Project acronym ANALYTIC
Project ANALYTIC PROPERTIES OF INFINITE GROUPS:
limits, curvature, and randomness
Researcher (PI) Gulnara Arzhantseva
Host Institution (HI) UNIVERSITAT WIEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967.
My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behavior of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic?
My motivation is two-fold:
- Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious.
- In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood.
The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behavior.
Summary
The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967.
My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behavior of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic?
My motivation is two-fold:
- Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious.
- In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood.
The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behavior.
Max ERC Funding
1 065 500 €
Duration
Start date: 2011-04-01, End date: 2016-03-31
Project acronym D-END
Project Telomeres: from the DNA end replication problem to the control of cell proliferation
Researcher (PI) Maria Teresa Teixeira Fernandes Bernardo
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), LS1, ERC-2010-StG_20091118
Summary Linear chromosomes of eukaryotes end with telomeres that ensure their stability. Because of the inability of semi-conservative DNA replication machinery to fully replicate DNA ends, telomeres require dedicated mechanisms to be duplicated and their length is eroded at each cell division. For this reason, telomeres constitute molecular clocks that determine cell proliferation potential in eukaryotes. Strikingly, we have shown recently that it is the shortest telomere in the cell that determines the onset of replicative senescence. This project aims a complete and detailed dissection of the in vivo DNA-end replication problem and the deep understanding of its impact for cell division capability. Specifically my goals are (1) the determination of the exact structures that result from the replication of DNA extremities, (2) the examination of the activities operating at the shortest telomere that triggers replicative senescence and (3) the investigation of the correspondence between telomere molecular structure and cell proliferation state at individual cell scale. To achieve this, I will undertake in Saccharomyces cerevisiae original and innovative single-molecule and single-cell approaches, that, in combination with genome-wide screens and sophisticated cellular settings, will allow to track and challenge a specified telomere of defined length. I anticipate that this work will lead to an in-depth understanding of how telomeres are replicated and how they enable the control of cell proliferation in eukaryotic cells, a matter at the intersection of the fundamentals of molecular genetics, cell biology of aging and oncology.
Summary
Linear chromosomes of eukaryotes end with telomeres that ensure their stability. Because of the inability of semi-conservative DNA replication machinery to fully replicate DNA ends, telomeres require dedicated mechanisms to be duplicated and their length is eroded at each cell division. For this reason, telomeres constitute molecular clocks that determine cell proliferation potential in eukaryotes. Strikingly, we have shown recently that it is the shortest telomere in the cell that determines the onset of replicative senescence. This project aims a complete and detailed dissection of the in vivo DNA-end replication problem and the deep understanding of its impact for cell division capability. Specifically my goals are (1) the determination of the exact structures that result from the replication of DNA extremities, (2) the examination of the activities operating at the shortest telomere that triggers replicative senescence and (3) the investigation of the correspondence between telomere molecular structure and cell proliferation state at individual cell scale. To achieve this, I will undertake in Saccharomyces cerevisiae original and innovative single-molecule and single-cell approaches, that, in combination with genome-wide screens and sophisticated cellular settings, will allow to track and challenge a specified telomere of defined length. I anticipate that this work will lead to an in-depth understanding of how telomeres are replicated and how they enable the control of cell proliferation in eukaryotic cells, a matter at the intersection of the fundamentals of molecular genetics, cell biology of aging and oncology.
Max ERC Funding
1 498 504 €
Duration
Start date: 2010-11-01, End date: 2015-10-31
Project acronym DISPEQ
Project Qualitative study of nonlinear dispersive equations
Researcher (PI) Nikolay Tzvetkov
Host Institution (HI) UNIVERSITE DE CERGY-PONTOISE
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary We plan to further improve the understanding of the nonlinear dispersive wave propagation phenomena. In particular we plan to develop tools allowing to make a statistical description of the corresponding flows and methods to study transverse stability independently of the very particular arguments based on the inverse scattering. We also plan to study critical problems in strongly non Euclidean geometries.
Summary
We plan to further improve the understanding of the nonlinear dispersive wave propagation phenomena. In particular we plan to develop tools allowing to make a statistical description of the corresponding flows and methods to study transverse stability independently of the very particular arguments based on the inverse scattering. We also plan to study critical problems in strongly non Euclidean geometries.
Max ERC Funding
880 270 €
Duration
Start date: 2010-10-01, End date: 2015-09-30
Project acronym HYRAX
Project Rock Hyrax Middens and Climate Change in Southern Africa during the last 50,000 years
Researcher (PI) Brian Mc Kee Chase
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), PE10, ERC-2010-StG_20091028
Summary In stark contrast to the abundance of high quality palaeoenvironmental records obtained from the temperate regions of the northern hemisphere, terrestrial palaeoenvironmental information from southern Africa's drylands comes from discontinuous deposits with poor absolute age control and ambiguous palaeoclimatic significance. Confronted with the possibility of future environmental and social disruption as a result of climate change, the need for reliable records from southern Africa has never been so acute. This project seeks to develop rock hyrax middens as novel palaeoenvironmental archives to investigate long-term climate change. Hyrax middens (fossilised accumulations of urine and faecal pellets) contain a range of palaeoenvironmental proxies, including fossil pollen and stable isotopes. As part of a pilot study, I have created new collection and sampling methodologies, establishing the proof of principle and showing that middens provide continuous sub-annual to multi-decadal multi-proxy records of environmental change spanning the last 50,000 years. This work has been exceptional in terms of its ability to elucidate long-term climate dynamics at the local scale, and I now intend to apply my techniques to studying environmental change across the whole of southern Africa, a climatically sensitive, but poorly understood region of the globe. Developing new sites, proxies and analytical techniques, HYRAX will provide the first opportunity to study rapid climate change events, the extent and phasing of major climatic phenomena, and the direction and potential impacts of future climate change.
Summary
In stark contrast to the abundance of high quality palaeoenvironmental records obtained from the temperate regions of the northern hemisphere, terrestrial palaeoenvironmental information from southern Africa's drylands comes from discontinuous deposits with poor absolute age control and ambiguous palaeoclimatic significance. Confronted with the possibility of future environmental and social disruption as a result of climate change, the need for reliable records from southern Africa has never been so acute. This project seeks to develop rock hyrax middens as novel palaeoenvironmental archives to investigate long-term climate change. Hyrax middens (fossilised accumulations of urine and faecal pellets) contain a range of palaeoenvironmental proxies, including fossil pollen and stable isotopes. As part of a pilot study, I have created new collection and sampling methodologies, establishing the proof of principle and showing that middens provide continuous sub-annual to multi-decadal multi-proxy records of environmental change spanning the last 50,000 years. This work has been exceptional in terms of its ability to elucidate long-term climate dynamics at the local scale, and I now intend to apply my techniques to studying environmental change across the whole of southern Africa, a climatically sensitive, but poorly understood region of the globe. Developing new sites, proxies and analytical techniques, HYRAX will provide the first opportunity to study rapid climate change events, the extent and phasing of major climatic phenomena, and the direction and potential impacts of future climate change.
Max ERC Funding
1 484 046 €
Duration
Start date: 2010-11-01, End date: 2016-10-31
Project acronym IONOSENSE
Project Exploitation of Organic Electrochemical Transistors for Biological Ionsensing
Researcher (PI) Roisin Meabh Owens
Host Institution (HI) ASSOCIATION POUR LA RECHERCHE ET LE DEVELOPPEMENT DES METHODES ET PROCESSUS INDUSTRIELS
Call Details Starting Grant (StG), PE7, ERC-2010-StG_20091028
Summary In biological systems many tissue types have evolved a barrier function to selectively allow the transport of matter from the lumen to tissue beneath. Characterization of these barriers is very important as their disruption or malfunction is often indicative of toxicity/disease. The degree of barrier integrity is also a key indicator of the appropriateness of in vitro models for use in toxicology/drug screening. The advent of organic electronics has created a unique opportunity to interface the worlds of electronics and biology, using devices such as the organic electrochemical transistor (OECT), that provides a very sensitive way to detect minute ionic currents. This proposal aims to integrate the barrier function of biological systems with OECTs to yield devices that can detect minute disruptions in barrier function. Specifically, OECTs will be integrated with cell monolayers that form tight junctions and with membranes that incorporate ion channels. A disruption in tight junctions or a change in permeability of ion channels will be detected by the OECT. These devices will have unprecedented sensitivity, in a format that can be mass produced at low-cost. The potential benefits of this multidisciplinary project are numerous: It will be a vehicle for fundamental research in life sciences and the development of new in vitro models for toxicology screening of disruptive agents and the development of drugs to treat disorders linked with barrier tissue malfunction (e.g. mutations in ion channels). Moreover, through the use of various cell lines and ion channels, this platform will also lead to the engineering of new sensors and biomedical instrumentation, with a host of applications in medical diagnostics, food/water safety, homeland security and environmental protection.
Summary
In biological systems many tissue types have evolved a barrier function to selectively allow the transport of matter from the lumen to tissue beneath. Characterization of these barriers is very important as their disruption or malfunction is often indicative of toxicity/disease. The degree of barrier integrity is also a key indicator of the appropriateness of in vitro models for use in toxicology/drug screening. The advent of organic electronics has created a unique opportunity to interface the worlds of electronics and biology, using devices such as the organic electrochemical transistor (OECT), that provides a very sensitive way to detect minute ionic currents. This proposal aims to integrate the barrier function of biological systems with OECTs to yield devices that can detect minute disruptions in barrier function. Specifically, OECTs will be integrated with cell monolayers that form tight junctions and with membranes that incorporate ion channels. A disruption in tight junctions or a change in permeability of ion channels will be detected by the OECT. These devices will have unprecedented sensitivity, in a format that can be mass produced at low-cost. The potential benefits of this multidisciplinary project are numerous: It will be a vehicle for fundamental research in life sciences and the development of new in vitro models for toxicology screening of disruptive agents and the development of drugs to treat disorders linked with barrier tissue malfunction (e.g. mutations in ion channels). Moreover, through the use of various cell lines and ion channels, this platform will also lead to the engineering of new sensors and biomedical instrumentation, with a host of applications in medical diagnostics, food/water safety, homeland security and environmental protection.
Max ERC Funding
1 496 539 €
Duration
Start date: 2011-04-01, End date: 2016-03-31
Project acronym MAD-ESEC
Project Magmas at Depth: an Experimental Study at Extreme Conditions
Researcher (PI) Chrystèle Sanloup
Host Institution (HI) UNIVERSITE PIERRE ET MARIE CURIE - PARIS 6
Call Details Starting Grant (StG), PE10, ERC-2010-StG_20091028
Summary Magmas, i.e. silicate melts, have played a key role in the chemical and thermal evolution of the Earth and other planets. The Earth's interior today is the outcome of mass transfers which occurred primarily in its early history and still occur now via magmatic events. Present day magmatic and volcanic processes are controlled by the properties of molten silicate at high pressure, considering that magmas are produced at depth. However, the physical properties of molten silicates remain largely unexplored across the broad range of relevant P-T conditions, and their chemical properties are very often assumed constant and equal to those known at ambient conditions. This blurs out our understanding of planetary differentiation and current magmatic processes.
The aim of this proposal is to place fundamental constraints on magma generation and transport in planetary interiors by measuring the properties of silicate melts in their natural high pressures (P) and high temperatures (T) conditions using a broad range of in situ key diagnostic probes (X-ray and neutron scattering techniques, X-ray absorption, radiography, Raman spectroscopy). The completion of this proposal will result in a comprehensive key database in the composition-P-T space that will form the foundation for modelling planetary formation and differentiation, and will provide answers to the very fundamental questions on magma formation, ascent or trapping at depth in the current and past Earth.
This experimental program is allowed by the recent advancements in in situ high P-T techniques, and comes in conjunction with a large and fruitful theoretical effort; time has thus come to understand Earth's melts and their keys to Earth's evolution.
Summary
Magmas, i.e. silicate melts, have played a key role in the chemical and thermal evolution of the Earth and other planets. The Earth's interior today is the outcome of mass transfers which occurred primarily in its early history and still occur now via magmatic events. Present day magmatic and volcanic processes are controlled by the properties of molten silicate at high pressure, considering that magmas are produced at depth. However, the physical properties of molten silicates remain largely unexplored across the broad range of relevant P-T conditions, and their chemical properties are very often assumed constant and equal to those known at ambient conditions. This blurs out our understanding of planetary differentiation and current magmatic processes.
The aim of this proposal is to place fundamental constraints on magma generation and transport in planetary interiors by measuring the properties of silicate melts in their natural high pressures (P) and high temperatures (T) conditions using a broad range of in situ key diagnostic probes (X-ray and neutron scattering techniques, X-ray absorption, radiography, Raman spectroscopy). The completion of this proposal will result in a comprehensive key database in the composition-P-T space that will form the foundation for modelling planetary formation and differentiation, and will provide answers to the very fundamental questions on magma formation, ascent or trapping at depth in the current and past Earth.
This experimental program is allowed by the recent advancements in in situ high P-T techniques, and comes in conjunction with a large and fruitful theoretical effort; time has thus come to understand Earth's melts and their keys to Earth's evolution.
Max ERC Funding
1 332 160 €
Duration
Start date: 2011-06-01, End date: 2017-05-31
Project acronym MATHANA
Project Mathematical modeling of anaesthetic action
Researcher (PI) Axel Hutt
Host Institution (HI) INSTITUT NATIONAL DE RECHERCHE ENINFORMATIQUE ET AUTOMATIQUE
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary General anaesthesia is an important method in today's hospital practice and especially in surgery. To supervise the depth of anaesthesia during surgery, the anaesthesist applies electroencephalography (EEG) and monitors the brain activity of the subject on the scalp. The applied monitoring machine calculates the change of the power spectrum of the brain signals to indicate the anaesthetic depth. This procedure is based on the finding that the concentration increase of the anaesthetic drug changes the EEG-power spectrum in a significant way. Although this procedure is applied world-wide, the underlying neural mechanism of the spectrum change is still unknown. The project aims to elucidate the underlying neural mechanism by a detailed investigating a mathematical model of neural populations.
The investigation is based on analytical calculations in a neural population model of the cortex involving intrinsic neural properties of brain areas and feedback loops to other areas, such as the loop between the cortex and the thalamus. Currently, there are two proposed mechanisms for the charactertisic change of the power spectrum: a highly nonlinear jump in the activation (so-called phase transition) and a linear behavior. The project mainly focusses on the nonlinear jump to finally rule it out or support it. A subsequent comparison to previous experimenta results aims to fit the physiological parameters. Since the cortex population is embedded into a network of other cortical areas and the thalamus, the corresponding analytical investigations takes into account external stochastic (from other brain areas) and time-periodic (thalamic) forces. To this end it is necessary to develop several novel nonlinear analysis technique of neural populations to derive the power spectrum close to the phase transition and conditions for physiological parameters.
Summary
General anaesthesia is an important method in today's hospital practice and especially in surgery. To supervise the depth of anaesthesia during surgery, the anaesthesist applies electroencephalography (EEG) and monitors the brain activity of the subject on the scalp. The applied monitoring machine calculates the change of the power spectrum of the brain signals to indicate the anaesthetic depth. This procedure is based on the finding that the concentration increase of the anaesthetic drug changes the EEG-power spectrum in a significant way. Although this procedure is applied world-wide, the underlying neural mechanism of the spectrum change is still unknown. The project aims to elucidate the underlying neural mechanism by a detailed investigating a mathematical model of neural populations.
The investigation is based on analytical calculations in a neural population model of the cortex involving intrinsic neural properties of brain areas and feedback loops to other areas, such as the loop between the cortex and the thalamus. Currently, there are two proposed mechanisms for the charactertisic change of the power spectrum: a highly nonlinear jump in the activation (so-called phase transition) and a linear behavior. The project mainly focusses on the nonlinear jump to finally rule it out or support it. A subsequent comparison to previous experimenta results aims to fit the physiological parameters. Since the cortex population is embedded into a network of other cortical areas and the thalamus, the corresponding analytical investigations takes into account external stochastic (from other brain areas) and time-periodic (thalamic) forces. To this end it is necessary to develop several novel nonlinear analysis technique of neural populations to derive the power spectrum close to the phase transition and conditions for physiological parameters.
Max ERC Funding
856 500 €
Duration
Start date: 2011-01-01, End date: 2015-10-31
Project acronym MERCURY ISOTOPES
Project Exploring the isotopic dimension of the global mercury cycle
Researcher (PI) Jeroen Sonke
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), PE10, ERC-2010-StG_20091028
Summary Mass-independent fractionation (MIF) of isotopes in terrestrial geochemical processes was first observed in 1983 for oxygen and in 2000 for sulfur isotopes. Recently mercury (Hg) was added to this shortlist when isotopic anomalies were observed for Hg s two odd isotopes, 199Hg and 201Hg in biological tissues. The objective of the MERCURY ISOTOPES project is to take Hg MIF beyond the initial discovery, and use it to address major outstanding scientific questions of societal and philosophical interest. Similar to the profound insights that carbon and oxygen isotope systematics have brought to climate research, we propose to use variations in Hg isotopic compositions to fingerprint natural and anthropogenic sources, quantify isotope fractionation processes, and provide new constraints on models of mercury cycling.
The MERCURY ISOTOPES project centres on the use of mercury MIF to understand global Hg dynamics at different time scales, from the Pleistocene to modern times. Three main themes will be investigated: 1. the modern Hg cycle focusing on Asian urban-industrial emissions related to coal burning, 2. recent atmospheric Hg deposition in the Arctic, recent Arctic Ocean Hg records from archived biological tissues, and post-glacial Hg deposition from 10,000 yr old ombrotrophic peat records along a mid-latitude sub-Arctic gradient. 3 Continuous atmospheric Hg speciation and isotopic monitoring at the Pic du Midi Observatory (Pyrenees).
By tapping information from the isotopic dimension of Hg cycling, including revolutionary mass-independent effects, I expect a maximum scientific impact while supporting a socially relevant and urgently needed investigation at the frontier of isotope geosciences.
Summary
Mass-independent fractionation (MIF) of isotopes in terrestrial geochemical processes was first observed in 1983 for oxygen and in 2000 for sulfur isotopes. Recently mercury (Hg) was added to this shortlist when isotopic anomalies were observed for Hg s two odd isotopes, 199Hg and 201Hg in biological tissues. The objective of the MERCURY ISOTOPES project is to take Hg MIF beyond the initial discovery, and use it to address major outstanding scientific questions of societal and philosophical interest. Similar to the profound insights that carbon and oxygen isotope systematics have brought to climate research, we propose to use variations in Hg isotopic compositions to fingerprint natural and anthropogenic sources, quantify isotope fractionation processes, and provide new constraints on models of mercury cycling.
The MERCURY ISOTOPES project centres on the use of mercury MIF to understand global Hg dynamics at different time scales, from the Pleistocene to modern times. Three main themes will be investigated: 1. the modern Hg cycle focusing on Asian urban-industrial emissions related to coal burning, 2. recent atmospheric Hg deposition in the Arctic, recent Arctic Ocean Hg records from archived biological tissues, and post-glacial Hg deposition from 10,000 yr old ombrotrophic peat records along a mid-latitude sub-Arctic gradient. 3 Continuous atmospheric Hg speciation and isotopic monitoring at the Pic du Midi Observatory (Pyrenees).
By tapping information from the isotopic dimension of Hg cycling, including revolutionary mass-independent effects, I expect a maximum scientific impact while supporting a socially relevant and urgently needed investigation at the frontier of isotope geosciences.
Max ERC Funding
1 176 924 €
Duration
Start date: 2010-12-01, End date: 2015-11-30
Project acronym MNIQS
Project Mathematics and Numerics of Infinite Quantum Systems
Researcher (PI) Mathieu Lewin
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The purpose of the project is to study linear and nonlinear models arising in quantum mechanics and which are used to describe
matter at the microscopic and nanoscopic scales. The project focuses on physically-oriented questions (rigorous derivation of a
given model from first principles), analytic problems (existence and properties of bound states, study of solutions to timedependent
equations) and numerical issues (development of reliable algorithmic strategies). Most of the models are nonlinear and
describe physical systems possessing an infinite number of quantum particles, leading to specific difficulties.
The first part of the project is devoted to the study of relativistic atoms and molecules, while taking into account quantum
electrodynamics effects like the polarization of the vacuum. The models are all based on the Dirac operator.
The second part is focused on the study of quantum crystals. The goal is to develop new strategies for describing their behavior in
the presence of defects and local deformations. Both insulators, semiconductors and metals are considered (including graphene).
In the third part, attractive systems are considered (like stars or a few nucleons interacting via strong forces in a nucleus). The
project aims at rigorously understanding some of their specific properties, like Cooper pairing or the possible dynamical collapse of
massive gravitational objects.
Finally, the last part is devoted to general properties of infinite quantum systems, in particular the proof of the existence of the
thermodynamic limit
Summary
The purpose of the project is to study linear and nonlinear models arising in quantum mechanics and which are used to describe
matter at the microscopic and nanoscopic scales. The project focuses on physically-oriented questions (rigorous derivation of a
given model from first principles), analytic problems (existence and properties of bound states, study of solutions to timedependent
equations) and numerical issues (development of reliable algorithmic strategies). Most of the models are nonlinear and
describe physical systems possessing an infinite number of quantum particles, leading to specific difficulties.
The first part of the project is devoted to the study of relativistic atoms and molecules, while taking into account quantum
electrodynamics effects like the polarization of the vacuum. The models are all based on the Dirac operator.
The second part is focused on the study of quantum crystals. The goal is to develop new strategies for describing their behavior in
the presence of defects and local deformations. Both insulators, semiconductors and metals are considered (including graphene).
In the third part, attractive systems are considered (like stars or a few nucleons interacting via strong forces in a nucleus). The
project aims at rigorously understanding some of their specific properties, like Cooper pairing or the possible dynamical collapse of
massive gravitational objects.
Finally, the last part is devoted to general properties of infinite quantum systems, in particular the proof of the existence of the
thermodynamic limit
Max ERC Funding
905 700 €
Duration
Start date: 2010-10-01, End date: 2015-09-30
Project acronym PAGAP
Project Periods in Algebraic Geometry and Physics
Researcher (PI) Francis Clement Sais Brown
Host Institution (HI) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary Periods are the integrals of algebraic differential forms over domains defined by polynomial inequalities, and are ubiquitous in mathematics and physics. One of the simplest classes of periods are given by multiple zeta values, which are the periods of moduli spaces M_{0,n} of curves of genus zero. They have recently undergone a huge revival of interest, and occur in number theory, the theory of mixed Tate motives, knot invariants, quantum groups, deformation quantization and many more branches of mathematics and physics.
Remarkably, it has been observed experimentally that Feynman amplitudes in quantum field theories typically evaluate numerically to multiple zeta values and polylogarithms (which are the iterated integrals on M_{0,n}), and a huge amount of effort is presently devoted to computations of such amplitudes in order to provide predictions for particle collider experiments. A deeper understanding of the reason for the appearance of the same mathematical objects in algebraic geometry and physics is essential to streamline these computations, and ultimately tackle the outstanding problems in particle physics.
The proposal has two parts: firstly to undertake a systematic study of the periods and iterated integrals on higher genus moduli spaces M_{g,n} and related varieties, and secondly to relate these fundamental mathematical objects to quantum field theories, bringing to bear modern techniques from algebraic geometry, Hodge theory, and motives to this emerging interdisciplinary area. Part of this would involve the implementation (with the assistance of future postdoc. team members) of an algorithm for the evaluation of Feynman diagrams which is due to the author and goes several orders beyond what has previously been possible, in order eventually to deduce concrete predictions for the Large Hadron Collider.
Summary
Periods are the integrals of algebraic differential forms over domains defined by polynomial inequalities, and are ubiquitous in mathematics and physics. One of the simplest classes of periods are given by multiple zeta values, which are the periods of moduli spaces M_{0,n} of curves of genus zero. They have recently undergone a huge revival of interest, and occur in number theory, the theory of mixed Tate motives, knot invariants, quantum groups, deformation quantization and many more branches of mathematics and physics.
Remarkably, it has been observed experimentally that Feynman amplitudes in quantum field theories typically evaluate numerically to multiple zeta values and polylogarithms (which are the iterated integrals on M_{0,n}), and a huge amount of effort is presently devoted to computations of such amplitudes in order to provide predictions for particle collider experiments. A deeper understanding of the reason for the appearance of the same mathematical objects in algebraic geometry and physics is essential to streamline these computations, and ultimately tackle the outstanding problems in particle physics.
The proposal has two parts: firstly to undertake a systematic study of the periods and iterated integrals on higher genus moduli spaces M_{g,n} and related varieties, and secondly to relate these fundamental mathematical objects to quantum field theories, bringing to bear modern techniques from algebraic geometry, Hodge theory, and motives to this emerging interdisciplinary area. Part of this would involve the implementation (with the assistance of future postdoc. team members) of an algorithm for the evaluation of Feynman diagrams which is due to the author and goes several orders beyond what has previously been possible, in order eventually to deduce concrete predictions for the Large Hadron Collider.
Max ERC Funding
1 068 540 €
Duration
Start date: 2010-11-01, End date: 2015-10-31
