Project acronym BirNonArchGeom
Project Birational and non-archimedean geometries
Researcher (PI) Michael TEMKIN
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Consolidator Grant (CoG), PE1, ERC-2017-COG
Summary 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.
Summary
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.
Max ERC Funding
1 365 600 €
Duration
Start date: 2018-05-01, End date: 2023-04-30
Project acronym CSG
Project C° symplectic geometry
Researcher (PI) Lev Buhovski
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Starting Grant (StG), PE1, ERC-2017-STG
Summary "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."
Summary
"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."
Max ERC Funding
1 345 282 €
Duration
Start date: 2017-10-01, End date: 2022-09-30
Project acronym DMMCA
Project Discrete Mathematics: methods, challenges and applications
Researcher (PI) Noga Alon
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary Discrete Mathematics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. Some of the main reasons for this growth are the broad applications of tools and techniques from extremal and probabilistic combinatorics in the rapid development of theoretical Computer Science, in the spectacular recent results in Additive Number Theory and in the study of basic questions in Information Theory. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools, like the probabilistic method, algebraic, topological and geometric techniques. The work of the principal investigator, partly jointly with several collaborators and students, and partly in individual efforts, has played a significant role in the introduction of powerful algebraic, probabilistic, spectral and geometric techniques that influenced the development of modern combinatorics. In the present project he aims to try and further develop such tools, trying to tackle some basic open problems in Combinatorics, as well as significant questions in Additive Combinatorics, Information Theory, and theoretical Computer Science. Progress on the problems mentioned in this proposal, and the study of related ones, is expected to provide new insights on these problems and to lead to the development of novel fruitful techniques that are likely to be useful in Discrete Mathematics as well as in related areas.
Summary
Discrete Mathematics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. Some of the main reasons for this growth are the broad applications of tools and techniques from extremal and probabilistic combinatorics in the rapid development of theoretical Computer Science, in the spectacular recent results in Additive Number Theory and in the study of basic questions in Information Theory. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools, like the probabilistic method, algebraic, topological and geometric techniques. The work of the principal investigator, partly jointly with several collaborators and students, and partly in individual efforts, has played a significant role in the introduction of powerful algebraic, probabilistic, spectral and geometric techniques that influenced the development of modern combinatorics. In the present project he aims to try and further develop such tools, trying to tackle some basic open problems in Combinatorics, as well as significant questions in Additive Combinatorics, Information Theory, and theoretical Computer Science. Progress on the problems mentioned in this proposal, and the study of related ones, is expected to provide new insights on these problems and to lead to the development of novel fruitful techniques that are likely to be useful in Discrete Mathematics as well as in related areas.
Max ERC Funding
1 061 300 €
Duration
Start date: 2008-12-01, End date: 2013-11-30
Project acronym EXPANDERS
Project Expander Graphs in Pure and Applied Mathematics
Researcher (PI) Alexander Lubotzky
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary Expander graphs are finite graphs which play a fundamental role in many areas of computer science such as: communication networks, algorithms and more. Several areas of deep mathematics have been used in order to give explicit constructions of such graphs e.g. Kazhdan property (T) from representation theory of semisimple Lie groups, Ramanujan conjecture from the theory of automorphic forms and more. In recent years, computer science has started to pay its debt to mathematics: expander graphs are playing an increasing role in several areas of pure mathematics. The goal of the current research plan is to deepen these connections in both directions with special emphasis of the more recent and surprising application of expanders to group theory, the geometry of 3-manifolds and number theory.
Summary
Expander graphs are finite graphs which play a fundamental role in many areas of computer science such as: communication networks, algorithms and more. Several areas of deep mathematics have been used in order to give explicit constructions of such graphs e.g. Kazhdan property (T) from representation theory of semisimple Lie groups, Ramanujan conjecture from the theory of automorphic forms and more. In recent years, computer science has started to pay its debt to mathematics: expander graphs are playing an increasing role in several areas of pure mathematics. The goal of the current research plan is to deepen these connections in both directions with special emphasis of the more recent and surprising application of expanders to group theory, the geometry of 3-manifolds and number theory.
Max ERC Funding
1 082 504 €
Duration
Start date: 2008-10-01, End date: 2014-09-30
Project acronym FACT
Project Factorizing the wave function of large quantum systems
Researcher (PI) Eberhard Gross
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Advanced Grant (AdG), PE3, ERC-2017-ADG
Summary This proposal puts forth a novel strategy to tackle large quantum systems. A variety of highly sophisticated methods such as quantum Monte Carlo, configuration interaction, coupled cluster, tensor networks, Feynman diagrams, dynamical mean-field theory, density functional theory, and semi-classical techniques have been developed to deal with the enormous complexity of the many-particle Schrödinger equation. The goal of our proposal is not to add another method to these standard techniques but, instead, we develop a systematic way of combining them. The essential ingredient is a novel way of decomposing the wave function without approximation into factors that describe subsystems of the full quantum system. This so-called exact factorization is asymmetric. In the case of two subsystems, one factor is a wave function satisfying a regular Schrödinger equation, while the other factor is a conditional probability amplitude satisfying a more complicated Schrödinger-like equation with a non-local, non-linear and non-Hermitian “Hamiltonian”. Since each subsystem is necessarily smaller than the full system, the above standard techniques can be applied more efficiently and, most importantly, different standard techniques can be applied to different subsystems. The power of the exact factorization lies in its versatility. Here we apply the technique to five different scenarios: The first two deal with non-adiabatic effects in (i) molecules and (ii) solids. Here the natural subsystems are electrons and nuclei. The third scenario deals with nuclear motion in (iii) molecules attached to semi-infinite metallic leads, requiring three subsystems: the electrons, the nuclei in the leads which ultimately reduce to a phonon bath, and the molecular nuclei which may perform large-amplitude movements, such as current-induced isomerization, (iv) purely electronic correlations, and (v) the interaction of matter with the quantized electromagnetic field, i.e., electrons, nuclei and photons.
Summary
This proposal puts forth a novel strategy to tackle large quantum systems. A variety of highly sophisticated methods such as quantum Monte Carlo, configuration interaction, coupled cluster, tensor networks, Feynman diagrams, dynamical mean-field theory, density functional theory, and semi-classical techniques have been developed to deal with the enormous complexity of the many-particle Schrödinger equation. The goal of our proposal is not to add another method to these standard techniques but, instead, we develop a systematic way of combining them. The essential ingredient is a novel way of decomposing the wave function without approximation into factors that describe subsystems of the full quantum system. This so-called exact factorization is asymmetric. In the case of two subsystems, one factor is a wave function satisfying a regular Schrödinger equation, while the other factor is a conditional probability amplitude satisfying a more complicated Schrödinger-like equation with a non-local, non-linear and non-Hermitian “Hamiltonian”. Since each subsystem is necessarily smaller than the full system, the above standard techniques can be applied more efficiently and, most importantly, different standard techniques can be applied to different subsystems. The power of the exact factorization lies in its versatility. Here we apply the technique to five different scenarios: The first two deal with non-adiabatic effects in (i) molecules and (ii) solids. Here the natural subsystems are electrons and nuclei. The third scenario deals with nuclear motion in (iii) molecules attached to semi-infinite metallic leads, requiring three subsystems: the electrons, the nuclei in the leads which ultimately reduce to a phonon bath, and the molecular nuclei which may perform large-amplitude movements, such as current-induced isomerization, (iv) purely electronic correlations, and (v) the interaction of matter with the quantized electromagnetic field, i.e., electrons, nuclei and photons.
Max ERC Funding
2 443 932 €
Duration
Start date: 2019-09-01, End date: 2024-08-31
Project acronym FQHE
Project Statistics of Fractionally Charged Quasi-Particles
Researcher (PI) Mordehai (Moty) Heiblum
Host Institution (HI) WEIZMANN INSTITUTE OF SCIENCE
Call Details Advanced Grant (AdG), PE3, ERC-2008-AdG
Summary The discovery of the fractional quantum Hall effect created a revolution in solid state research by introducing a new state of matter resulting from strong electron interactions. The new state is characterized by excitations (quasi-particles) that carry fractional charge, which are expected to obey fractional statistics. While odd denominator fractional states are expected to have an abelian statistics, the newly discovered 5/2 even denominator fractional state is expected to have a non-abelian statistics. Moreover, a large number of emerging proposals predict that the latter state can be employed for topological quantum computing ( Station Q was founded by Microsoft Corp. in order to pursue this goal). This proposal aims at studying the abelian and non-abelian fractional charges, and in particular to observe their peculiar statistics. While charges are preferably determined by measuring quantum shot noise, their statistics must be determined via interference experiments, where one particle goes around another. The experiments are very demanding since the even denominator fractions turn to be very fragile and thus can be observed only in the purest possible two dimensional electron gas and at the lowest temperatures. While until very recently such high quality samples were available only by a single grower (in the USA), we have the capability now to grow extremely pure samples with profound even denominator states. As will be detailed in the proposal, we have all the necessary tools to study charge and statistics of these fascinating excitations, due to our experience in crystal growth, shot noise and interferometry measurements.
Summary
The discovery of the fractional quantum Hall effect created a revolution in solid state research by introducing a new state of matter resulting from strong electron interactions. The new state is characterized by excitations (quasi-particles) that carry fractional charge, which are expected to obey fractional statistics. While odd denominator fractional states are expected to have an abelian statistics, the newly discovered 5/2 even denominator fractional state is expected to have a non-abelian statistics. Moreover, a large number of emerging proposals predict that the latter state can be employed for topological quantum computing ( Station Q was founded by Microsoft Corp. in order to pursue this goal). This proposal aims at studying the abelian and non-abelian fractional charges, and in particular to observe their peculiar statistics. While charges are preferably determined by measuring quantum shot noise, their statistics must be determined via interference experiments, where one particle goes around another. The experiments are very demanding since the even denominator fractions turn to be very fragile and thus can be observed only in the purest possible two dimensional electron gas and at the lowest temperatures. While until very recently such high quality samples were available only by a single grower (in the USA), we have the capability now to grow extremely pure samples with profound even denominator states. As will be detailed in the proposal, we have all the necessary tools to study charge and statistics of these fascinating excitations, due to our experience in crystal growth, shot noise and interferometry measurements.
Max ERC Funding
2 000 000 €
Duration
Start date: 2009-01-01, End date: 2013-12-31
Project acronym HD-App
Project New horizons in homogeneous dynamics and its applications
Researcher (PI) Uri SHAPIRA
Host Institution (HI) TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY
Call Details Starting Grant (StG), PE1, ERC-2017-STG
Summary We present a large variety of novel lines of research in Homogeneous Dynamics with emphasis on the dynamics of the diagonal group. Both new and classical applications are suggested, most notably to
• Number Theory
• Geometry of Numbers
• Diophantine approximation.
Emphasis is given to applications in
• Diophantine properties of algebraic numbers.
The proposal is built of 4 sections.
(1) In the first section we discuss questions pertaining to topological and distributional aspects of periodic orbits of the diagonal group in the space of lattices in Euclidean space. These objects encode deep information regarding Diophantine properties of algebraic numbers. We demonstrate how these questions are closely related to, and may help solve, some of the central open problems in the geometry of numbers and Diophantine approximation.
(2) In the second section we discuss Minkowski's conjecture regarding integral values of products of linear forms. For over a century this central conjecture is resisting a general solution and a novel and promising strategy for its resolution is presented.
(3) In the third section, a novel conjecture regarding limiting distribution of infinite-volume-orbits is presented, in analogy with existing results regarding finite-volume-orbits. Then, a variety of applications and special cases are discussed, some of which give new results regarding classical concepts such as continued fraction expansion of rational numbers.
(4) In the last section we suggest a novel strategy to attack one of the most notorious open problems in Diophantine approximation, namely: Do cubic numbers have unbounded continued fraction expansion? This novel strategy leads us to embark on a systematic study of an area in homogeneous dynamics which has not been studied yet. Namely, the dynamics in the space of discrete subgroups of rank k in R^n (identified up to scaling).
Summary
We present a large variety of novel lines of research in Homogeneous Dynamics with emphasis on the dynamics of the diagonal group. Both new and classical applications are suggested, most notably to
• Number Theory
• Geometry of Numbers
• Diophantine approximation.
Emphasis is given to applications in
• Diophantine properties of algebraic numbers.
The proposal is built of 4 sections.
(1) In the first section we discuss questions pertaining to topological and distributional aspects of periodic orbits of the diagonal group in the space of lattices in Euclidean space. These objects encode deep information regarding Diophantine properties of algebraic numbers. We demonstrate how these questions are closely related to, and may help solve, some of the central open problems in the geometry of numbers and Diophantine approximation.
(2) In the second section we discuss Minkowski's conjecture regarding integral values of products of linear forms. For over a century this central conjecture is resisting a general solution and a novel and promising strategy for its resolution is presented.
(3) In the third section, a novel conjecture regarding limiting distribution of infinite-volume-orbits is presented, in analogy with existing results regarding finite-volume-orbits. Then, a variety of applications and special cases are discussed, some of which give new results regarding classical concepts such as continued fraction expansion of rational numbers.
(4) In the last section we suggest a novel strategy to attack one of the most notorious open problems in Diophantine approximation, namely: Do cubic numbers have unbounded continued fraction expansion? This novel strategy leads us to embark on a systematic study of an area in homogeneous dynamics which has not been studied yet. Namely, the dynamics in the space of discrete subgroups of rank k in R^n (identified up to scaling).
Max ERC Funding
1 432 730 €
Duration
Start date: 2018-10-01, End date: 2023-09-30
Project acronym INNOSTOCH
Project INNOVATIONS IN STOCHASTIC ANALYSIS AND APPLICATIONS with emphasis on STOCHASTIC CONTROL AND INFORMATION
Researcher (PI) Bernt Karsten Øksendal
Host Institution (HI) UNIVERSITETET I OSLO
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary "For almost all kinds of dynamic systems modeling real processes in nature or society, most of the mathematical models we can formulate are - at best - inaccurate, and subject to random fluctuations and other types of ""noise"". Therefore it is important to be able to deal with such noisy models in a mathematically rigorous way. This rigorous theory is stochastic analysis. Theoretical progress in stochastic analysis will lead to new and improved applications in a wide range of fields.
The main purpose of this proposal is to establish a research environment which enhances the creation of new ideas and methods in the research of stochastic analysis and its applications. The emphasis is more on innovation, new models and challenges in the research frontiers, rather than small variations and minor improvements of already established theories and results. We will concentrate on applications in finance and biology, but the theoretical results may as well apply to several other areas.
Utilizing recent results and achievements by PI and a large group of distinguished coworkers, the natural extensions from the present knowledge is to concentrate on the mathematical theory of the interplay between stochastic analysis, stochastic control and information. More precisely, we have ambitions to make fundamental progress in the general theory of stochastic control of random systems and applications in finance and biology, and the explicit relation between the optimal performance and the amount of information available to the controller. Explicit examples of special interest include optimal control under partial or delayed information, and optimal control under inside or advanced information. A success of the present proposal will represent a substantial breakthrough, and in turn bring us a significant step forward in our attempts to understand various aspects of the world better, and it will help us to find optimal, sustainable ways to influence it."
Summary
"For almost all kinds of dynamic systems modeling real processes in nature or society, most of the mathematical models we can formulate are - at best - inaccurate, and subject to random fluctuations and other types of ""noise"". Therefore it is important to be able to deal with such noisy models in a mathematically rigorous way. This rigorous theory is stochastic analysis. Theoretical progress in stochastic analysis will lead to new and improved applications in a wide range of fields.
The main purpose of this proposal is to establish a research environment which enhances the creation of new ideas and methods in the research of stochastic analysis and its applications. The emphasis is more on innovation, new models and challenges in the research frontiers, rather than small variations and minor improvements of already established theories and results. We will concentrate on applications in finance and biology, but the theoretical results may as well apply to several other areas.
Utilizing recent results and achievements by PI and a large group of distinguished coworkers, the natural extensions from the present knowledge is to concentrate on the mathematical theory of the interplay between stochastic analysis, stochastic control and information. More precisely, we have ambitions to make fundamental progress in the general theory of stochastic control of random systems and applications in finance and biology, and the explicit relation between the optimal performance and the amount of information available to the controller. Explicit examples of special interest include optimal control under partial or delayed information, and optimal control under inside or advanced information. A success of the present proposal will represent a substantial breakthrough, and in turn bring us a significant step forward in our attempts to understand various aspects of the world better, and it will help us to find optimal, sustainable ways to influence it."
Max ERC Funding
1 864 800 €
Duration
Start date: 2009-09-01, End date: 2014-08-31
Project acronym LEGOTOP
Project From Local Elements to Globally Ordered TOPological states of matter
Researcher (PI) Yuval OREG
Host Institution (HI) WEIZMANN INSTITUTE OF SCIENCE
Call Details Advanced Grant (AdG), PE3, ERC-2017-ADG
Summary We present a novel constructive approach for realizations of topological states of matter. Our approach starts with well-understood building blocks, and proceeds towards coupling them to create the desired states. This approach promises both a guide for a tunable experimental realization of states which have not been observed so far, and a theoretical tool for deeper understanding of different topological states, their dualities and inter-relations.
We will apply the constructive approach in two different directions. In the first direction our goal will be the construction of topological superconductors. Our tool will be Josephson junctions in which superconductors are coupled by two- and three-dimensional electronic non-superconducting systems. Two dimensional examples include transition metal dichalcogenides, Quantum Hall states, Quantum Anomalous Hall states, and the (111) bi-layer state, which may be viewed as a fractionalized electron-hole condensate. Three dimensional examples include Weyl semi-metals and weak topological insulators.
In the second direction our goal is the construction of fractionalized spin liquid states. Our building block will be a Majorana-Cooper box, which is a superconducting quantum dot coupled to semi-conducting wires that host Majorana zero modes. We will consider arrays of such boxes. The ratio of the box's charging energy to inter-box tunnel-coupling determines whether the array is superconducting or insulating. We will aim to use insulating arrays for realizing fractionalized and non-abelian spin liquids, study the transition to the superconducting state, and search for possible relations between the topological properties on both sides of the transition.
A deeper comprehension and a feasible path for realization of these states would have a profound effect on the field of topological matter and will open novel avenues for universal topological quantum computers.
Summary
We present a novel constructive approach for realizations of topological states of matter. Our approach starts with well-understood building blocks, and proceeds towards coupling them to create the desired states. This approach promises both a guide for a tunable experimental realization of states which have not been observed so far, and a theoretical tool for deeper understanding of different topological states, their dualities and inter-relations.
We will apply the constructive approach in two different directions. In the first direction our goal will be the construction of topological superconductors. Our tool will be Josephson junctions in which superconductors are coupled by two- and three-dimensional electronic non-superconducting systems. Two dimensional examples include transition metal dichalcogenides, Quantum Hall states, Quantum Anomalous Hall states, and the (111) bi-layer state, which may be viewed as a fractionalized electron-hole condensate. Three dimensional examples include Weyl semi-metals and weak topological insulators.
In the second direction our goal is the construction of fractionalized spin liquid states. Our building block will be a Majorana-Cooper box, which is a superconducting quantum dot coupled to semi-conducting wires that host Majorana zero modes. We will consider arrays of such boxes. The ratio of the box's charging energy to inter-box tunnel-coupling determines whether the array is superconducting or insulating. We will aim to use insulating arrays for realizing fractionalized and non-abelian spin liquids, study the transition to the superconducting state, and search for possible relations between the topological properties on both sides of the transition.
A deeper comprehension and a feasible path for realization of these states would have a profound effect on the field of topological matter and will open novel avenues for universal topological quantum computers.
Max ERC Funding
1 532 163 €
Duration
Start date: 2018-10-01, End date: 2023-09-30
Project acronym NANOSQUID
Project Scanning Nano-SQUID on a Tip
Researcher (PI) Eli Zeldov
Host Institution (HI) WEIZMANN INSTITUTE OF SCIENCE
Call Details Advanced Grant (AdG), PE3, ERC-2008-AdG
Summary At the boundaries of physics research it is constantly necessary to introduce new tools and methods to expand the horizons and address fundamental issues. In this proposal, we will develop and then apply radically new tools that will enable groundbreaking progress in the field of vortex matter in superconductors and will be of great importance to condensed matter physics and nanoscience. We propose a new scanning magnetic imaging method based on self-aligned fabrication of Josephson junctions with characteristic sizes of 10 nm and superconducting quantum interference devices (SQUID) with typical diameter of 100 nm on the end of a pulled quartz tip. Such nano-SQUID on a tip will provide high-sensitivity high-bandwidth mapping of static and dynamic magnetic fields on nanometer scale that is significantly beyond the state of the art. We will develop a new washboard frequency dynamic microscopy for imaging of site-dependent vortex velocities over a remarkable range of over six orders of magnitude in velocity that is expected to reveal the most interesting dynamic phenomena in vortex mater that could not be investigated so far. Our study will provide a novel bottom-up comprehension of microscopic vortex dynamics from single vortex up to numerous predicted dynamic phase transitions, including disorder-dependent depinning processes, plastic deformations, channel flow, metastabilities and memory effects, moving smectic, moving Bragg glass, and dynamic melting. We will also develop a hybrid technology that combines a single electron transistor with nano-SQUID which will provide an unprecedented simultaneous nanoscale imaging of magnetic and electric fields. Using these tools we will carry out innovative studies of additional nano-systems and exciting quantum phenomena, including quantum tunneling in molecular magnets, spin injection and magnetic domain wall dynamics, vortex charge, unconventional superconductivity, and coexistence of superconductivity and ferromagnetism.
Summary
At the boundaries of physics research it is constantly necessary to introduce new tools and methods to expand the horizons and address fundamental issues. In this proposal, we will develop and then apply radically new tools that will enable groundbreaking progress in the field of vortex matter in superconductors and will be of great importance to condensed matter physics and nanoscience. We propose a new scanning magnetic imaging method based on self-aligned fabrication of Josephson junctions with characteristic sizes of 10 nm and superconducting quantum interference devices (SQUID) with typical diameter of 100 nm on the end of a pulled quartz tip. Such nano-SQUID on a tip will provide high-sensitivity high-bandwidth mapping of static and dynamic magnetic fields on nanometer scale that is significantly beyond the state of the art. We will develop a new washboard frequency dynamic microscopy for imaging of site-dependent vortex velocities over a remarkable range of over six orders of magnitude in velocity that is expected to reveal the most interesting dynamic phenomena in vortex mater that could not be investigated so far. Our study will provide a novel bottom-up comprehension of microscopic vortex dynamics from single vortex up to numerous predicted dynamic phase transitions, including disorder-dependent depinning processes, plastic deformations, channel flow, metastabilities and memory effects, moving smectic, moving Bragg glass, and dynamic melting. We will also develop a hybrid technology that combines a single electron transistor with nano-SQUID which will provide an unprecedented simultaneous nanoscale imaging of magnetic and electric fields. Using these tools we will carry out innovative studies of additional nano-systems and exciting quantum phenomena, including quantum tunneling in molecular magnets, spin injection and magnetic domain wall dynamics, vortex charge, unconventional superconductivity, and coexistence of superconductivity and ferromagnetism.
Max ERC Funding
2 000 000 €
Duration
Start date: 2008-12-01, End date: 2013-11-30
Project acronym PolSymAGA
Project Polarity and Central-Symmetry in Asymptotic Geometric Analysis
Researcher (PI) Shiri ARTSTEIN
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Consolidator Grant (CoG), PE1, ERC-2017-COG
Summary Asymptotic Geometric Analysis is a relatively new field, the young finite dimensional cousin of Banach Space theory, functional analysis and classical convexity. It concerns the {\em geometric} study of high, but finite, dimensional objects, where the disorder of many parameters and many dimensions is "regularized" by convexity assumptions.
The proposed research is composed of several connected innovative studies in the frontier of Asymptotic Geometric Analysis, pertaining to the deeper understanding of two fundamental notions: Polarity and Central-Symmetry.
While the main drive comes from Asymptotic Convex Geometry, the applications extend throughout many mathematical fields from analysis, probability and symplectic geometry to combinatorics and computer science. The project will concern: The polarity map for functions, functional covering numbers, measures of Symmetry, Godbersen's conjecture, Mahler's conjecture, Minkowski billiard dynamics and caustics.
My research objectives are twofold. First, to progress towards a solution of the open research questions described in the proposal, which I consider to be pivotal in the field, including Mahler's conjecture, Viterbo's conjecture and Godberesen's conjecture. Some of these questions have already been studied intensively, and the solution is yet to found; progress toward solving them would be of high significance. Secondly, as the studies in this proposal lie at the meeting point of several mathematical fields, and use Asymptotic Geometric Analysis in order to address major questions in other fields, such as Symplectic Geometry and Optimal transport theory, my second goal is to deepen these connections, creating a powerful framework that will lead to a deeper understanding, and the formulation, and resolution, of interesting questions currently unattainable.
Summary
Asymptotic Geometric Analysis is a relatively new field, the young finite dimensional cousin of Banach Space theory, functional analysis and classical convexity. It concerns the {\em geometric} study of high, but finite, dimensional objects, where the disorder of many parameters and many dimensions is "regularized" by convexity assumptions.
The proposed research is composed of several connected innovative studies in the frontier of Asymptotic Geometric Analysis, pertaining to the deeper understanding of two fundamental notions: Polarity and Central-Symmetry.
While the main drive comes from Asymptotic Convex Geometry, the applications extend throughout many mathematical fields from analysis, probability and symplectic geometry to combinatorics and computer science. The project will concern: The polarity map for functions, functional covering numbers, measures of Symmetry, Godbersen's conjecture, Mahler's conjecture, Minkowski billiard dynamics and caustics.
My research objectives are twofold. First, to progress towards a solution of the open research questions described in the proposal, which I consider to be pivotal in the field, including Mahler's conjecture, Viterbo's conjecture and Godberesen's conjecture. Some of these questions have already been studied intensively, and the solution is yet to found; progress toward solving them would be of high significance. Secondly, as the studies in this proposal lie at the meeting point of several mathematical fields, and use Asymptotic Geometric Analysis in order to address major questions in other fields, such as Symplectic Geometry and Optimal transport theory, my second goal is to deepen these connections, creating a powerful framework that will lead to a deeper understanding, and the formulation, and resolution, of interesting questions currently unattainable.
Max ERC Funding
1 514 125 €
Duration
Start date: 2018-09-01, End date: 2023-08-31
Project acronym RMAST
Project Random Models in Arithmetic and Spectral Theory
Researcher (PI) Zeev Rudnick
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Advanced Grant (AdG), PE1, ERC-2017-ADG
Summary The proposal studies deterministic problems in the spectral theory of the Laplacian and in analytic number theory by using random models. I propose two projects in spectral theory on this theme, both with a strong arithmetic ingredient, the first about minimal gaps between the eigenvalues of the Laplacian, where I seek a fit with the corresponding quantities for the various conjectured universality classes (Poisson/GUE/GOE), and the second about curvature measures of nodal lines of eigenfunctions of the Laplacian, where I seek to determine the size of the curvature measures for the large eigenvalue limit. The third project originates in analytic number theory, on angular distribution of prime ideals in number fields, function field analogues and connections with Random Matrix Theory, where I raise new conjectures and problems on a very classical subject, and aim to resolve them at least in the function field setting.
Summary
The proposal studies deterministic problems in the spectral theory of the Laplacian and in analytic number theory by using random models. I propose two projects in spectral theory on this theme, both with a strong arithmetic ingredient, the first about minimal gaps between the eigenvalues of the Laplacian, where I seek a fit with the corresponding quantities for the various conjectured universality classes (Poisson/GUE/GOE), and the second about curvature measures of nodal lines of eigenfunctions of the Laplacian, where I seek to determine the size of the curvature measures for the large eigenvalue limit. The third project originates in analytic number theory, on angular distribution of prime ideals in number fields, function field analogues and connections with Random Matrix Theory, where I raise new conjectures and problems on a very classical subject, and aim to resolve them at least in the function field setting.
Max ERC Funding
1 840 625 €
Duration
Start date: 2019-02-01, End date: 2024-01-31
Project acronym ThermoQuantumImage
Project Thermal imaging of nano and atomic-scale dissipation in quantum states of matter
Researcher (PI) Elia ZELDOV
Host Institution (HI) WEIZMANN INSTITUTE OF SCIENCE
Call Details Advanced Grant (AdG), PE3, ERC-2017-ADG
Summary Energy dissipation is a fundamental process governing the dynamics of physical, chemical and biological systems and is of major importance in condensed matter physics where scattering, loss of quantum information, and even breakdown of topological protection are deeply linked to intricate details of how and where the dissipation occurs. But despite its vital importance, dissipation is currently not a readily measurable microscopic quantity. The aim of this proposal is to launch a new discipline of nanoscale dissipation imaging and spectroscopy and to apply it to study of quantum systems and novel states of matter. The proposed scanning thermal microscopy will be revolutionary in three aspects: the first-ever cryogenic thermal imaging; improvement of thermal sensitivity by five orders of magnitude over the state of the art; and imaging and spectroscopy of dissipation of single atomic defects. We will develop a superconducting quantum interference nano-thermometer on the apex of a sharp tip which will provide non-contact non-invasive low-temperature scanning thermal microscopy with unprecedented target sensitivity of 100 nK/Hz1/2 at 4 K. These advances will enable hitherto impossible direct thermal imaging of the most elemental processes such as phonon emission from a single atomic defect due to inelastic electron scattering, relaxation mechanisms in topological surface and edge states, and variation in dissipation in individual quantum dots due to single electron changes in their occupation. We will utilize this trailblazing tool to uncover nanoscale processes that lead to energy dissipation in novel systems including resonant quasi-bound edge states in graphene, helical surface states in topological insulators, and chiral anomaly in Weyl semimetals, and to provide groundbreaking insight into nonlocal dissipation and transport properties in mesoscopic systems and in 2D topological states of matter including quantum Hall, quantum anomalous Hall, and quantum spin Hall.
Summary
Energy dissipation is a fundamental process governing the dynamics of physical, chemical and biological systems and is of major importance in condensed matter physics where scattering, loss of quantum information, and even breakdown of topological protection are deeply linked to intricate details of how and where the dissipation occurs. But despite its vital importance, dissipation is currently not a readily measurable microscopic quantity. The aim of this proposal is to launch a new discipline of nanoscale dissipation imaging and spectroscopy and to apply it to study of quantum systems and novel states of matter. The proposed scanning thermal microscopy will be revolutionary in three aspects: the first-ever cryogenic thermal imaging; improvement of thermal sensitivity by five orders of magnitude over the state of the art; and imaging and spectroscopy of dissipation of single atomic defects. We will develop a superconducting quantum interference nano-thermometer on the apex of a sharp tip which will provide non-contact non-invasive low-temperature scanning thermal microscopy with unprecedented target sensitivity of 100 nK/Hz1/2 at 4 K. These advances will enable hitherto impossible direct thermal imaging of the most elemental processes such as phonon emission from a single atomic defect due to inelastic electron scattering, relaxation mechanisms in topological surface and edge states, and variation in dissipation in individual quantum dots due to single electron changes in their occupation. We will utilize this trailblazing tool to uncover nanoscale processes that lead to energy dissipation in novel systems including resonant quasi-bound edge states in graphene, helical surface states in topological insulators, and chiral anomaly in Weyl semimetals, and to provide groundbreaking insight into nonlocal dissipation and transport properties in mesoscopic systems and in 2D topological states of matter including quantum Hall, quantum anomalous Hall, and quantum spin Hall.
Max ERC Funding
3 075 000 €
Duration
Start date: 2018-06-01, End date: 2023-05-31
