ERC FUNDED PROJECTS

In a typical learning problem one tries to approximate an unknown function by a function from a given class using random data, sampled according to an unknown measure. In this project we will be interested in parameters that govern the complexity of a learning problem. It turns out that this complexity is determined by the geometry of certain sets in high dimension that are connected to the given class (random coordinate projections of the class). Thus, one has to understand the structure of these sets as a function of the dimension - which is given by the cardinality of the random sample. The resulting analysis leads to many theoretical questions in Asymptotic Geometric Analysis, Probability (most notably, Empirical Processes Theory) and Combinatorics, which are of independent interest beyond the application to Learning Theory. Our main goal is to describe the role of various complexity parameters involved in a learning problem, to analyze the connections between them and to investigate the way they determine the geometry of the relevant high dimensional sets. Some of the questions we intend to tackle are well known open problems and making progress towards their solution will have a significant theoretical impact. Moreover, this project should lead to a more complete theory of learning and is likely to have some practical impact, for example, in the design of more efficient learning algorithms.

750 000 €

Start date: 2009-03-01, End date: 2014-02-28

When triggered by nutrient limitation, the Gram-positive bacterium Bacillus subtilis and its relatives enter a pathway of cellular differentiation culminating in the formation of a dormant cell type called a spore, the most resilient cell type known. Bacterial spores can survive for long periods of time and are able to endure extremes of heat, radiation and chemical assault. Remarkably, dormant spores can rapidly convert back to actively growing cells by a process called germination. Consequently, spore forming bacteria, including dangerous pathogens, (such as C. botulinum and B. anthracis) are highly resistant to antibacterial treatments and difficult to eradicate. Despite significant advances in our understanding of the process of spore formation, little is known about the nature of the mature spore. It is unrevealed how dormancy is maintained within the spore and how it is ceased, as the organization and the dynamics of the spore macromolecules remain obscure. The unusual biochemical and biophysical characteristics of the dormant spore make it a challenging biological system to investigate using conventional methods, and thus set the need to develop innovative approaches to study spore biology. We propose to explore the nature of spores by using B. subtilis as a primary experimental system. We intend to: (1) define the architecture of the spore chromosome, (2) track the complexity and fate of mRNA and protein molecules during sporulation, dormancy and germination, (3) revisit the basic notion of the spore dormancy (is it metabolically inert?), (4) compare the characteristics of bacilli spores from diverse ecophysiological groups, (5) investigate the features of spores belonging to distant bacterial genera, (6) generate an integrative database that categorizes the molecular features of spores. Our study will provide original insights and introduce novel concepts to the field of spore biology and may help devise innovative ways to combat spore forming pathogens.

1 630 000 €

Start date: 2008-10-01, End date: 2013-09-30

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

The generation of animals by nuclear transfer demonstrated that the epigenetic state of somatic cells could be reset to an embryonic state, capable of directing the development of a new organism. The nuclear cloning technology is of interest for transplantation medicine, but any application is hampered by the inefficiency and ethical problems. A breakthrough solving these issues has been the in vitro derivation of reprogrammed Induced Pluripotent Stem “iPS” cells by the ectopic expression of defined transcription factors in somatic cells. iPS cells recapitulate all defining features of embryo-derived pluripotent stem cells, including the ability to differentiate into all somatic cell types. Further, recent publications have demonstrated the ability to directly trans-differentiate somatic cell types by ectopic expression of lineage specification factors. Thus, it is becoming increasingly clear that an ultimate goal in the stem cell field is to enable scientists to have the power to safely manipulate somatic cells by “reprogramming” their behavior at will. However, to frame this challenge, we must understand the basic mechanisms underlying the generation of reprogrammed cells in parallel to designing strategies for their medical application and their use in human disease specific research. In this ERC Starting Grant proposal, I describe comprehensive lines of experimentation that I plan to conduct in my new lab scheduled to open in April 2011 at the Weizmann Institute of Science. We will utilize exacting transgenic mammalian models and high throughput sequencing and genomic screening tools for in depth characterization of the molecular “rules” of rewiring the epigenome of somatic and pluripotent cell states. The proposed research endeavors will not only contribute to the development of safer strategies for cell reprogramming, but will also help decipher how diverse gene expression programs lead to cellular specification during normal development.

1 960 000 €

Start date: 2011-11-01, End date: 2016-10-31

"In recent years, the importance of superimposing the contribution of the measure to that of the metric, in determining the underlying space's (generalized Ricci) curvature, has been clarified in the works of Lott, Sturm, Villani and others, following the definition of Curvature-Dimension introduced by Bakry and Emery. We wish to systematically incorporate this important idea of considering the measure and metric in tandem, in the study of questions pertaining to isoperimetric and concentration properties of convex domains in high-dimensional Euclidean space, where a-priori there is only a trivial metric (Euclidean) and trivial measure (Lebesgue). The first step of enriching the class of uniform measures on convex domains to that of non-negatively curved (""log-concave"") measures in Euclidean space has been very successfully implemented in the last decades, leading to substantial progress in our understanding of volumetric properties of convex domains, mostly regarding concentration of linear functionals. However, the potential advantages of altering the Euclidean metric into a more general Riemannian one or exploiting related Riemannian structures have not been systematically explored. Our main paradigm is that in order to progress in non-linear questions pertaining to concentration in Euclidean space, it is imperative to cast and study these problems in the more general Riemannian context. As witnessed by our own work over the last years, we expect that broadening the scope and incorporating tools from the Riemannian world will lead to significant progress in our understanding of the qualitative and quantitative structure of isoperimetric minimizers in the purely Euclidean setting. Such progress would have dramatic impact on long-standing fundamental conjectures regarding concentration of measure on high-dimensional convex domains, as well as other closely related fields such as Probability Theory, Learning Theory, Random Matrix Theory and Algorithmic Geometry."

1 194 190 €

Start date: 2015-10-01, End date: 2020-09-30

"The objective of this proposal is to study ""continuous"" (or C^0) objects, as well as C^0 properties of smooth objects, in the field of symplectic geometry and topology. C^0 symplectic geometry has seen spectacular progress in recent years, drawing attention of mathematicians from various background. The proposed study aims to discover new fascinating C^0 phenomena in symplectic geometry. One circle of questions concerns symplectic and Hamiltonian homeomorphisms. Recent studies indicate that these objects possess both rigidity and flexibility, appearing in surprising and counter-intuitive ways. Our understanding of symplectic and Hamiltonian homeomorphisms is far from being satisfactory, and here we intend to study questions related to action of symplectic homeomorphisms on submanifolds. Some other questions are about Hamiltonian homeomorphisms in relation to the celebrated Arnold conjecture. The PI suggests to study spectral invariants of continuous Hamiltonian flows, which allow to formulate the C^0 Arnold conjecture in higher dimensions. Another central problem that the PI will work on is the C^0 flux conjecture. A second circle of questions is about the Poisson bracket operator, and its functional-theoretic properties. The first question concerns the lower bound for the Poisson bracket invariant of a cover, conjectured by L. Polterovich who indicated relations between this problem and quantum mechanics. Another direction aims to study the C^0 rigidity versus flexibility of the L_p norm of the Poisson bracket. Despite a recent progress in dimension two showing rigidity, very little is known in higher dimensions. The PI proposes to use combination of tools from topology and from hard analysis in order to address this question, whose solution will be a big step towards understanding functional-theoretic properties of the Poisson bracket operator."

1 345 282 €

Start date: 2017-10-01, End date: 2022-09-30

Model theory deals with general classes of structures (called models). Specific examples of such classes are: the class of rings or the class of algebraically closed fields. It turns out that counting the so-called complete types over models in the class has an important role in the development of model theory in general and stability theory in particular. Stable classes are those with relatively few complete types (over structures from the class); understanding stable classes has been central in model theory and its applications. Recently, I have proved a new dichotomy among the unstable classes: Instead of counting all the complete types, they are counted up to conjugacy. Classes which have few types up to conjugacy are proved to be so-called ``dependent'' classes (which have also been called NIP classes). I have developed (under reasonable restrictions) a ``recounting theorem'', parallel to the basic theorems of stability theory. I have started to develop some of the basic properties of this new approach. The goal of the current project is to develop systematically the theory of dependent classes. The above mentioned results give strong indication that this new theory can be eventually as useful as the (by now the classical) stability theory. In particular, it covers many well known classes which stability theory cannot treat.

1 748 000 €

Start date: 2014-03-01, End date: 2019-02-28

There are many parallels between Lie group actions on homogeneous spaces and the action of $\SL_2(\R)$ and its subgroups on strata of translation or half-translation surfaces. I propose to investigate these two spaces in parallel, focusing on the dynamical behavior, and more specifically, the description of orbit-closures. I intend to utilize existing and emerging measure rigidity results, and to develop new topological approaches. These should also shed light on the geometry and topology of the spaces. I propose to apply results concerning these spaces to the study of diophantine approximations (approximation on fractals), geometry of numbers (Minkowski's conjecture), interval exchanges, and rational billiards.

850 000 €

Start date: 2011-10-01, End date: 2016-09-30

Tame geometry studies structures in which every definable set has a finite geometric complexity. The study of tame geometry spans several interrelated mathematical fields, including semialgebraic, subanalytic, and o-minimal geometry. The past decade has seen the emergence of a spectacular link between tame geometry and arithmetic following the discovery of the fundamental Pila-Wilkie counting theorem and its applications in unlikely diophantine intersections. The P-W theorem itself relies crucially on the Yomdin-Gromov theorem, a classical result of tame geometry with fundamental applications in smooth dynamics. It is natural to ask whether the complexity of a tame set can be estimated effectively in terms of the defining formulas. While a large body of work is devoted to answering such questions in the semialgebraic case, surprisingly little is known concerning more general tame structures - specifically those needed in recent applications to arithmetic. The nature of the link between tame geometry and arithmetic is such that any progress toward effectivizing the theory of tame structures will likely lead to effective results in the domain of unlikely intersections. Similarly, a more effective version of the Yomdin-Gromov theorem is known to imply important consequences in smooth dynamics. The proposed research will approach effectivity in tame geometry from a fundamentally new direction, bringing to bear methods from the theory of differential equations which have until recently never been used in this context. Toward this end, our key goals will be to gain insight into the differential algebraic and complex analytic structure of tame sets; and to apply this insight in combination with results from the theory of differential equations to effectivize key results in tame geometry and its applications to arithmetic and dynamics. I believe that my preliminary work in this direction amply demonstrates the feasibility and potential of this approach.

1 155 027 €

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