are known. Furthermore, we can establish cell-free in vitro systems that reconstitute Pi-regulated transport and storage processes, providing an excellent basis for identifying signaling complexes and studying their dynamics. We will (A) generate novel tools to uncouple, individually manipulate and measure key parameters for the INPHORS pathway; (B) identify its components, study their interactions and regulation; (C) elucidate how acidocalcisomes are targeted by INPHORS and how they contribute to Pi homeostasis; (D) study the crosstalk between INPHORS and Pi-regulated transcriptional responses; (E) test the relevance of INPHORS for Pi homeostasis in mammalian cells.

2 499 998 €

Start date: 2018-09-01, End date: 2023-08-31

This proposal is concerned with the mathematical theory of inverse problems. This is a vibrant research field at the intersection of pure and applied mathematics, drawing techniques from PDE, geometry, and harmonic analysis as well as generating new research questions inspired by applications. Prominent questions include the Calderón problem related to electrical imaging, the Gel'fand problem related to seismic imaging, and geometric inverse problems such as inversion of the geodesic X-ray transform. Recently, exciting new connections between these different topics have begun to emerge in the work of the PI and others, such as: - The explicit appearance of the geodesic X-ray transform in the Calderón problem. - An unexpected connection between the Calderón and Gel’fand problems involving control theory. - Pseudo-linearization as a potential unifying principle for reducing nonlinear problems to linear ones. - The introduction of microlocal normal forms in inverse problems for PDE. These examples strongly suggest that there is a larger picture behind various different inverse problems, which remains to be fully revealed. This project will explore the possibility of a unified theory for several inverse boundary problems. Particular objectives include: 1. The use of normal forms and pseudo-linearization as a unified point of view, including reductions to questions in integral geometry and control theory. 2. The solution of integral geometry problems, including the analysis of convex foliations, invertibility of ray transforms, and a systematic Carleman estimate approach to uniqueness results. 3. A theory of inverse problems for nonlocal models based on control theory arguments. Such a unified theory could have remarkable consequences even in other fields of mathematics, including controllability methods in transport theory, a solution of the boundary rigidity problem in geometry, or a general pseudo-linearization approach for solving nonlinear operator equations.

920 880 €

Start date: 2018-05-01, End date: 2023-04-30

Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the 20th century, several basic links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The virtual fundamental class plays an essential role in these connections. I propose to study the algebraic cycle theory of basic moduli spaces. The guiding questions are: What are the most important cycles? What is the structure of the algebra of cycles? How can the classes of geometric loci be expressed? The virtual fundamental class and the associated invariants often control the answers. A combination of virtual localization, degeneration, and R-matrix methods together with new ideas from log geometry will be used in the study. Most of the basic moduli spaces in algebraic geometry related to varieties of dimension at most 3 -- including the moduli of curves, the moduli of maps, the moduli of surfaces, and the moduli of sheaves on 3-folds -- will be considered. The current state of the study of the algebraic cycle theory in these cases varies from rather advanced (for the moduli of curves) to much less so (for the moduli of surfaces). There is a range of rich open questions which I will attack: Pixton's conjectures for the moduli of curves, the structure of the ring of Noether-Lefschetz loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten theory, and conjectures governing descendents for the moduli of sheaves. The dimension 3 restriction is often necessary for a good deformation theory and the existence of a virtual fundamental class.

2 496 055 €

Start date: 2018-09-01, End date: 2023-08-31

During the development of the central nervous system (CNS), neurons extend axons through a crowded environment along well-defined pathways to reach their distant targets. It isA start date of 1st June 2018 is being requested to enable the PI to complete a number of current commitments and put the necessary arrangements in place to enable an efficient start up phase of the project. evident that attractive and repulsive guidance cues in the tissue provide important biochemical signals to guide growing axons along their paths. This can only be part of the story, however, as it is still not possible to predict axonal growth patterns in vivo. In a recent breakthrough discovery, we provided in vivo evidence that neurons also respond to mechanical cues, such as local tissue stiffness, suggesting that mechanical signals are likely an important missing part of the puzzle. However, mechanically activated signaling pathways are currently poorly understood, and how neurons integrate mechanical and chemical signals to result in proper outgrowth is unknown. By investigating how mechanical signals control neuronal growth and pathfinding, this proposal will close this comprehension gap. By combining state-of-the-art approaches in physics, engineering and biology, we will, for the first time, identify mechanosensitive molecular mechanisms that regulate neuronal growth and guidance in vitro and in vivo. In particular, we will investigate how mechanotransduction cascades (1) directly modulate axon growth by inducing local changes in cytoskeletal dynamics, and (2) indirectly lead to alterations in axon outgrowth by modulating chemical signalling pathways. Ultimately, we will develop a computational model based on our findings, which will lead to a predictive framework for understanding axon pathfinding in the developing brain. The proposed research challenges current concepts in developmental biology and is very relevant to many other areas in biology. Our results will not only shed new light on the complex control mechanisms of cellular growth and motility, but could also lead to novel biomedical approaches aimed at facilitating neuronal re-growth and regeneration in the damaged CNS.

2 468 520 €

Start date: 2018-06-01, End date: 2023-05-31

Nervous system evolution involved multiple cellular innovations, enabling fast synaptic transmission, novel sensory modalities, and complex forms of neural connectivity. These innovations were distributed to an ever-increasing number of neural cell types, each specified and maintained by the combinatorial activity of transcription factors. When, where, and how did this complexity arise? This proposal aims at resolving the history of cell type diversification that led to this complexity, from the birth of the first neurons to the many families of neuron types that exist today. Our emphasis will be on reconstructing the cellular diversity in the ancestor of bilaterian animals and finding the key molecular innovations that drove early nervous system complexity. Towards this aim, we will first use whole-body single-cell RNAseq, in combination with a spatial expression atlas at cellular resolution, to comprehensively characterise cell types in the model annelid Platynereis dumerilii, a genetically tractable, slow-evolving bilaterian. We will then generate and compare similar datasets from diverse bilaterians and non-bilaterian outgroups to map the history of neuronal cell type diversification and infer the key regulatory and functional innovations that gave rise to the first bilaterian nervous system. For several such regulatory innovations, we will experimentally validate transcription factor binding to effector gene loci via superresolution microscopy and chromatin immunoprecipitation. We will also investigate neuron family-specific protein complexes, their subcellular localization, and neural functions via biochemical and proteomics approaches, correlative microscopy and loss-of-function analyses. This analysis of neuronal cell type diversity will for the first time trace the evolutionary history of nervous system complexity, unravelling when, where and how key neuronal innovations have driven the success of bilaterian nervous systems.

2 500 000 €

Start date: 2018-06-01, End date: 2023-05-31

Noise-sensitivity of a Boolean function with iid random input bits means that resampling a tiny proportion of the input makes the output unpredictable. This notion arises naturally in computer science, but perhaps the most striking example comes from statistical physics, in large part due to the PI: the macroscopic geometry of planar percolation is very sensitive to noise. This can be recast in terms of Fourier analysis on the hypercube: a function is noise sensitive iff most of its Fourier weight is on "high energy" eigenfunctions of the random walk operator. We propose to use noise sensitivity ideas in three main directions: (A) Address some outstanding questions in the classical case of iid inputs: universality in critical planar percolation; the Friedgut-Kalai conjecture on Fourier Entropy vs Influence; noise in First Passage Percolation. (B) In statistical physics, a key example is the critical planar FK-Ising model, with noise being Glauber dynamics. One task is to prove noise sensitivity of the macroscopic structure. A key obstacle is that hypercontractivity of the critical dynamics is not known. (C) Babai’s conjecture says that random walk on any finite simple group, with any generating set, mixes in time poly-logarithmic in the volume. Two key open cases are the alternating groups and the linear groups SL(n,F2). We will approach these questions by first proving fast mixing for certain macroscopic structures. For permutation groups, this is the cycle structure, and it is related to a conjecture of Tóth on the interchange process, motivated by a phase transition question in quantum mechanics. We will apply ideas of statistical physics to group theory in other novel ways: using near-critical FK-percolation models to prove a conjecture of Gaboriau connecting the first ell2-Betti number of a group to its cost, and using random walk in random environment to prove the amenability of the interval exchange transformation group, refuting a conjecture of Katok.

1 386 364 €

Start date: 2018-02-01, End date: 2023-01-31

Epithelial polarity is one of the most fundamental types of cellular organization, and correct cellular polarization is vital for all epithelial tissue. Failure to establish polarity leads to severe phenotypes, from catastrophic developmental deficiencies to life-threatening diseases such as cancer. Despite knowing much about the signalling and trafficking machinery vital for polarity, we lack quantitative knowledge about the intracellular mechanical processes which organize and stabilize epithelial polarity. This presents a critical knowledge gap, as any elaborated understanding of intracellular organization needs to include the forces and viscoelastic mechanical properties that position organelles and proteins. As such, the main aim of POLARIZEME is to determine the intracellular mechanical processes relevant for epithelial polarization, thus providing a mechanical understanding of polarity. We will combine advanced optical tweezers technology with cutting-edge molecular biology tools to rigorously test new intracellular transport concepts such as the active, diffusion-like forces that can position organelles or the recently introduced cortical actin flows that can drag polarity-defining proteins around the cell. Thus we propose (i) to quantify active forces and intracellular mechanics and their relation to organelle positioning, (ii) to quantify polarized cortical and cytoplasmic flows, and (iii) to measure the forces and mechanical obstacles relevant for direct vesicle trafficking. These quantitative biophysics experiments will be supported by mathematical modelling and the development of two new instruments which (a) allow for automated intracellular mechanics measurements over extended time periods and (b) combine multi-view light-sheet microscopy with optical tweezers and UV ablation. Overall, we will provide a new access to understand and describe polarity by merging the physical and biological aspects of its initiation, maintenance and stability.

1 995 564 €

Start date: 2018-03-01, End date: 2023-02-28

The systematic investigation of random discrete structures and processes was initiated by Erdős and Rényi in a seminal paper about random graphs in 1960. Since then the study of such objects has become an important topic that has remarkable applications not only in combinatorics, but also in computer science and statistical physics. Random discrete objects have two striking characteristics. First, they often exhibit phase transitions, meaning that only small changes in some typically local control parameter result in dramatic changes of the global structure. Second, several statistics of the models concentrate, that is, although the support of the underlying distribution is large, the random variables usually take values in a small set only. A central topic is the investigation of the fine behaviour, namely the determination of the limiting distribution. Although the current knowledge about random discrete structures is broad, there are many fundamental and long-standing questions with respect to the two key characteristics. In particular, up to a small number of notable exceptions, several well-studied models undoubtedly exhibit phase transitions, but we are not able to understand them from a mathematical viewpoint nor to investigate their fine properties. The goal of the proposed project is to study some prominent open problems whose solution will improve significantly our general understanding of phase transitions and of the fine behaviour in random discrete structures. The objectives include the establishment of phase transitions in random constraint satisfaction problems and the analysis of the limiting distribution of central parameters, like the chromatic number in dense random graphs. All these problems are known to be difficult and fundamental, and the results of this project will open up new avenues for the study of random discrete objects, both sparse and dense.

1 219 462 €

Start date: 2018-04-01, End date: 2023-03-31

Arbuscular mycorrhiza (AM) is an ancient plant-fungus symbiosis that is wide-spread in the plant kingdom. AM improves plant nutrition, stress resistance and general plant performance and thus represents a promising addition to sustainable agricultural practices. Mineral nutrients are released from the fungus to the plant at highly branched hyphal structures, the arbuscules, which form inside root cortex cells. Like the cells of other multicellular eukaryotes, plant cells show a remarkable developmental plasticity. Single cell re-differentiation is a fascinating process during arbuscule development, which can be conceptually separated into distinct stages controlled by the plant cell which precisely guide the step-wise formation of different parts of the arbuscule. It involves cell autonomous transcriptional reprogramming and subcellular remodelling, leading to repositioning of subcellular structures, cell polarization and multiplication of organelles. It is currently unknown how cell-autonomous reprogramming during arbuscule development is regulated. RECEIVE utilizes an integrated strategy combining transcriptional profiling, transcription factor identification, interaction network analysis with reverse genetics and cell biological techniques to understand the coordinated step-wise progression of arbuscule development. RECEIVE builds on the hypothesis that each stage of arbuscule development is accompanied by a stage-specific wave of gene expression and that transcriptional regulation is a key determinant of the developmental progress from stage to stage. The characterisation of these waves and the identification of the underlying transcriptional regulatory nodes is the focus of this project. RECEIVE aims to bridge a major knowledge gap about the molecular basis of one of the most important symbioses on earth.

1 499 625 €

Start date: 2018-02-01, End date: 2023-01-31

"Plant growth and development is governed by finely tuned, highly regulated hormone gradients. Impressive progress has been made in understanding plant hormone signaling, but knowledge on the mechanisms underlying their precise localization at the tissue and subcellular levels is still very limited. We and others have recently identified the first bona fide GA transporters in plants as members of the NPF protein family. Proteins from the ABC family were shown to transport the CK, ABA, and auxin hormones. Although these studies suggested specialized functions for members of these large protein families, progress in understanding their level of specialization has been limited by the scarcity of loss-of-function phenotypes, masked by the highly redundant plant genome. The goal of this proposal is to reveal the robust and specialized function of the NPF and ABC plant hormone transporter families. The project places key technological challenges that require multi-disciplinary expertise to examine how plants balance redundancy and specialization to tightly regulate hormone localization. Broad and targeted transportome screens using multi-targeted artificial miRNAs and CRISPR technology in Arabidopsis and tomato, respectively, are designed to unveil novel plant hormone transporters, with an emphasis on subcellular localized transporters and the missing GA exporters. Specialization and robustness of candidate transporters will be evaluated by integrating in vitro transport assays with in vivo growth and development experiments. I believe that the proposed ""redundant-free"" populations will lead to new paradigms in plant genetics and would explain how gene families have developed robustness together with unique and diverse specialization. Importantly, our combined genetic and organelle-specific hormone profiling approaches will establish fundamental new concepts regarding plant hormone localization, activity, and specificity at the subcellular level."

1 500 000 €

Start date: 2018-01-01, End date: 2022-12-31

The emergence of singularities, such as oscillations and concentrations, is at the heart of some of the most intriguing problems in the theory of nonlinear PDEs. Rich sources of these phenomena can be found for instance in the equations of mathematical material science and hyperbolic conservation laws. Building on recent pioneering work of the PI, The SINGULARITY project will investigate singularities through innovative strategies and tools that combine geometric measure theory with harmonic analysis. The potential of this approach is far-reaching and has already led to the resolution of several long-standing conjectures as well as opened up new avenues to understand the fine structure of singularities. The project comprises three inter-connected themes: Theme I investigates condensated singularities, i.e. singular parts of (vector) measures solving a PDE. A powerful structure theorem was recently established by the PI and De Philippis, which will be developed into a fine structure theory for PDE-constrained measures. Theme II is concerned with the development of a compensated compactness theory for sequences of solutions to a PDE, which is capable of dealing with concentrations. The central aim is to study in detail the (non-)compactness properties of such sequences in the presence of asymptotic singularities, for instance in relation to the Bouchitt ́e Conjecture in shape optimization. Theme III investigates higher-order microstructure, i.e. nested periodic oscillations in sequences, such as laminates. The main objective is to understand the effective properties of such microstructures and to make progress on pressing open problems in homogenization theory and on the fundamental Morrey Conjecture. We will employ the promising tool of microlocal compactness forms, recently invented by the PI. All three themes tackle challenging and important open questions, which will serve as guiding lights towards a robust framework for the effective study of singularities.

1 483 943 €

Start date: 2018-04-01, End date: 2023-03-31

I will introduce new techniques to address two big open questions in the theory of derived/triangulated categories and their many applications in algebraic geometry. The first one concerns the theory of Bridgeland stability conditions, which provides a notion of stability for complexes in the derived category. The problem of showing that the space parametrizing stability conditions is non-empty is one of the most difficult and challenging ones. Once we know that such stability conditions exist, it remains to prove that the corresponding moduli spaces of stable objects have an interesting geometry (e.g. they are projective varieties). This is a deep and intricate problem. On the more foundational side, the most successful approach to avoid the many problematic aspects of the theory of triangulated categories consisted in considering higher categorical enhancements of triangulated categories. On the one side, a big open question concerns the uniqueness and canonicity of these enhancements. On the other side, this approach does not give a solution to the problem of describing all exact functors, leaving this as a completely open question. We need a completely new and comprehensive approach to these fundamental questions. I intend to address these two sets of problems in the following innovative long-term projects: 1. Develop a theory of stability conditions for semiorthogonal decompositions and its applications to moduli problems. The main applications concern cubic fourfolds, Calabi-Yau threefolds and Calabi-Yau categories. 2. Apply these new results to the study of moduli spaces of rational normal curves on cubic fourfolds and their deep relations to hyperkaehler geometry. 3. Investigate the uniqueness of dg enhancements for the category of perfect complexes and, most prominently, of admissible subcategories of derived categories. 4. Develop a new theory for an effective description of exact functors in order to prove some related conjectures.

785 866 €

Start date: 2018-02-01, End date: 2023-01-31

How complex but stereotyped tissues are formed, maintained and regenerated through local growth, differentiation and remodeling is a fundamental open question in biology. Understanding how single cell behaviors are coordinated on the population level and how population-level dynamics is coupled to tissue architecture is required to resolve this question as well as to develop stem cell (SC) therapies and effective treatments against cancers. As a self-renewing organ maintained by multiple distinct SC populations, the epidermis represents an outstanding, clinically highly relevant research paradigm to address this question. A key epidermal SC population are the hair follicle stem cells (HFSCs) that fuel hair follicle regeneration, repair epidermal injuries and, when deregulated, initiate carcinogenesis. The major obstacle in mechanistic understanding of HFSC regulation has been the lack of an in vitro culture system enabling their precise monitoring and manipulation. We have overcome this barrier by developing a method for long-term maintenance of multipotent HFSCs that recapitulates the complexity of HFSC fate decisions and dynamic crosstalk between HFSCs and their progeny. This breakthrough invention puts me in the unique position to investigate how HFSCs self-organize into a network of SCs and progenitors through population-level signaling crosstalk and phenotypic plasticity. This project will uncover the spatiotemporal dynamics of HFSCs fate decisions and establish the role of the niche in this process (Aim1), decipher key gene-regulatory networks and epigenetic barriers that control phenotypic plasticity (Aim2), and discover druggable signaling networks that drive bi-directional reprogramming of HFSCs and their progeny (Aim3). By deconstructing complex tissue-level behaviors at an unprecedented spatiotemporal resolution this study has the potential to transform the fundaments of adult SC biology with immediate implications to regenerative medicine.

1 999 918 €

Start date: 2018-05-01, End date: 2023-04-30

"In the study of symplectic and contact manifolds, a decisive role has been played by the theory of pseudoholomorphic curves, introduced by Gromov in 1985. One major drawback of this theory is the fundamental conflict between ""genericity"" and ""symmetry"", which for instance causes moduli spaces of holomorphic curves to be singular or have the wrong dimension whenever multiply covered curves are present. Most traditional solutions to this problem involve abstract perturbations of the Cauchy-Riemann equation, but recently there has been progress in tackling the transversality problem more directly, leading in particular to a proof of the ""super-rigidity"" conjecture on symplectic Calabi-Yau 6-manifolds. The overriding goal of the proposed project is to unravel the full implications of these new transversality techniques for problems in symplectic topology and neighboring fields. Examples of applications to be explored include: (1) Understanding the symplectic field theory of unit cotangent bundles for manifolds with negative or nonpositive curvature, with applications to the nearby Lagrangian conjecture and dynamical questions in Riemannian geometry; (2) Developing a comprehensive bifurcation theory for Reeb orbits and holomorphic curves in symplectic cobordisms, leading e.g. to a proof that planar contact structures are ""quasiflexible""; (3) Completing the analytical foundations of Hutchings's embedded contact homology (ECH), a 3-dimensional holomorphic curve theory with important applications to dynamics and symplectic embedding problems; (4) Developing new refinements of the Gromov-Witten invariants based on super-rigidity and bifurcation theory; (5) Defining higher-dimensional analogues of ECH; (6) Proving integrality relations in the setting of 6-dimensional symplectic cobordisms, analogous to the Gopakumar-Vafa formula for Calabi-Yau 3-folds."

1 624 500 €

Start date: 2018-09-01, End date: 2023-08-31