Project acronym BSD
Project Euler systems and the conjectures of Birch and Swinnerton-Dyer, Bloch and Kato
Researcher (PI) Victor Rotger cerdà
Host Institution (HI) UNIVERSITAT POLITECNICA DE CATALUNYA
Call Details Consolidator Grant (CoG), PE1, ERC-2015-CoG
Summary In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).
Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:
I. BSD over totally real number fields. I plan to prove new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.
Summary
In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $1.000.000 Prize Problems. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960's. The main object of this proposal is developing innovative and unconventional strategies for proving groundbreaking results towards the resolution of this problem and their generalizations by Bloch and Kato (BK).
Breakthroughs on BSD were achieved by Coates-Wiles, Gross, Zagier and Kolyvagin, and Kato. Since then, there have been nearly no new ideas on how to tackle BSD. Only very recently, three independent revolutionary approaches have seen the light: the works of (1) the Fields medalist Bhargava, (2) Skinner and Urban, and (3) myself and my collaborators. In spite of that, our knowledge of BSD is rather poor. In my proposal I suggest innovating strategies for approaching new horizons in BSD and BK that I aim to develop with the team of PhD and postdoctoral researchers that the CoG may allow me to consolidate. The results I plan to prove represent a departure from the achievements obtained with my coauthors during the past years:
I. BSD over totally real number fields. I plan to prove new ground-breaking instances of BSD in rank 0 for elliptic curves over totally real number fields, generalizing the theorem of Kato (by providing a new proof) and covering many new scenarios that have never been considered before.
II. BSD in rank r=2. Most of the literature on BSD applies when r=0 or 1. I expect to prove p-adic versions of the theorems of Gross-Zagier and Kolyvagin in rank 2.
III. Darmon's 2000 conjecture on Stark-Heegner points. I plan to prove Darmon’s striking conjecture announced at the ICM2000 by recasting it in terms of special values of p-adic L-functions.
Max ERC Funding
1 428 588 €
Duration
Start date: 2016-09-01, End date: 2021-08-31
Project acronym DYCON
Project Dynamic Control and Numerics of Partial Differential Equations
Researcher (PI) Enrique Zuazua
Host Institution (HI) FUNDACION DEUSTO
Call Details Advanced Grant (AdG), PE1, ERC-2015-AdG
Summary This project aims at making a breakthrough contribution in the broad area of Control of Partial Differential Equations (PDE) and their numerical approximation methods by addressing key unsolved issues appearing systematically in real-life applications.
To this end, we pursue three objectives: 1) to contribute with new key theoretical methods and results, 2) to develop the corresponding numerical tools, and 3) to build up new computational software, the DYCON-COMP computational platform, thereby bridging the gap to applications.
The field of PDEs, together with numerical approximation and simulation methods and control theory, have evolved significantly in the last decades in a cross-fertilization process, to address the challenging demands of industrial and cross-disciplinary applications such as, for instance, the management of natural resources, meteorology, aeronautics, oil industry, biomedicine, human and animal collective behaviour, etc. Despite these efforts, some of the key issues still remain unsolved, either because of a lack of analytical understanding, of the absence of efficient numerical solvers, or of a combination of both.
This project identifies and focuses on six key topics that play a central role in most of the processes arising in applications, but which are still poorly understood: control of parameter dependent problems; long time horizon control; control under constraints; inverse design of time-irreversible models; memory models and hybrid PDE/ODE models, and finite versus infinite-dimensional dynamical systems.
These topics cannot be handled by superposing the state of the art in the various disciplines, due to the unexpected interactive phenomena that may emerge, for instance, in the fine numerical approximation of control problems. The coordinated and focused effort that we aim at developing is timely and much needed in order to solve these issues and bridge the gap from modelling to control, computer simulations and applications.
Summary
This project aims at making a breakthrough contribution in the broad area of Control of Partial Differential Equations (PDE) and their numerical approximation methods by addressing key unsolved issues appearing systematically in real-life applications.
To this end, we pursue three objectives: 1) to contribute with new key theoretical methods and results, 2) to develop the corresponding numerical tools, and 3) to build up new computational software, the DYCON-COMP computational platform, thereby bridging the gap to applications.
The field of PDEs, together with numerical approximation and simulation methods and control theory, have evolved significantly in the last decades in a cross-fertilization process, to address the challenging demands of industrial and cross-disciplinary applications such as, for instance, the management of natural resources, meteorology, aeronautics, oil industry, biomedicine, human and animal collective behaviour, etc. Despite these efforts, some of the key issues still remain unsolved, either because of a lack of analytical understanding, of the absence of efficient numerical solvers, or of a combination of both.
This project identifies and focuses on six key topics that play a central role in most of the processes arising in applications, but which are still poorly understood: control of parameter dependent problems; long time horizon control; control under constraints; inverse design of time-irreversible models; memory models and hybrid PDE/ODE models, and finite versus infinite-dimensional dynamical systems.
These topics cannot be handled by superposing the state of the art in the various disciplines, due to the unexpected interactive phenomena that may emerge, for instance, in the fine numerical approximation of control problems. The coordinated and focused effort that we aim at developing is timely and much needed in order to solve these issues and bridge the gap from modelling to control, computer simulations and applications.
Max ERC Funding
2 065 125 €
Duration
Start date: 2016-10-01, End date: 2021-09-30
Project acronym RESTRICTION
Project Restriction of the Fourier transform with applications to the Schrödinger and wave equations
Researcher (PI) Keith Mckenzie Rogers
Host Institution (HI) AGENCIA ESTATAL CONSEJO SUPERIOR DEINVESTIGACIONES CIENTIFICAS
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary In 1967, Stein proved that the Fourier transform of functions in L^p could be meaningfully restricted to the sphere for certain p>1. The restriction conjecture, which asserts the maximal range of such p, was solved by Fefferman in two dimensions, but the conjecture remains open in higher dimensions. Strichartz considered the same question but with the sphere replaced by the paraboloid or the cone, and a great deal of progress has been made in the last two decades by Bourgain, Wolff and Tao, among others. Due to the fact that the adjoint operators of the restriction operators to the paraboloid and cone correspond to the Schrödinger and wave evolution operators, respectively, this work has been hugely influential. The main goal of this proposal is to improve the state of the art for the mixed norm analogues of these conjectures.
Summary
In 1967, Stein proved that the Fourier transform of functions in L^p could be meaningfully restricted to the sphere for certain p>1. The restriction conjecture, which asserts the maximal range of such p, was solved by Fefferman in two dimensions, but the conjecture remains open in higher dimensions. Strichartz considered the same question but with the sphere replaced by the paraboloid or the cone, and a great deal of progress has been made in the last two decades by Bourgain, Wolff and Tao, among others. Due to the fact that the adjoint operators of the restriction operators to the paraboloid and cone correspond to the Schrödinger and wave evolution operators, respectively, this work has been hugely influential. The main goal of this proposal is to improve the state of the art for the mixed norm analogues of these conjectures.
Max ERC Funding
950 000 €
Duration
Start date: 2011-09-01, End date: 2017-08-31
Project acronym SPIA
Project Magnetic connectivity through the Solar Partially Ionized Atmosphere
Researcher (PI) Olena Khomenko
Host Institution (HI) INSTITUTO DE ASTROFISICA DE CANARIAS
Call Details Starting Grant (StG), PE9, ERC-2011-StG_20101014
Summary The broad scientific objective of the SPIA proposal is to understand the magnetism of the Sun and stars and to establish connections between the magnetic activity in sub-surface layers and its manifestation in the outer atmosphere. The complex interactions in magnetized stellar plasmas are best studied via numerical simulations, a new powerful method of research that appeared in astrophysics with the development of large supercomputer facilities. With a coming era of large aperture solar telescopes, ATST and EST, spectropolarimetric observations of the Sun will become available at extraordinary high spatial and temporal resolutions. New modelling tools are required to understand the plasma behaviour at these scales. I propose to consolidate a research group of bright scientists around the PI to explore a novel promising approach for the description solar atmospheric plasma under multi-fluid approximation. The degree of plasma ionization in the photosphere and chromosphere of the Sun is extremely low and significant deviations from the classical magneto-hydrodynamic description are expected. A major development of the SPIA proposal will be the implementation of a multi-fluid plasma description, appropriate for a partially ionized medium, relaxing approximations of classical magneto-hydrodynamics. With the inclusion of standard radiative transfer into the three-dimensional multi-fluid code to be developed by the project team, it will be possible to perform simulations of solar sub-photospheric and photospheric regions, up to the low chromosphere, with a realism not achieved before. The importance of the non-ideal plasma effect for the energy balance of the solar chromosphere will be evaluated, and three-dimensional time-dependent models of multi-fluid magneto-convection will be created. This effort will produce a significant step toward the solution of the long-standing question of the origin of solar chromosphere, one of the most poorly understood regions of the Sun.
Summary
The broad scientific objective of the SPIA proposal is to understand the magnetism of the Sun and stars and to establish connections between the magnetic activity in sub-surface layers and its manifestation in the outer atmosphere. The complex interactions in magnetized stellar plasmas are best studied via numerical simulations, a new powerful method of research that appeared in astrophysics with the development of large supercomputer facilities. With a coming era of large aperture solar telescopes, ATST and EST, spectropolarimetric observations of the Sun will become available at extraordinary high spatial and temporal resolutions. New modelling tools are required to understand the plasma behaviour at these scales. I propose to consolidate a research group of bright scientists around the PI to explore a novel promising approach for the description solar atmospheric plasma under multi-fluid approximation. The degree of plasma ionization in the photosphere and chromosphere of the Sun is extremely low and significant deviations from the classical magneto-hydrodynamic description are expected. A major development of the SPIA proposal will be the implementation of a multi-fluid plasma description, appropriate for a partially ionized medium, relaxing approximations of classical magneto-hydrodynamics. With the inclusion of standard radiative transfer into the three-dimensional multi-fluid code to be developed by the project team, it will be possible to perform simulations of solar sub-photospheric and photospheric regions, up to the low chromosphere, with a realism not achieved before. The importance of the non-ideal plasma effect for the energy balance of the solar chromosphere will be evaluated, and three-dimensional time-dependent models of multi-fluid magneto-convection will be created. This effort will produce a significant step toward the solution of the long-standing question of the origin of solar chromosphere, one of the most poorly understood regions of the Sun.
Max ERC Funding
969 600 €
Duration
Start date: 2012-01-01, End date: 2016-12-31
