Project acronym ANPROB
Project Analytic-probabilistic methods for borderline singular integrals
Researcher (PI) Tuomas Pentinpoika Hytönen
Host Institution (HI) HELSINGIN YLIOPISTO
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary The proposal consists of an extensive research program to advance the understanding of singular integral operators of Harmonic Analysis in various situations on the borderline of the existing theory. This is to be achieved by a creative combination of techniques from Analysis and Probability. On top of the standard arsenal of modern Harmonic Analysis, the main probabilistic tools are the martingale transform inequalities of Burkholder, and random geometric constructions in the spirit of the random dyadic cubes introduced to Nonhomogeneous Analysis by Nazarov, Treil and Volberg.
The problems to be addressed fall under the following subtitles, with many interconnections and overlap: (i) sharp weighted inequalities; (ii) nonhomogeneous singular integrals on metric spaces; (iii) local Tb theorems with borderline assumptions; (iv) functional calculus of rough differential operators; and (v) vector-valued singular integrals.
Topic (i) is a part of Classical Analysis, where new methods have led to substantial recent progress, culminating in my solution in July 2010 of a celebrated problem on the linear dependence of the weighted operator norm on the Muckenhoupt norm of the weight. The proof should be extendible to several related questions, and the aim is to also address some outstanding open problems in the area.
Topics (ii) and (v) deal with extensions of the theory of singular integrals to functions with more general domain and range spaces, allowing them to be abstract metric and Banach spaces, respectively. In case (ii), I have recently been able to relax the requirements on the space compared to the established theories, opening a new research direction here. Topics (iii) and (iv) are concerned with weakening the assumptions on singular integrals in the usual Euclidean space, to allow certain applications in the theory of Partial Differential Equations. The goal is to maintain a close contact and exchange of ideas between such abstract and concrete questions.
Summary
The proposal consists of an extensive research program to advance the understanding of singular integral operators of Harmonic Analysis in various situations on the borderline of the existing theory. This is to be achieved by a creative combination of techniques from Analysis and Probability. On top of the standard arsenal of modern Harmonic Analysis, the main probabilistic tools are the martingale transform inequalities of Burkholder, and random geometric constructions in the spirit of the random dyadic cubes introduced to Nonhomogeneous Analysis by Nazarov, Treil and Volberg.
The problems to be addressed fall under the following subtitles, with many interconnections and overlap: (i) sharp weighted inequalities; (ii) nonhomogeneous singular integrals on metric spaces; (iii) local Tb theorems with borderline assumptions; (iv) functional calculus of rough differential operators; and (v) vector-valued singular integrals.
Topic (i) is a part of Classical Analysis, where new methods have led to substantial recent progress, culminating in my solution in July 2010 of a celebrated problem on the linear dependence of the weighted operator norm on the Muckenhoupt norm of the weight. The proof should be extendible to several related questions, and the aim is to also address some outstanding open problems in the area.
Topics (ii) and (v) deal with extensions of the theory of singular integrals to functions with more general domain and range spaces, allowing them to be abstract metric and Banach spaces, respectively. In case (ii), I have recently been able to relax the requirements on the space compared to the established theories, opening a new research direction here. Topics (iii) and (iv) are concerned with weakening the assumptions on singular integrals in the usual Euclidean space, to allow certain applications in the theory of Partial Differential Equations. The goal is to maintain a close contact and exchange of ideas between such abstract and concrete questions.
Max ERC Funding
1 100 000 €
Duration
Start date: 2011-11-01, End date: 2016-10-31
Project acronym AROMA-CFD
Project Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics
Researcher (PI) Gianluigi Rozza
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary The aim of AROMA-CFD is to create a team of scientists at SISSA for the development of Advanced Reduced Order Modelling techniques with a focus in Computational Fluid Dynamics (CFD), in order to face and overcome many current limitations of the state of the art and improve the capabilities of reduced order methodologies for more demanding applications in industrial, medical and applied sciences contexts. AROMA-CFD deals with strong methodological developments in numerical analysis, with a special emphasis on mathematical modelling and extensive exploitation of computational science and engineering. Several tasks have been identified to tackle important problems and open questions in reduced order modelling: study of bifurcations and instabilities in flows, increasing Reynolds number and guaranteeing stability, moving towards turbulent flows, considering complex geometrical parametrizations of shapes as computational domains into extended networks. A reduced computational and geometrical framework will be developed for nonlinear inverse problems, focusing on optimal flow control, shape optimization and uncertainty quantification. Further, all the advanced developments in reduced order modelling for CFD will be delivered for applications in multiphysics, such as fluid-structure interaction problems and general coupled phenomena involving inviscid, viscous and thermal flows, solids and porous media. The advanced developed framework within AROMA-CFD will provide attractive capabilities for several industrial and medical applications (e.g. aeronautical, mechanical, naval, off-shore, wind, sport, biomedical engineering, and cardiovascular surgery as well), combining high performance computing (in dedicated supercomputing centers) and advanced reduced order modelling (in common devices) to guarantee real time computing and visualization. A new open source software library for AROMA-CFD will be created: ITHACA, In real Time Highly Advanced Computational Applications.
Summary
The aim of AROMA-CFD is to create a team of scientists at SISSA for the development of Advanced Reduced Order Modelling techniques with a focus in Computational Fluid Dynamics (CFD), in order to face and overcome many current limitations of the state of the art and improve the capabilities of reduced order methodologies for more demanding applications in industrial, medical and applied sciences contexts. AROMA-CFD deals with strong methodological developments in numerical analysis, with a special emphasis on mathematical modelling and extensive exploitation of computational science and engineering. Several tasks have been identified to tackle important problems and open questions in reduced order modelling: study of bifurcations and instabilities in flows, increasing Reynolds number and guaranteeing stability, moving towards turbulent flows, considering complex geometrical parametrizations of shapes as computational domains into extended networks. A reduced computational and geometrical framework will be developed for nonlinear inverse problems, focusing on optimal flow control, shape optimization and uncertainty quantification. Further, all the advanced developments in reduced order modelling for CFD will be delivered for applications in multiphysics, such as fluid-structure interaction problems and general coupled phenomena involving inviscid, viscous and thermal flows, solids and porous media. The advanced developed framework within AROMA-CFD will provide attractive capabilities for several industrial and medical applications (e.g. aeronautical, mechanical, naval, off-shore, wind, sport, biomedical engineering, and cardiovascular surgery as well), combining high performance computing (in dedicated supercomputing centers) and advanced reduced order modelling (in common devices) to guarantee real time computing and visualization. A new open source software library for AROMA-CFD will be created: ITHACA, In real Time Highly Advanced Computational Applications.
Max ERC Funding
1 656 579 €
Duration
Start date: 2016-05-01, End date: 2021-04-30
Project acronym CAVE
Project Challenges and Advancements in Virtual Elements
Researcher (PI) Lourenco Beirao da veiga
Host Institution (HI) UNIVERSITA' DEGLI STUDI DI MILANO-BICOCCA
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes.
The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/de-refinement procedure along the propagation front, to an exact satisfaction of conservation laws).
Summary
The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes.
The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/de-refinement procedure along the propagation front, to an exact satisfaction of conservation laws).
Max ERC Funding
980 634 €
Duration
Start date: 2016-07-01, End date: 2021-06-30
Project acronym COSMOS
Project Semiparametric Inference for Complex and Structural Models in Survival Analysis
Researcher (PI) Ingrid VAN KEILEGOM
Host Institution (HI) KATHOLIEKE UNIVERSITEIT LEUVEN
Call Details Advanced Grant (AdG), PE1, ERC-2015-AdG
Summary In survival analysis investigators are interested in modeling and analysing the time until an event happens. It often happens that the available data are right censored, which means that only a lower bound of the time of interest is observed. This feature complicates substantially the statistical analysis of this kind of data. The aim of this project is to solve a number of open problems related to time-to-event data, that would represent a major step forward in the area of survival analysis.
The project has three objectives:
[1] Cure models take into account that a certain fraction of the subjects under study will never experience the event of interest. Because of the complex nature of these models, many problems are still open and rigorous theory is rather scarce in this area. Our goal is to fill this gap, which will be a challenging but important task.
[2] Copulas are nowadays widespread in many areas in statistics. However, they can contribute more substantially to resolving a number of the outstanding issues in survival analysis, such as in quantile regression and dependent censoring. Finding answers to these open questions, would open up new horizons for a wide variety of problems.
[3] We wish to develop new methods for doing correct inference in some of the common models in survival analysis in the presence of endogeneity or measurement errors. The present methodology has serious shortcomings, and we would like to propose, develop and validate new methods, that would be a major breakthrough if successful.
The above objectives will be achieved by using mostly semiparametric models. The development of mathematical properties under these models is often a challenging task, as complex tools from the theory on empirical processes and semiparametric efficiency are required. The project will therefore require an innovative combination of highly complex mathematical skills and cutting edge results from modern theory for semiparametric models.
Summary
In survival analysis investigators are interested in modeling and analysing the time until an event happens. It often happens that the available data are right censored, which means that only a lower bound of the time of interest is observed. This feature complicates substantially the statistical analysis of this kind of data. The aim of this project is to solve a number of open problems related to time-to-event data, that would represent a major step forward in the area of survival analysis.
The project has three objectives:
[1] Cure models take into account that a certain fraction of the subjects under study will never experience the event of interest. Because of the complex nature of these models, many problems are still open and rigorous theory is rather scarce in this area. Our goal is to fill this gap, which will be a challenging but important task.
[2] Copulas are nowadays widespread in many areas in statistics. However, they can contribute more substantially to resolving a number of the outstanding issues in survival analysis, such as in quantile regression and dependent censoring. Finding answers to these open questions, would open up new horizons for a wide variety of problems.
[3] We wish to develop new methods for doing correct inference in some of the common models in survival analysis in the presence of endogeneity or measurement errors. The present methodology has serious shortcomings, and we would like to propose, develop and validate new methods, that would be a major breakthrough if successful.
The above objectives will be achieved by using mostly semiparametric models. The development of mathematical properties under these models is often a challenging task, as complex tools from the theory on empirical processes and semiparametric efficiency are required. The project will therefore require an innovative combination of highly complex mathematical skills and cutting edge results from modern theory for semiparametric models.
Max ERC Funding
2 318 750 €
Duration
Start date: 2016-09-01, End date: 2021-08-31
Project acronym DASTCO
Project Developing and Applying Structural Techniques for Combinatorial Objects
Researcher (PI) Paul Joseph Wollan
Host Institution (HI) UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary The proposed project will tackle a series of fundamental problems in discrete mathematics by studying labeled graphs, a generalization of graphs which readily apply to problems beyond graph theory. To achieve these goals will require both developing new graph theoretic tools and techniques as well as further expounding upon known methodologies.
The specific problems to be studied can be grouped into a series of semi-independent projects. The first focuses on signed graphs with applications to a conjecture of Seymour concerning 1-flowing binary matroids and a related conjecture on the intregality of polyhedra defined by a class of binary matrices. The second proposes to develop a theory of minors for directed graphs. Finally, the project looks at topological questions arising from graphs embedding in a surface and the classic problem of efficiently identifying the trivial knot. The range of topics considered will lead to the development of tools and techniques applicable to questions in discrete mathematics beyond those under direct study.
The project will create a research group incorporating graduate students and post doctoral researchers lead by the PI. Each area to be studied offers the potential for ground-breaking results at the same time offering numerous intermediate opportunities for scientific progress.
Summary
The proposed project will tackle a series of fundamental problems in discrete mathematics by studying labeled graphs, a generalization of graphs which readily apply to problems beyond graph theory. To achieve these goals will require both developing new graph theoretic tools and techniques as well as further expounding upon known methodologies.
The specific problems to be studied can be grouped into a series of semi-independent projects. The first focuses on signed graphs with applications to a conjecture of Seymour concerning 1-flowing binary matroids and a related conjecture on the intregality of polyhedra defined by a class of binary matrices. The second proposes to develop a theory of minors for directed graphs. Finally, the project looks at topological questions arising from graphs embedding in a surface and the classic problem of efficiently identifying the trivial knot. The range of topics considered will lead to the development of tools and techniques applicable to questions in discrete mathematics beyond those under direct study.
The project will create a research group incorporating graduate students and post doctoral researchers lead by the PI. Each area to be studied offers the potential for ground-breaking results at the same time offering numerous intermediate opportunities for scientific progress.
Max ERC Funding
850 000 €
Duration
Start date: 2011-12-01, End date: 2017-09-30
Project acronym HEVO
Project Holomorphic Evolution Equations
Researcher (PI) Filippo Bracci
Host Institution (HI) UNIVERSITA DEGLI STUDI DI ROMA TOR VERGATA
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary The scope of this project is to study holomorphic evolution equations and the associated dynamical systems, both from the local and the global point of view. In particular we aim to study general Loewner equations (both autonomous and non-autonomous) and applications to dynamical systems, in one and several complex variables. In one variable we plan to develop a general version of SLE's for non-slit evolutions and apply to physical and others problems. In several variables we plan to develop the general theory, together with applications.
Summary
The scope of this project is to study holomorphic evolution equations and the associated dynamical systems, both from the local and the global point of view. In particular we aim to study general Loewner equations (both autonomous and non-autonomous) and applications to dynamical systems, in one and several complex variables. In one variable we plan to develop a general version of SLE's for non-slit evolutions and apply to physical and others problems. In several variables we plan to develop the general theory, together with applications.
Max ERC Funding
700 000 €
Duration
Start date: 2011-11-01, End date: 2016-10-31
Project acronym PASCAL
Project Probabilistic And Statistical methods for Cosmological AppLications
Researcher (PI) Domenico Marinucci
Host Institution (HI) UNIVERSITA DEGLI STUDI DI ROMA TOR VERGATA
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary This is an interdisciplinary project at the interface between Mathematical Statistics, Probability and Cosmological Applications. The principal focus is on the harmonic analysis for isotropic, mean square continuous spherical random fields, with a view to applications to Cosmic Microwave Background radiation data analysis. We will focus in particular on the following issues:
1) the characterizations of isotropy in harmonic space, the analysis of higher order polyspectra and their applications to non-Gaussianity analysis;
2) the constructions of spherical needlets/wavelets and their stochastic properties;
3) the analysis of random sections of spin fiber bundles on the sphere, as motivated by the analysis of CMB polarization and weak gravitational lensing
4) adaptive density estimation for directional data, as motivated by Cosmic Rays data analysis.
Summary
This is an interdisciplinary project at the interface between Mathematical Statistics, Probability and Cosmological Applications. The principal focus is on the harmonic analysis for isotropic, mean square continuous spherical random fields, with a view to applications to Cosmic Microwave Background radiation data analysis. We will focus in particular on the following issues:
1) the characterizations of isotropy in harmonic space, the analysis of higher order polyspectra and their applications to non-Gaussianity analysis;
2) the constructions of spherical needlets/wavelets and their stochastic properties;
3) the analysis of random sections of spin fiber bundles on the sphere, as motivated by the analysis of CMB polarization and weak gravitational lensing
4) adaptive density estimation for directional data, as motivated by Cosmic Rays data analysis.
Max ERC Funding
1 193 000 €
Duration
Start date: 2011-11-01, End date: 2016-10-31
Project acronym QUADYNEVOPRO
Project Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture
Researcher (PI) Gianni Dal Maso
Host Institution (HI) SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Call Details Advanced Grant (AdG), PE1, ERC-2011-ADG_20110209
Summary This research project deals with nonlinear evolution problems that arise in the study of the inelastic behaviour of solids, in particular in plasticity and fracture. The project will focus on selected problems, grouped into three main topics, namely:
1. Plasticity with hardening and softening, 2. Quasistatic crack growth, 3. Dynamic fracture mechanics.
The analysis of the models of these mechanical problems leads to deep mathematical questions originated by two common features: the energies are not convex and the solutions exhibit discontinuities both with respect to space and time. In addition, plasticity problems often lead to concentration of the strains, whose mathematical description requires singular measures. Most of these problems have a variational structure and are governed by partial differential equations. Therefore, the construction of consistent models and their analysis need advanced mathematical tools from the calculus of variations, from measure theory and geometric measure theory, and also from the theory of nonlinear elliptic and parabolic partial differential equations. The models of dynamic crack growth considered in the project also need results from the theory of linear hyperbolic equations.
Our goal is to develop new mathematical tools in these areas for the study of the selected problems. Quasistatic evolution problems in plasticity with hardening and softening will be studied through a vanishing viscosity approach, that has been successfully used by the P.I. in the study of the Cam-Clay model in soil mechanics. Quasistatic models of crack growth will be developed under different assumptions on the elastic response of the material and on the mechanisms of crack formation. For the problem of crack growth in the dynamic regime our aim is to develop a model that predicts the crack path as well as the time evolution of the crack along its path, taking into account all inertial effects.
Summary
This research project deals with nonlinear evolution problems that arise in the study of the inelastic behaviour of solids, in particular in plasticity and fracture. The project will focus on selected problems, grouped into three main topics, namely:
1. Plasticity with hardening and softening, 2. Quasistatic crack growth, 3. Dynamic fracture mechanics.
The analysis of the models of these mechanical problems leads to deep mathematical questions originated by two common features: the energies are not convex and the solutions exhibit discontinuities both with respect to space and time. In addition, plasticity problems often lead to concentration of the strains, whose mathematical description requires singular measures. Most of these problems have a variational structure and are governed by partial differential equations. Therefore, the construction of consistent models and their analysis need advanced mathematical tools from the calculus of variations, from measure theory and geometric measure theory, and also from the theory of nonlinear elliptic and parabolic partial differential equations. The models of dynamic crack growth considered in the project also need results from the theory of linear hyperbolic equations.
Our goal is to develop new mathematical tools in these areas for the study of the selected problems. Quasistatic evolution problems in plasticity with hardening and softening will be studied through a vanishing viscosity approach, that has been successfully used by the P.I. in the study of the Cam-Clay model in soil mechanics. Quasistatic models of crack growth will be developed under different assumptions on the elastic response of the material and on the mechanisms of crack formation. For the problem of crack growth in the dynamic regime our aim is to develop a model that predicts the crack path as well as the time evolution of the crack along its path, taking into account all inertial effects.
Max ERC Funding
968 500 €
Duration
Start date: 2012-03-01, End date: 2017-02-28
Project acronym SIMPLELCGPS
Project Simple locally compact groups: exploring the boundaries of the linear world
Researcher (PI) Pierre-Emmanuel Caprace
Host Institution (HI) UNIVERSITE CATHOLIQUE DE LOUVAIN
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary The theory of locally compact groups stretches out between two antipodes: on one hand, connected groups whose structure, according to the solution to Hilbert fifth problem, is governed by Lie theory and is thus relatively rigid, and on the other hand, discrete groups, which are subject to a spectacular variety of behaviours, going from the most stringent rigidity properties to the most intriguing pathological ones. The goal of this research program is to explore the wide space lying between these two extremes.
The entire program is built around two major open problems: performing an exhaustive study of compactly generated simple locally compact groups, and finding an algebraic characterization of those locally compact groups which are linear. Although these problems do not seem to be directly approachable given the current state of knowledge, they are nevertheless considered as guidelines suggesting a number of specific questions and conjectures which are envisaged in detail under various perspectives of algebraic, geometric, arithmetic and analytic nature. Each of these specific questions presents independent interest; answers to any of them will moreover provide insight into the guiding problems.
Summary
The theory of locally compact groups stretches out between two antipodes: on one hand, connected groups whose structure, according to the solution to Hilbert fifth problem, is governed by Lie theory and is thus relatively rigid, and on the other hand, discrete groups, which are subject to a spectacular variety of behaviours, going from the most stringent rigidity properties to the most intriguing pathological ones. The goal of this research program is to explore the wide space lying between these two extremes.
The entire program is built around two major open problems: performing an exhaustive study of compactly generated simple locally compact groups, and finding an algebraic characterization of those locally compact groups which are linear. Although these problems do not seem to be directly approachable given the current state of knowledge, they are nevertheless considered as guidelines suggesting a number of specific questions and conjectures which are envisaged in detail under various perspectives of algebraic, geometric, arithmetic and analytic nature. Each of these specific questions presents independent interest; answers to any of them will moreover provide insight into the guiding problems.
Max ERC Funding
848 640 €
Duration
Start date: 2011-12-01, End date: 2016-11-30
Project acronym StableChaoticPlanetM
Project Stable and Chaotic Motions in the Planetary Problem
Researcher (PI) Gabriella Pinzari
Host Institution (HI) UNIVERSITA DEGLI STUDI DI PADOVA
Call Details Starting Grant (StG), PE1, ERC-2015-STG
Summary The planetary problem consists in determining the motions of n planets, interacting among themselves and with a sun, via gravity only. Its deep comprehension has relevant consequences in Mathematics, Physics, Astronomy and Astrophysics.
The problem is by its nature perturbative, being well approximated by the much easier (and in fact exactly solved since the XVII century) problem where each planet interacts only with the sun. However, when the mutual interactions among planets are taken into account, the dynamics of the system is much richer and, up to nowadays, essentially unsolved. Stable and unstable motions coexist as well.
In general, perturbation theory allows to describe qualitative aspects of the motion, but it does not apply directly to the problem, because of its deep degeneracies.
During my PhD, I obtained important results on the stability of the problem, based on a new symplectic description, that allowed me to write, for the first time, in the framework of close to be integrable systems, the Hamilton equations governing the dynamics of the problem, made free of its integral of motions, and degeneracies related. By such results, I was an invited speaker to the ICM of 2014, in Seoul.
The goal of this research is to use such recent tools, develop techniques, ideas and wide collaborations, also by means of the creation of post-doc positions, assistant professorships (non-tenure track), workshops and advanced schools, in order to find results concerning the long-time stability of the problem, as well as unstable or diffusive motions.
Summary
The planetary problem consists in determining the motions of n planets, interacting among themselves and with a sun, via gravity only. Its deep comprehension has relevant consequences in Mathematics, Physics, Astronomy and Astrophysics.
The problem is by its nature perturbative, being well approximated by the much easier (and in fact exactly solved since the XVII century) problem where each planet interacts only with the sun. However, when the mutual interactions among planets are taken into account, the dynamics of the system is much richer and, up to nowadays, essentially unsolved. Stable and unstable motions coexist as well.
In general, perturbation theory allows to describe qualitative aspects of the motion, but it does not apply directly to the problem, because of its deep degeneracies.
During my PhD, I obtained important results on the stability of the problem, based on a new symplectic description, that allowed me to write, for the first time, in the framework of close to be integrable systems, the Hamilton equations governing the dynamics of the problem, made free of its integral of motions, and degeneracies related. By such results, I was an invited speaker to the ICM of 2014, in Seoul.
The goal of this research is to use such recent tools, develop techniques, ideas and wide collaborations, also by means of the creation of post-doc positions, assistant professorships (non-tenure track), workshops and advanced schools, in order to find results concerning the long-time stability of the problem, as well as unstable or diffusive motions.
Max ERC Funding
900 000 €
Duration
Start date: 2016-03-01, End date: 2021-02-28
