Project acronym COMPASP
Project Complex analysis and statistical physics
Researcher (PI) Stanislav Smirnov
Host Institution (HI) UNIVERSITE DE GENEVE
Call Details Advanced Grant (AdG), PE1, ERC-2013-ADG
Summary "The goal of this project is to achieve breakthroughs in a few fundamental questions in 2D statistical physics, using techniques from complex analysis, probability, dynamical systems, geometric measure theory and theoretical physics.
Over the last decade, we significantly expanded our understanding of 2D lattice models of statistical physics, their conformally invariant scaling limits and related random geometries. However, there seem to be serious obstacles, preventing further development and requiring novel ideas. We plan to attack those, in particular we intend to:
(A) Describe new scaling limits by Schramm’s SLE curves and their generalizations,
(B) Study discrete complex structures and use them to describe more 2D models,
(C) Describe the scaling limits of random planar graphs by the Liouville Quantum Gravity,
(D) Understand universality and lay framework for the Renormalization Group Formalism,
(E) Go beyond the current setup of spin models and SLEs.
These problems are known to be very difficult, but fundamental questions, which have the potential to lead to significant breakthroughs in our understanding of phase transitions, allowing for further progresses. In resolving them, we plan to exploit interactions of different subjects, and recent advances are encouraging."
Summary
"The goal of this project is to achieve breakthroughs in a few fundamental questions in 2D statistical physics, using techniques from complex analysis, probability, dynamical systems, geometric measure theory and theoretical physics.
Over the last decade, we significantly expanded our understanding of 2D lattice models of statistical physics, their conformally invariant scaling limits and related random geometries. However, there seem to be serious obstacles, preventing further development and requiring novel ideas. We plan to attack those, in particular we intend to:
(A) Describe new scaling limits by Schramm’s SLE curves and their generalizations,
(B) Study discrete complex structures and use them to describe more 2D models,
(C) Describe the scaling limits of random planar graphs by the Liouville Quantum Gravity,
(D) Understand universality and lay framework for the Renormalization Group Formalism,
(E) Go beyond the current setup of spin models and SLEs.
These problems are known to be very difficult, but fundamental questions, which have the potential to lead to significant breakthroughs in our understanding of phase transitions, allowing for further progresses. In resolving them, we plan to exploit interactions of different subjects, and recent advances are encouraging."
Max ERC Funding
1 995 900 €
Duration
Start date: 2014-01-01, End date: 2018-12-31
Project acronym COMPLEXDATA
Project Statistics for Complex Data: Understanding Randomness, Geometry and Complexity with a view Towards Biophysics
Researcher (PI) Victor Michael Panaretos
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The ComplexData project aims at advancing our understanding of the statistical treatment of varied types of complex data by generating new theory and methods, and to obtain progress in concrete current biophysical problems through the implementation of the new tools developed. Complex Data constitute data where the basic object of observation cannot be described in the standard Euclidean context of statistics, but rather needs to be thought of as an element of an abstract mathematical space with special properties. Scientific progress has, in recent years, begun to generate an increasing number of new and complex types of data that require statistical understanding and analysis. Four such types of data that are arising in the context of current scientific research and that the project will be focusing on are: random integral transforms, random unlabelled shapes, random flows of functions, and random tensor fields. In these unconventional contexts for statistics, the strategy of the project will be to carefully exploit the special aspects involved due to geometry, dimension and randomness in order to be able to either adapt and synthesize existing statistical methods, or to generate new statistical ideas altogether. However, the project will not restrict itself to merely studying the theoretical aspects of complex data, but will be truly interdisciplinary. The connecting thread among all the above data types is that their study is motivated by, and will be applied to concrete practical problems arising in the study of biological structure, dynamics, and function: biophysics. For this reason, the programme will be in interaction with local and international contacts from this field. In particular, the theoretical/methodological output of the four programme research foci will be applied to gain insights in the following corresponding four application areas: electron microscopy, protein homology, DNA molecular dynamics, brain imaging.
Summary
The ComplexData project aims at advancing our understanding of the statistical treatment of varied types of complex data by generating new theory and methods, and to obtain progress in concrete current biophysical problems through the implementation of the new tools developed. Complex Data constitute data where the basic object of observation cannot be described in the standard Euclidean context of statistics, but rather needs to be thought of as an element of an abstract mathematical space with special properties. Scientific progress has, in recent years, begun to generate an increasing number of new and complex types of data that require statistical understanding and analysis. Four such types of data that are arising in the context of current scientific research and that the project will be focusing on are: random integral transforms, random unlabelled shapes, random flows of functions, and random tensor fields. In these unconventional contexts for statistics, the strategy of the project will be to carefully exploit the special aspects involved due to geometry, dimension and randomness in order to be able to either adapt and synthesize existing statistical methods, or to generate new statistical ideas altogether. However, the project will not restrict itself to merely studying the theoretical aspects of complex data, but will be truly interdisciplinary. The connecting thread among all the above data types is that their study is motivated by, and will be applied to concrete practical problems arising in the study of biological structure, dynamics, and function: biophysics. For this reason, the programme will be in interaction with local and international contacts from this field. In particular, the theoretical/methodological output of the four programme research foci will be applied to gain insights in the following corresponding four application areas: electron microscopy, protein homology, DNA molecular dynamics, brain imaging.
Max ERC Funding
681 146 €
Duration
Start date: 2011-06-01, End date: 2016-05-31
Project acronym LIQRISK
Project Liquidity and Risk in Macroeconomic Models
Researcher (PI) Philippe Jean Louis Bacchetta
Host Institution (HI) UNIVERSITE DE LAUSANNE
Call Details Advanced Grant (AdG), SH1, ERC-2010-AdG_20100407
Summary The proposal is motivated by the need to incorporate financial realities into macroeconomic models. The objective is to introduce leverage and liquidity in standard dynamic general equilibrium models and analyze their macroeconomic implications. The proposal is divided into two sub-projects and analyzes two different aspects of liquidity. The first deals with leverage and market liquidity in developed financial economies. The second examines the demand for liquid assets by emerging countries and its global implications. In the first sub-project, the proposal breaks new ground in the understanding of the dynamics of risk and in explaining some important features of the recent crisis. The project particularly emphasizes the role of self-fulfilling changes in expectations that can lead to sudden large shifts in risk. This can take the form of a financial panic with a big drop in asset prices. Various extensions will investigate the empirical implications as well as the implications for international capital flows, exchange rates, macroeconomic activity and policy recommendations. In the second sub-project, the objective is to formalize and analyze different degrees of liquidity in international capital flows. The project will innovate in finding ways to model liquidity in dynamic open economy models. This will allow a better understanding of the recent pattern in international capital flows, where less developed countries lend to richer economies. It will also shed light on the evolution of global imbalances before and after the crisis.
Summary
The proposal is motivated by the need to incorporate financial realities into macroeconomic models. The objective is to introduce leverage and liquidity in standard dynamic general equilibrium models and analyze their macroeconomic implications. The proposal is divided into two sub-projects and analyzes two different aspects of liquidity. The first deals with leverage and market liquidity in developed financial economies. The second examines the demand for liquid assets by emerging countries and its global implications. In the first sub-project, the proposal breaks new ground in the understanding of the dynamics of risk and in explaining some important features of the recent crisis. The project particularly emphasizes the role of self-fulfilling changes in expectations that can lead to sudden large shifts in risk. This can take the form of a financial panic with a big drop in asset prices. Various extensions will investigate the empirical implications as well as the implications for international capital flows, exchange rates, macroeconomic activity and policy recommendations. In the second sub-project, the objective is to formalize and analyze different degrees of liquidity in international capital flows. The project will innovate in finding ways to model liquidity in dynamic open economy models. This will allow a better understanding of the recent pattern in international capital flows, where less developed countries lend to richer economies. It will also shed light on the evolution of global imbalances before and after the crisis.
Max ERC Funding
2 070 570 €
Duration
Start date: 2011-08-01, End date: 2016-07-31
Project acronym MODFLAT
Project "Moduli of flat connections, planar networks and associators"
Researcher (PI) Anton Alekseev
Host Institution (HI) UNIVERSITE DE GENEVE
Call Details Advanced Grant (AdG), PE1, ERC-2013-ADG
Summary "The project lies at the crossroads between three different topics in Mathematics: moduli spaces of flat connections on surfaces in Differential Geometry and Topology, the Kashiwara-Vergne problem and Drinfeld associators in Lie theory, and combinatorics of planar networks in the theory of Total Positivity.
The time is ripe to establish deep connections between these three theories. The main factors are the recent progress in the Kashiwara-Vergne theory (including the proof of the Kashiwara-Vergne conjecture by Alekseev-Meinrenken), the discovery of a link between the Horn problem on eigenvalues of sums of Hermitian matrices and planar network combinatorics, and intimate links with the Topological Quantum Field Theory shared by the three topics.
The scientific objectives of the project include answering the following questions:
1) To find a universal non-commutative volume formula for moduli of flat connections which would contain the Witten’s volume formula, the Verlinde formula, and the Moore-Nekrasov-Shatashvili formula as particular cases.
2) To show that all solutions of the Kashiwara-Vergne problem come from Drinfeld associators. If the answer is indeed positive, it will have applications to the study of the Gothendieck-Teichmüller Lie algebra grt.
3) To find a Gelfand-Zeiltin type integrable system for the symplectic group Sp(2n). This question is open since 1983.
To achieve these goals, one needs to use a multitude of techniques. Here we single out the ones developed by the author:
- Quasi-symplectic and quasi-Poisson Geometry and the theory of group valued moment maps.
- The linearization method for Poisson-Lie groups relating the additive problem z=x+y and the multiplicative problem Z=XY.
- Free Lie algebra approach to the Kashiwara-Vergne theory, including the non-commutative divergence and Jacobian cocylces.
- Non-abelian topical calculus which establishes a link between the multiplicative problem and combinatorics of planar networks."
Summary
"The project lies at the crossroads between three different topics in Mathematics: moduli spaces of flat connections on surfaces in Differential Geometry and Topology, the Kashiwara-Vergne problem and Drinfeld associators in Lie theory, and combinatorics of planar networks in the theory of Total Positivity.
The time is ripe to establish deep connections between these three theories. The main factors are the recent progress in the Kashiwara-Vergne theory (including the proof of the Kashiwara-Vergne conjecture by Alekseev-Meinrenken), the discovery of a link between the Horn problem on eigenvalues of sums of Hermitian matrices and planar network combinatorics, and intimate links with the Topological Quantum Field Theory shared by the three topics.
The scientific objectives of the project include answering the following questions:
1) To find a universal non-commutative volume formula for moduli of flat connections which would contain the Witten’s volume formula, the Verlinde formula, and the Moore-Nekrasov-Shatashvili formula as particular cases.
2) To show that all solutions of the Kashiwara-Vergne problem come from Drinfeld associators. If the answer is indeed positive, it will have applications to the study of the Gothendieck-Teichmüller Lie algebra grt.
3) To find a Gelfand-Zeiltin type integrable system for the symplectic group Sp(2n). This question is open since 1983.
To achieve these goals, one needs to use a multitude of techniques. Here we single out the ones developed by the author:
- Quasi-symplectic and quasi-Poisson Geometry and the theory of group valued moment maps.
- The linearization method for Poisson-Lie groups relating the additive problem z=x+y and the multiplicative problem Z=XY.
- Free Lie algebra approach to the Kashiwara-Vergne theory, including the non-commutative divergence and Jacobian cocylces.
- Non-abelian topical calculus which establishes a link between the multiplicative problem and combinatorics of planar networks."
Max ERC Funding
2 148 211 €
Duration
Start date: 2014-02-01, End date: 2019-01-31
Project acronym RIGIDITY
Project Rigidity: Groups, Geometry and Cohomology
Researcher (PI) Nicolas Monod
Host Institution (HI) ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary "Our proposal has three components:
1. Unitarizable representations.
2. Spaces and groups of non-positive curvature.
3. Bounds for characteristic classes.
The three parts are independent and each one is justified by major well-known conjectures and/or ambitious goals. Nevertheless, there is a unifying theme: Group Theory and its relations to Geometry, Dynamics and Analysis.
In the first part, we study the Dixmier Unitarizability Problem. Even though it has remained open for 60 years, it has witnessed deep results in the last 10 years. More recently, the PI and co-authors have obtained new progress. Related questions include the Kadison Conjecture. Our methods are as varied as ergodic theory, random graphs, L2-invariants.
In the second part, we study CAT(0) spaces and groups. The first motivation is that this framework encompasses classical objects such as S-arithmetic groups and algebraic groups; indeed, the PI obtained new extensions of Margulis' superrigidity and arithmeticity theorems. We are undertaking an in-depth study of the subject, notably with Caprace, aiming at constructing the full ""semi-simple theory"" in the most general setting. This has many new consequences even for the most classical objects such as matrix groups, and we propose several conjectures as well as the likely methods to attack them.
In the last part, we study bounded characteristic classes. One motivation is the outstanding Chern Conjecture, according to which closed affine manifolds have zero Euler characteristic. We propose a strategy using a range of techniques in order to either attack the problem or at least obtain new results on simplicial volumes."
Summary
"Our proposal has three components:
1. Unitarizable representations.
2. Spaces and groups of non-positive curvature.
3. Bounds for characteristic classes.
The three parts are independent and each one is justified by major well-known conjectures and/or ambitious goals. Nevertheless, there is a unifying theme: Group Theory and its relations to Geometry, Dynamics and Analysis.
In the first part, we study the Dixmier Unitarizability Problem. Even though it has remained open for 60 years, it has witnessed deep results in the last 10 years. More recently, the PI and co-authors have obtained new progress. Related questions include the Kadison Conjecture. Our methods are as varied as ergodic theory, random graphs, L2-invariants.
In the second part, we study CAT(0) spaces and groups. The first motivation is that this framework encompasses classical objects such as S-arithmetic groups and algebraic groups; indeed, the PI obtained new extensions of Margulis' superrigidity and arithmeticity theorems. We are undertaking an in-depth study of the subject, notably with Caprace, aiming at constructing the full ""semi-simple theory"" in the most general setting. This has many new consequences even for the most classical objects such as matrix groups, and we propose several conjectures as well as the likely methods to attack them.
In the last part, we study bounded characteristic classes. One motivation is the outstanding Chern Conjecture, according to which closed affine manifolds have zero Euler characteristic. We propose a strategy using a range of techniques in order to either attack the problem or at least obtain new results on simplicial volumes."
Max ERC Funding
1 332 710 €
Duration
Start date: 2011-03-01, End date: 2016-02-29
