ERC FUNDED PROJECTS

Optical microscopy, perhaps the most important tool in biomedical investigation and clinical diagnostics, is currently held back by the assumption that it is not possible to noninvasively image microscopic structures more than a fraction of a millimeter deep inside tissue. The governing paradigm is that high-resolution information carried by light is lost due to random scattering in complex samples such as tissue. While non-optical imaging techniques, employing non-ionizing radiation such as ultrasound, allow deeper investigations, they possess drastically inferior resolution and do not permit microscopic studies of cellular structures, crucial for accurate diagnosis of cancer and other diseases. I propose a new kind of microscope, one that can peer deep inside visually opaque samples, combining the sub-micron resolution of light with the penetration depth of ultrasound. My novel approach is based on our discovery that information on microscopic structures is contained in random scattered-light patterns. It breaks current limits by exploiting the randomness of scattered light rather than struggling to fight it. We will transform this concept into a breakthrough imaging platform by combining ultrasonic probing and modulation of light with advanced digital signal processing algorithms, extracting the hidden microscopic structure by two complementary approaches: 1) By exploiting the stochastic dynamics of scattered light using methods developed to surpass the diffraction limit in optical nanoscopy and for compressive sampling, harnessing nonlinear effects. 2) Through the analysis of intrinsic correlations in scattered light that persist deep inside scattering tissue. This proposal is formed by bringing together novel insights on the physics of light in complex media, advanced microscopy techniques, and ultrasound-mediated imaging. It is made possible by the new ability to digitally process vast amounts of scattering data, and has the potential to impact many fields.

1 500 000 €

Start date: 2016-04-01, End date: 2021-03-31

"Expander graphs have been playing a fundamental role in many areas of computer science. During the last 15 years they have also found important and unexpected applications in pure mathematics. The goal of the current research is to develop systematically high-dimensional (HD) theory of expanders, i.e., simplicial complexes and hypergraphs which resemble in dimension d, the role of expander graphs for d = 1. There are several motivations for developing such a theory, some from pure mathematics and some from computer science. For example, Ramanujan complexes (the HD versions of the ""optimal"" expanders, the Ramanujan graphs) have already been useful for extremal hypergraph theory. One of the main goals of this research is to use them to solve other problems, such as Gromov's problem: are there bounded degree simplicial complexes with the topological overlapping property (""topological expanders""). Other directions of HD expanders have applications in property testing, a very important subject in theoretical computer science. Moreover they can be a tool for the construction of locally testable codes, an important question of theoretical and practical importance in the theory of error correcting codes. In addition, the study of these simplicial complexes suggests new quantum error correcting codes (QECC). It is hoped that it will lead to such codes which are also low density parity check (LDPC). The huge success and impact of the theory of expander graphs suggests that the high dimensional theory will also bring additional unexpected applications beside those which can be foreseen as of now."

1 592 500 €

Start date: 2016-08-01, End date: 2021-07-31

Mathematical statistical physics has seen spectacular progress in recent years. Existing problems which were previously unattainable were solved, opening a way to approach some of the classical open questions in the field. The proposed research focuses on phenomena of localization and long-range order in physical systems of large size, identifying several fundamental questions lying at the interface of Statistical Physics, Probability Theory and Combinatorics. One circle of questions concerns the fluctuation behavior of random surfaces, where the PI has recently proved delocalization in two dimensions answering a 1975 question of Brascamp, Lieb and Lebowitz. A main goal of the research is to establish some of the long-standing universality conjectures for random surfaces. This study is also tied to the localization features of random operators, such as random Schrodinger operators and band matrices, as well as those of reinforced random walks. The PI intends to develop this connection further to bring the state-of-the-art to the conjectured thresholds. A second circle of questions regards long-range order in high-dimensional systems. This phenomenon is predicted to encompass many models of statistical physics but rigorous results are quite limited. A notable example is the PI’s proof of Kotecky’s 1985 conjecture on the rigidity of proper 3-colorings in high dimensions. The methods used in this context are not limited to high dimensions and were recently used by the PI to prove the analogue for the loop O(n) model of Polyakov’s 1975 prediction that the 2D Heisenberg model and its higher spin versions exhibit exponential decay of correlations at any temperature. Lastly, statistical physics methods are proposed for solving purely combinatorial problems. The PI has applied this approach successfully to solve questions of existence and asymptotics for combinatorial structures and intends to develop it further to answer some of the tantalizing open questions in the field.

1 136 904 €

Start date: 2016-01-01, End date: 2020-12-31

In today’s electronics, the information storage and processing are performed by independent technologies. The information-processing is based on semiconductor (silicon) devices, while non-volatile data storage relies on ferromagnetic metals. Integrating these tasks on a single chip and within the same material technology would enable disruptively new device concepts opening the way towards ultra-high speed electronic circuits. Due to the unique versatility of its electronic and magnetic properties, graphene has a strong potential as a platform for the implementation of such devices. By engineering their structure at the atomic level, graphene nanostructures of metallic, semiconducting, as well as magnetic properties can be realized. Here we propose that the unmatched precision and full edge orientation control of our STM-based nanofabrication technique enables the reliable implementation of such graphene nanostructures, as well as their complex, functional networks. In particular, we propose to experimentally demonstrate the feasibility of (1) semiconductor graphene nanostructures based on the quantum confinement effect, (2) spin-based devices from graphene nanostructures with magnetic edges, as well as (3) novel operation principles based on the interplay of the electronic and spin-degrees of freedom. We propose to demonstrate the electrical control of magnetism in graphene nanostructures, as well as a novel switching mechanism for graphene field effect transistors induced by the transition between two magnetic edge configurations. Exploiting such novel operation mechanisms in graphene nanostructure engineered at the atomic scale is expected to lay the foundations of disruptively new device concepts combining electronic and spin-based mechanisms that can overcome some of the fundamental limitations of today’s electronics.

1 496 500 €

Start date: 2016-07-01, End date: 2021-06-30

The objective of this proposal is an investigation of the geometric structure of random spaces that arise in critical models of statistical physics. The proposal is motivated by inspiring yet non-rigorous predictions from the physics community and the models studied are some of the most popular models in contemporary probability theory such as percolation, random planar maps and random walks. One set of problems are on the topic of random planar maps and quantum gravity, a thriving field on the intersection of probability, statistical physics, combinatorics and complex analysis. Our goal is to develop a rigorous theory of these maps viewed as surfaces (rather than metric spaces) via their circle packing. The circle packing structure was recently used by the PI and Gurel-Gurevich to show that these maps are a.s. recurrent, resolving a major conjecture in this area. Among other consequences, this research will hopefully lead to progress on the most important open problem in this field: a rigorous proof of the mysterious KPZ correspondence, a conjectural formula from the physics literature allowing to compute dimensions of certain random sets in the usual square lattice from the corresponding dimension in the random geometry. Such a program will hopefully lead to the solution of the most central problems in two-dimensional statistical physics, such as finding the typical displacement of the self-avoiding walk, proving conformal invariance for percolation on the square lattice and many others. Another set of problems is investigating aspects of universality in critical percolation in various high-dimensional graphs. These graphs include lattices in dimension above 6, Cayley graphs of finitely generated non-amenable groups and also finite graphs such as the complete graph, the Hamming hypercube and expanders. It is believed that critical percolation on these graphs is universal in the sense that the resulting percolated clusters exhibit the same mean-field geometry.

1 286 150 €

Start date: 2016-01-01, End date: 2020-12-31