Project acronym AGEnTh
Project Atomic Gauge and Entanglement Theories
Researcher (PI) Marcello DALMONTE
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Starting Grant (StG), PE2, ERC-2017-STG
Summary AGEnTh is an interdisciplinary proposal which aims at theoretically investigating atomic many-body systems (cold atoms and trapped ions) in close connection to concepts from quantum information, condensed matter, and high energy physics. The main goals of this programme are to:
I) Find to scalable schemes for the measurements of entanglement properties, and in particular entanglement spectra, by proposing a shifting paradigm to access entanglement focused on entanglement Hamiltonians and field theories instead of probing density matrices;
II) Show how atomic gauge theories (including dynamical gauge fields) are ideal candidates for the realization of long-sought, highly-entangled states of matter, in particular topological superconductors supporting parafermion edge modes, and novel classes of quantum spin liquids emerging from clustering;
III) Develop new implementation strategies for the realization of gauge symmetries of paramount importance, such as discrete and SU(N)xSU(2)xU(1) groups, and establish a theoretical framework for the understanding of atomic physics experiments within the light-from-chaos scenario pioneered in particle physics.
These objectives are at the cutting-edge of fundamental science, and represent a coherent effort aimed at underpinning unprecedented regimes of strongly interacting quantum matter by addressing the basic aspects of probing, many-body physics, and implementations. The results are expected to (i) build up and establish qualitatively new synergies between the aforementioned communities, and (ii) stimulate an intense theoretical and experimental activity focused on both entanglement and atomic gauge theories.
In order to achieve those, AGEnTh builds: (1) on my background working at the interface between atomic physics and quantum optics from one side, and many-body theory on the other, and (2) on exploratory studies which I carried out to mitigate the conceptual risks associated with its high-risk/high-gain goals.
Summary
AGEnTh is an interdisciplinary proposal which aims at theoretically investigating atomic many-body systems (cold atoms and trapped ions) in close connection to concepts from quantum information, condensed matter, and high energy physics. The main goals of this programme are to:
I) Find to scalable schemes for the measurements of entanglement properties, and in particular entanglement spectra, by proposing a shifting paradigm to access entanglement focused on entanglement Hamiltonians and field theories instead of probing density matrices;
II) Show how atomic gauge theories (including dynamical gauge fields) are ideal candidates for the realization of long-sought, highly-entangled states of matter, in particular topological superconductors supporting parafermion edge modes, and novel classes of quantum spin liquids emerging from clustering;
III) Develop new implementation strategies for the realization of gauge symmetries of paramount importance, such as discrete and SU(N)xSU(2)xU(1) groups, and establish a theoretical framework for the understanding of atomic physics experiments within the light-from-chaos scenario pioneered in particle physics.
These objectives are at the cutting-edge of fundamental science, and represent a coherent effort aimed at underpinning unprecedented regimes of strongly interacting quantum matter by addressing the basic aspects of probing, many-body physics, and implementations. The results are expected to (i) build up and establish qualitatively new synergies between the aforementioned communities, and (ii) stimulate an intense theoretical and experimental activity focused on both entanglement and atomic gauge theories.
In order to achieve those, AGEnTh builds: (1) on my background working at the interface between atomic physics and quantum optics from one side, and many-body theory on the other, and (2) on exploratory studies which I carried out to mitigate the conceptual risks associated with its high-risk/high-gain goals.
Max ERC Funding
1 055 317 €
Duration
Start date: 2018-05-01, End date: 2023-04-30
Project acronym AISENS
Project New generation of high sensitive atom interferometers
Researcher (PI) Marco Fattori
Host Institution (HI) CONSIGLIO NAZIONALE DELLE RICERCHE
Call Details Starting Grant (StG), PE2, ERC-2010-StG_20091028
Summary Interferometers are fundamental tools for the study of nature laws and for the precise measurement and control of the physical world. In the last century, the scientific and technological progress has proceeded in parallel with a constant improvement of interferometric performances. For this reason, the challenge of conceiving and realizing new generations of interferometers with broader ranges of operation and with higher sensitivities is always open and actual.
Despite the introduction of laser devices has deeply improved the way of developing and performing interferometric measurements with light, the atomic matter wave analogous, i.e. the Bose-Einstein condensate (BEC), has not yet triggered any revolution in precision interferometry. However, thanks to recent improvements on the control of the quantum properties of ultra-cold atomic gases, and new original ideas on the creation and manipulation of quantum entangled particles, the field of atom interferometry is now mature to experience a big step forward.
The system I want to realize is a Mach-Zehnder spatial interferometer operating with trapped BECs. Undesired decoherence sources will be suppressed by implementing BECs with tunable interactions in ultra-stable optical potentials. Entangled states will be used to improve the sensitivity of the sensor beyond the standard quantum limit to ideally reach the ultimate, Heisenberg, limit set by quantum mechanics. The resulting apparatus will show unprecedented spatial resolution and will overcome state-of-the-art interferometers with cold (non condensed) atomic gases.
A successful completion of this project will lead to a new generation of interferometers for the immediate application to local inertial measurements with unprecedented resolution. In addition, we expect to develop experimental capabilities which might find application well beyond quantum interferometry and crucially contribute to the broader emerging field of quantum-enhanced technologies.
Summary
Interferometers are fundamental tools for the study of nature laws and for the precise measurement and control of the physical world. In the last century, the scientific and technological progress has proceeded in parallel with a constant improvement of interferometric performances. For this reason, the challenge of conceiving and realizing new generations of interferometers with broader ranges of operation and with higher sensitivities is always open and actual.
Despite the introduction of laser devices has deeply improved the way of developing and performing interferometric measurements with light, the atomic matter wave analogous, i.e. the Bose-Einstein condensate (BEC), has not yet triggered any revolution in precision interferometry. However, thanks to recent improvements on the control of the quantum properties of ultra-cold atomic gases, and new original ideas on the creation and manipulation of quantum entangled particles, the field of atom interferometry is now mature to experience a big step forward.
The system I want to realize is a Mach-Zehnder spatial interferometer operating with trapped BECs. Undesired decoherence sources will be suppressed by implementing BECs with tunable interactions in ultra-stable optical potentials. Entangled states will be used to improve the sensitivity of the sensor beyond the standard quantum limit to ideally reach the ultimate, Heisenberg, limit set by quantum mechanics. The resulting apparatus will show unprecedented spatial resolution and will overcome state-of-the-art interferometers with cold (non condensed) atomic gases.
A successful completion of this project will lead to a new generation of interferometers for the immediate application to local inertial measurements with unprecedented resolution. In addition, we expect to develop experimental capabilities which might find application well beyond quantum interferometry and crucially contribute to the broader emerging field of quantum-enhanced technologies.
Max ERC Funding
1 068 000 €
Duration
Start date: 2011-01-01, End date: 2015-12-31
Project acronym COMBOS
Project Collective phenomena in quantum and classical many body systems
Researcher (PI) Alessandro Giuliani
Host Institution (HI) UNIVERSITA DEGLI STUDI ROMA TRE
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The collective behavior of quantum and classical many body systems such as ultracold atomic gases, nanowires, cuprates and micromagnets are currently subject of an intense experimental and theoretical research worldwide. Understanding the fascinating phenomena of Bose-Einstein condensation, Luttinger liquid vs non-Luttinger liquid behavior, high temperature superconductivity, and spontaneous formation of periodic patterns in magnetic systems, is an exciting challenge for theoreticians. Most of these phenomena are still far from being fully understood, even from a heuristic point of view. Unveiling the exotic properties of such systems by rigorous mathematical analysis is an important and difficult challenge for mathematical physics. In the last two decades, substantial progress has been made on various aspects of many-body theory, including Fermi liquids, Luttinger liquids, perturbed Ising models at criticality, bosonization, trapped Bose gases and spontaneous formation of periodic patterns. The techniques successfully employed in this field are diverse, and range from constructive renormalization group to functional variational estimates. In this research project we propose to investigate a number of statistical mechanics models by a combination of different mathematical methods. The objective is, on the one hand, to understand crossover phenomena, phase transitions and low-temperature states with broken symmetry, which are of interest in the theory of condensed matter and that we believe to be accessible to the currently available methods; on the other, to develop new techiques combining different and complementary methods, such as multiscale analysis and localization bounds, or reflection positivity and cluster expansion, which may be useful to further progress on important open problems, such as Bose-Einstein condensation, conformal invariance in non-integrable models, existence of magnetic or superconducting long range order.
Summary
The collective behavior of quantum and classical many body systems such as ultracold atomic gases, nanowires, cuprates and micromagnets are currently subject of an intense experimental and theoretical research worldwide. Understanding the fascinating phenomena of Bose-Einstein condensation, Luttinger liquid vs non-Luttinger liquid behavior, high temperature superconductivity, and spontaneous formation of periodic patterns in magnetic systems, is an exciting challenge for theoreticians. Most of these phenomena are still far from being fully understood, even from a heuristic point of view. Unveiling the exotic properties of such systems by rigorous mathematical analysis is an important and difficult challenge for mathematical physics. In the last two decades, substantial progress has been made on various aspects of many-body theory, including Fermi liquids, Luttinger liquids, perturbed Ising models at criticality, bosonization, trapped Bose gases and spontaneous formation of periodic patterns. The techniques successfully employed in this field are diverse, and range from constructive renormalization group to functional variational estimates. In this research project we propose to investigate a number of statistical mechanics models by a combination of different mathematical methods. The objective is, on the one hand, to understand crossover phenomena, phase transitions and low-temperature states with broken symmetry, which are of interest in the theory of condensed matter and that we believe to be accessible to the currently available methods; on the other, to develop new techiques combining different and complementary methods, such as multiscale analysis and localization bounds, or reflection positivity and cluster expansion, which may be useful to further progress on important open problems, such as Bose-Einstein condensation, conformal invariance in non-integrable models, existence of magnetic or superconducting long range order.
Max ERC Funding
650 000 €
Duration
Start date: 2010-01-01, End date: 2014-12-31
Project acronym CONLAWS
Project Hyperbolic Systems of Conservation Laws: singular limits, properties of solutions and control problems
Researcher (PI) Stefano Bianchini
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The research program concerns various theoretic aspects of hyperbolic conservation laws. In first place we plan to study the existence and uniqueness of solutions to systems of equations of mathematical physics with physic viscosity. This is one of the main open problems within the theory of conservation laws in one space dimension, which cannot be tackled relying on the techniques developed in the case where the viscosity matrix is the identity. Furthermore, this represents a first step toward the analysis of more complex relaxation and kinetic models with a finite number of velocities as for Broadwell equation, or with a continuous distribution of velocities as for Boltzmann equation. A second research topic concerns the study of conservation laws with large data. Even in this case the basic model is provided by fluidodynamic equations. We wish to extend the results of existence, uniqueness and continuous dependence of solutions to the case of large (in BV or in L^infty) data, at least for the simplest systems of mathematical physics such as the isentropic gas dynamics. A third research topic that we wish to pursue concerns the analysis of fine properties of solutions to conservation laws. Many of such properties depend on the existence of one or more entropies of the system. In particular, we have in mind to study the regularity and the concentration of the dissipativity measure for an entropic solution of a system of conservation laws. Finally, we wish to continue the study of hyperbolic equations from the control theory point of view along two directions: (i) the analysis of controllability and asymptotic stabilizability properties; (ii) the study of optimal control problems related to hyperbolic systems.
Summary
The research program concerns various theoretic aspects of hyperbolic conservation laws. In first place we plan to study the existence and uniqueness of solutions to systems of equations of mathematical physics with physic viscosity. This is one of the main open problems within the theory of conservation laws in one space dimension, which cannot be tackled relying on the techniques developed in the case where the viscosity matrix is the identity. Furthermore, this represents a first step toward the analysis of more complex relaxation and kinetic models with a finite number of velocities as for Broadwell equation, or with a continuous distribution of velocities as for Boltzmann equation. A second research topic concerns the study of conservation laws with large data. Even in this case the basic model is provided by fluidodynamic equations. We wish to extend the results of existence, uniqueness and continuous dependence of solutions to the case of large (in BV or in L^infty) data, at least for the simplest systems of mathematical physics such as the isentropic gas dynamics. A third research topic that we wish to pursue concerns the analysis of fine properties of solutions to conservation laws. Many of such properties depend on the existence of one or more entropies of the system. In particular, we have in mind to study the regularity and the concentration of the dissipativity measure for an entropic solution of a system of conservation laws. Finally, we wish to continue the study of hyperbolic equations from the control theory point of view along two directions: (i) the analysis of controllability and asymptotic stabilizability properties; (ii) the study of optimal control problems related to hyperbolic systems.
Max ERC Funding
422 000 €
Duration
Start date: 2009-11-01, End date: 2013-10-31
Project acronym CRIPHERASY
Project Critical Phenomena in Random Systems
Researcher (PI) Giorgio Parisi
Host Institution (HI) UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA
Call Details Advanced Grant (AdG), PE2, ERC-2009-AdG
Summary This project aims to get a theoretical understanding of the most important large-scale phenomena in classical and quantum disordered systems. Thanks to the renormalization group approach the critical behaviour of pure systems is under very good control; however disordered systems are in many ways remarkably peculiar (think for example to non-perturbative phenomena like Griffiths singularities), often the conventional approach does not work and many crucial issues are still unclear. My work aims to fill this important hole in our understanding of disordered systems. I will concentrate my efforts on some of the most important and studied systems, i.e. spin glasses, random field ferromagnets (that are realized in nature as diluted antiferromagnets in a field), Anderson and Mott localization (with possible experimental applications to Bose-Einstein condensates and to electron glasses), surface growth in random media (KPZ and DLA models). In this project I want to pursue a new approach to these problems. I aim to compute in the most accurate way the properties of these systems using the original Wilson formulation of the renormalization group with a phase space cell analysis; this is equivalent to solving a statistical model on a hierarchical lattice (Dyson-Bleher-Sinai model). This is not an easy job. In the same conceptual frame we plan to use simultaneously very different techniques: probabilistic techniques, perturbative techniques at high orders, expansions around mean field on Bethe lattice and numerical techniques to evaluate the critical behaviour. I believe that even this restricted approach is very ambitious, but that the theoretical progresses that have been done in unveiling important features of disordered systems suggest that it will be possible to obtain solid results.
Summary
This project aims to get a theoretical understanding of the most important large-scale phenomena in classical and quantum disordered systems. Thanks to the renormalization group approach the critical behaviour of pure systems is under very good control; however disordered systems are in many ways remarkably peculiar (think for example to non-perturbative phenomena like Griffiths singularities), often the conventional approach does not work and many crucial issues are still unclear. My work aims to fill this important hole in our understanding of disordered systems. I will concentrate my efforts on some of the most important and studied systems, i.e. spin glasses, random field ferromagnets (that are realized in nature as diluted antiferromagnets in a field), Anderson and Mott localization (with possible experimental applications to Bose-Einstein condensates and to electron glasses), surface growth in random media (KPZ and DLA models). In this project I want to pursue a new approach to these problems. I aim to compute in the most accurate way the properties of these systems using the original Wilson formulation of the renormalization group with a phase space cell analysis; this is equivalent to solving a statistical model on a hierarchical lattice (Dyson-Bleher-Sinai model). This is not an easy job. In the same conceptual frame we plan to use simultaneously very different techniques: probabilistic techniques, perturbative techniques at high orders, expansions around mean field on Bethe lattice and numerical techniques to evaluate the critical behaviour. I believe that even this restricted approach is very ambitious, but that the theoretical progresses that have been done in unveiling important features of disordered systems suggest that it will be possible to obtain solid results.
Max ERC Funding
2 098 800 €
Duration
Start date: 2010-01-01, End date: 2014-12-31
Project acronym CRITIQUEUE
Project Critical queues and reflected stochastic processes
Researcher (PI) Johannes S.H. Van Leeuwaarden
Host Institution (HI) TECHNISCHE UNIVERSITEIT EINDHOVEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary Our primary motivation stems from queueing theory, the branch of applied probability that deals with congestion phenomena. Congestion levels are typically nonnegative, which is why reflected stochastic processes arise naturally in queueing theory. Other applications of reflected stochastic processes are in the fields of branching processes and random graphs.
We are particularly interested in critically-loaded queueing systems (close to 100% utilization), also referred to as queues in heavy traffic. Heavy-traffic analysis typically reduces complicated queueing processes to much simpler (reflected) limit processes or scaling limits. This makes the analysis of complex systems tractable, and from a mathematical point of view, these results are appealing since they can be made rigorous. Within the large
body of literature on heavy-traffic theory and critical stochastic processes, we launch two new research lines:
(i) Time-dependent analysis through scaling limits.
(ii) Dimensioning stochastic systems via refined scaling limits and optimization.
Both research lines involve mathematical techniques that combine stochastic theory with asymptotic theory, complex analysis, functional analysis, and modern probabilistic methods. It will provide a platform enabling collaborations between researchers in pure and applied probability and researchers in performance analysis of queueing systems. This will particularly be the case at TU/e, the host institution, and at
the affiliated institution EURANDOM.
Summary
Our primary motivation stems from queueing theory, the branch of applied probability that deals with congestion phenomena. Congestion levels are typically nonnegative, which is why reflected stochastic processes arise naturally in queueing theory. Other applications of reflected stochastic processes are in the fields of branching processes and random graphs.
We are particularly interested in critically-loaded queueing systems (close to 100% utilization), also referred to as queues in heavy traffic. Heavy-traffic analysis typically reduces complicated queueing processes to much simpler (reflected) limit processes or scaling limits. This makes the analysis of complex systems tractable, and from a mathematical point of view, these results are appealing since they can be made rigorous. Within the large
body of literature on heavy-traffic theory and critical stochastic processes, we launch two new research lines:
(i) Time-dependent analysis through scaling limits.
(ii) Dimensioning stochastic systems via refined scaling limits and optimization.
Both research lines involve mathematical techniques that combine stochastic theory with asymptotic theory, complex analysis, functional analysis, and modern probabilistic methods. It will provide a platform enabling collaborations between researchers in pure and applied probability and researchers in performance analysis of queueing systems. This will particularly be the case at TU/e, the host institution, and at
the affiliated institution EURANDOM.
Max ERC Funding
970 800 €
Duration
Start date: 2010-08-01, End date: 2016-07-31
Project acronym DAMESYFLA
Project Electroweak Symmetry Breaking, Flavor and Dark
Matter: One Solution for Three Mysteries
Researcher (PI) Guido Martinelli
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Advanced Grant (AdG), PE2, ERC-2010-AdG_20100224
Summary In the next five years, experiments will give us a unique opportunity to unravel the mysteries of Electroweak Symmetry Breaking, Flavor and Dark Matter. The LHC at CERN will push the Energy frontier well into the TeV region and shed light on electroweak symmetry breaking. The LHCb experiment, super-B factories and other dedicated experiments, also in the lepton sector, will push forward the Intensity frontier and test the Standard Model description of flavor and CP violation with unprecedented accuracy. Earth- and space-based experiments will push forward the Astroparticle frontier, in particular direct and indirect searches for Dark Matter. My goal is to identify a coherent explanation of the three mysteries, as complete and as unique as possible, by combining the vast information coming from the Energy, Intensity and Astroparticle frontiers. This requires a global strategy, making use of highly qualified competences in the relevant branches of theory and phenomenology. I will put together some of the leading particle theorists operating in SISSA, Padua and Rome into a unique and extraordinarily strong team. The variety of competences, ranging from phenomenological fits and data interpretation to unified models and fundamental theories, will be used to interpret the results coming from a wide range of experiments and to formulate a coherent framework to account for them. With the essential contribution of the researchers paid on the project funds, the project will catalyze results going much beyond what the team members could individually achieve. The main support requested to the ERC is for hiring six experienced researchers, the rest of the funds are for optimizing the effectiveness of the team and the research environment.
Summary
In the next five years, experiments will give us a unique opportunity to unravel the mysteries of Electroweak Symmetry Breaking, Flavor and Dark Matter. The LHC at CERN will push the Energy frontier well into the TeV region and shed light on electroweak symmetry breaking. The LHCb experiment, super-B factories and other dedicated experiments, also in the lepton sector, will push forward the Intensity frontier and test the Standard Model description of flavor and CP violation with unprecedented accuracy. Earth- and space-based experiments will push forward the Astroparticle frontier, in particular direct and indirect searches for Dark Matter. My goal is to identify a coherent explanation of the three mysteries, as complete and as unique as possible, by combining the vast information coming from the Energy, Intensity and Astroparticle frontiers. This requires a global strategy, making use of highly qualified competences in the relevant branches of theory and phenomenology. I will put together some of the leading particle theorists operating in SISSA, Padua and Rome into a unique and extraordinarily strong team. The variety of competences, ranging from phenomenological fits and data interpretation to unified models and fundamental theories, will be used to interpret the results coming from a wide range of experiments and to formulate a coherent framework to account for them. With the essential contribution of the researchers paid on the project funds, the project will catalyze results going much beyond what the team members could individually achieve. The main support requested to the ERC is for hiring six experienced researchers, the rest of the funds are for optimizing the effectiveness of the team and the research environment.
Max ERC Funding
1 439 400 €
Duration
Start date: 2011-04-01, End date: 2017-03-31
Project acronym DarkGRA
Project Unveiling the dark universe with gravitational waves: Black holes and compact stars as laboratories for fundamental physics
Researcher (PI) Paolo PANI
Host Institution (HI) UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA
Call Details Starting Grant (StG), PE2, ERC-2017-STG
Summary In recent years, our theoretical understanding of the strong-field regime of gravity has grown in parallel with the observational confirmations that culminated in the landmark detection of gravitational waves (GWs). This synergy of breakthroughs at the observational, technical, and conceptual level offers the unprecedented opportunity to merge traditionally disjoint areas, and to make strong gravity a precision tool to probe fundamental physics.
The aim of the DarkGRA project is to investigate novel effects related to strong gravitational sources -such as black holes (BHs) and compact stars- that can be used to turn these objects into cosmic labs, where matter in extreme conditions, particle physics, and the very foundations of Einstein's theory of gravity can be put to the test. In this context, we propose to explore some outstanding, cross-cutting problems in fundamental physics: the existence of extra light fields, the limits of classical gravity, the nature of BHs and of spacetime singularities, and the effects of dark matter near compact objects. Our ultimate goal is to probe fundamental physics in the most extreme gravitational settings and to devise new approaches for detection, complementary to laboratory searches. This groundbreaking research program -located at the interface between particle physics, astrophysics and gravitation- is now made possible by novel techniques to scrutinize astrophysical compact objects, by current and future GW and X-ray detectors, and by the astonishing precision of pulsar timing. If supported by a solid theoretical framework, these new observations can potentially lead to surprising discoveries and paradigm shifts in our understanding of the fundamental laws of nature at all scales.
Summary
In recent years, our theoretical understanding of the strong-field regime of gravity has grown in parallel with the observational confirmations that culminated in the landmark detection of gravitational waves (GWs). This synergy of breakthroughs at the observational, technical, and conceptual level offers the unprecedented opportunity to merge traditionally disjoint areas, and to make strong gravity a precision tool to probe fundamental physics.
The aim of the DarkGRA project is to investigate novel effects related to strong gravitational sources -such as black holes (BHs) and compact stars- that can be used to turn these objects into cosmic labs, where matter in extreme conditions, particle physics, and the very foundations of Einstein's theory of gravity can be put to the test. In this context, we propose to explore some outstanding, cross-cutting problems in fundamental physics: the existence of extra light fields, the limits of classical gravity, the nature of BHs and of spacetime singularities, and the effects of dark matter near compact objects. Our ultimate goal is to probe fundamental physics in the most extreme gravitational settings and to devise new approaches for detection, complementary to laboratory searches. This groundbreaking research program -located at the interface between particle physics, astrophysics and gravitation- is now made possible by novel techniques to scrutinize astrophysical compact objects, by current and future GW and X-ray detectors, and by the astonishing precision of pulsar timing. If supported by a solid theoretical framework, these new observations can potentially lead to surprising discoveries and paradigm shifts in our understanding of the fundamental laws of nature at all scales.
Max ERC Funding
1 337 481 €
Duration
Start date: 2017-10-01, End date: 2022-09-30
Project acronym DIOPHANTINE PROBLEMS
Project Integral and Algebraic Points on Varieties, Diophantine Problems on Number Fields and Function Fields
Researcher (PI) Umberto Zannier
Host Institution (HI) SCUOLA NORMALE SUPERIORE
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary Diophantine problems have always been a central topic in Number Theory, and have shown deep links with other basic mathematical topics, like Algebraic and Complex Geometry. Our research plan focuses on some issues in this realm, which are strictly interrelated. In the last years the PI and collaborators obtained several results on integral and algebraic points on varieties, which have inspired much subsequent research by others, and which we plan to develop further. In particular:
We plan a further study of integral points on varieties, and applications to Algebraic Dynamics, a possibility which has emerged recently.
We plan to study further the so-called `Unlikely intersections'. This theme contains celebrated issues like the Manin-Mumford conjecture. After work of the PI with Bombieri and Masser in the last 10 years, it has been the object of much recent work and also of new conjectures by R. Pink and B. Zilber. Here a new method has recently emerged in work of the PI with Masser and Pila, which also leads (as shown by Pila) to signi_cant new cases of the Andr_e-Oort conjecture. We intend to pursue in this kind of investigation, exploring further the range of the methods.
Finally, we plan further study of topics of Diophantine Approximation and Hilbert Irreducibility, connected with the above ones in the contents and in the methodology.
Summary
Diophantine problems have always been a central topic in Number Theory, and have shown deep links with other basic mathematical topics, like Algebraic and Complex Geometry. Our research plan focuses on some issues in this realm, which are strictly interrelated. In the last years the PI and collaborators obtained several results on integral and algebraic points on varieties, which have inspired much subsequent research by others, and which we plan to develop further. In particular:
We plan a further study of integral points on varieties, and applications to Algebraic Dynamics, a possibility which has emerged recently.
We plan to study further the so-called `Unlikely intersections'. This theme contains celebrated issues like the Manin-Mumford conjecture. After work of the PI with Bombieri and Masser in the last 10 years, it has been the object of much recent work and also of new conjectures by R. Pink and B. Zilber. Here a new method has recently emerged in work of the PI with Masser and Pila, which also leads (as shown by Pila) to signi_cant new cases of the Andr_e-Oort conjecture. We intend to pursue in this kind of investigation, exploring further the range of the methods.
Finally, we plan further study of topics of Diophantine Approximation and Hilbert Irreducibility, connected with the above ones in the contents and in the methodology.
Max ERC Funding
928 500 €
Duration
Start date: 2011-02-01, End date: 2016-01-31
Project acronym DISQUA
Project Disorder physics with ultracold quantum gases
Researcher (PI) Massimo Inguscio
Host Institution (HI) LABORATORIO EUROPEO DI SPETTROSCOPIE NON LINEARI
Call Details Advanced Grant (AdG), PE2, ERC-2009-AdG
Summary Disorder is ubiquitous in nature and has a strong impact on the behaviour of many physical systems. The most celebrated effect of disorder is Anderson localization of single particles, but many other more complex phenomena arise in interacting, many-body systems. A full understanding of how disorder affects the behavior of quantum systems is still missing, also because of the unavoidable presence of nonlinearities, dissipation and thermal effects that make a careful exploration of real condensed-matter systems very difficult. In this project we want to fully exploit the unprecedented potentialities offered by ultracold atomic quantum gases to explore some of the present challenges for our understanding of the physics of disorder. These systems offer indeed the possibility of controlling to a great extent crucial parameters such as the type of disorder, the nonlinearities due to interactions, the temperature and density, the dimensionality, the quantum statistics. A variety of advanced diagnostic techniques allow to gain detailed information on the static and dynamic properties of the system. The potentialities of atomic quantum gases for the study of disorder have already showed up in recent breakthrough experiments. The project aims at an experimental exploration, supported by advanced theory, of the current issues in disordered quantum systems. We will investigate a few frontier themes of general interest: 1) Anderson localization and the interplay of disorder and a weak interaction; 2) strongly correlated, disordered bosonic systems; 3) disordered, interacting fermionic systems. In the research we will employ atomic Bose and Fermi gases with tunable interactions and advanced diagnostic techniques that we have recently contributed to develop. A successful completion of the project will push forward our understanding of the behaviour of quantum systems with disorder, with a potentially large impact on many fields of physics.
Summary
Disorder is ubiquitous in nature and has a strong impact on the behaviour of many physical systems. The most celebrated effect of disorder is Anderson localization of single particles, but many other more complex phenomena arise in interacting, many-body systems. A full understanding of how disorder affects the behavior of quantum systems is still missing, also because of the unavoidable presence of nonlinearities, dissipation and thermal effects that make a careful exploration of real condensed-matter systems very difficult. In this project we want to fully exploit the unprecedented potentialities offered by ultracold atomic quantum gases to explore some of the present challenges for our understanding of the physics of disorder. These systems offer indeed the possibility of controlling to a great extent crucial parameters such as the type of disorder, the nonlinearities due to interactions, the temperature and density, the dimensionality, the quantum statistics. A variety of advanced diagnostic techniques allow to gain detailed information on the static and dynamic properties of the system. The potentialities of atomic quantum gases for the study of disorder have already showed up in recent breakthrough experiments. The project aims at an experimental exploration, supported by advanced theory, of the current issues in disordered quantum systems. We will investigate a few frontier themes of general interest: 1) Anderson localization and the interplay of disorder and a weak interaction; 2) strongly correlated, disordered bosonic systems; 3) disordered, interacting fermionic systems. In the research we will employ atomic Bose and Fermi gases with tunable interactions and advanced diagnostic techniques that we have recently contributed to develop. A successful completion of the project will push forward our understanding of the behaviour of quantum systems with disorder, with a potentially large impact on many fields of physics.
Max ERC Funding
2 500 000 €
Duration
Start date: 2010-03-01, End date: 2015-02-28