Project acronym DIOPHANTINE PROBLEMS
Project Integral and Algebraic Points on Varieties, Diophantine Problems on Number Fields and Function Fields
Researcher (PI) Umberto Zannier
Host Institution (HI) SCUOLA NORMALE SUPERIORE
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary Diophantine problems have always been a central topic in Number Theory, and have shown deep links with other basic mathematical topics, like Algebraic and Complex Geometry. Our research plan focuses on some issues in this realm, which are strictly interrelated. In the last years the PI and collaborators obtained several results on integral and algebraic points on varieties, which have inspired much subsequent research by others, and which we plan to develop further. In particular:
We plan a further study of integral points on varieties, and applications to Algebraic Dynamics, a possibility which has emerged recently.
We plan to study further the so-called `Unlikely intersections'. This theme contains celebrated issues like the Manin-Mumford conjecture. After work of the PI with Bombieri and Masser in the last 10 years, it has been the object of much recent work and also of new conjectures by R. Pink and B. Zilber. Here a new method has recently emerged in work of the PI with Masser and Pila, which also leads (as shown by Pila) to signi_cant new cases of the Andr_e-Oort conjecture. We intend to pursue in this kind of investigation, exploring further the range of the methods.
Finally, we plan further study of topics of Diophantine Approximation and Hilbert Irreducibility, connected with the above ones in the contents and in the methodology.
Summary
Diophantine problems have always been a central topic in Number Theory, and have shown deep links with other basic mathematical topics, like Algebraic and Complex Geometry. Our research plan focuses on some issues in this realm, which are strictly interrelated. In the last years the PI and collaborators obtained several results on integral and algebraic points on varieties, which have inspired much subsequent research by others, and which we plan to develop further. In particular:
We plan a further study of integral points on varieties, and applications to Algebraic Dynamics, a possibility which has emerged recently.
We plan to study further the so-called `Unlikely intersections'. This theme contains celebrated issues like the Manin-Mumford conjecture. After work of the PI with Bombieri and Masser in the last 10 years, it has been the object of much recent work and also of new conjectures by R. Pink and B. Zilber. Here a new method has recently emerged in work of the PI with Masser and Pila, which also leads (as shown by Pila) to signi_cant new cases of the Andr_e-Oort conjecture. We intend to pursue in this kind of investigation, exploring further the range of the methods.
Finally, we plan further study of topics of Diophantine Approximation and Hilbert Irreducibility, connected with the above ones in the contents and in the methodology.
Max ERC Funding
928 500 €
Duration
Start date: 2011-02-01, End date: 2016-01-31
Project acronym ENTROPHASE
Project Entropy formulation of evolutionary phase transitions
Researcher (PI) Elisabetta Rocca
Host Institution (HI) UNIVERSITA DEGLI STUDI DI PAVIA
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The ground-breaking nature of the project relies on the possibility of opening new horizons
with a novel mathematical formulation of physical problems.
The project aim is indeed to obtain relevant mathematical results in order to
get further insight into new models for phase transitions and the
corresponding evolution PDE systems. The new approach presented here turns
out to be particularly helpful within the investigation of issues like as existence, uniqueness,
control, and long-time behavior of the solutions for such evolutionary PDEs.
Moreover, the importance of the opportunity to apply such new theory to phase transitions lies
in the fact that such phenomena arise in a variety of applied problems like, e.g.,
melting and freezing in solid-liquid mixtures, phase changes in solids, crystal growth, soil freezing,
damage in elastic materials, plasticity, food conservation, collisions, and so on. From
the practical viewpoint, the possibility to describe these phenomena in a quantitative way
has deeply influenced the technological
development of our society, stimulating the related mathematical interest.
Summary
The ground-breaking nature of the project relies on the possibility of opening new horizons
with a novel mathematical formulation of physical problems.
The project aim is indeed to obtain relevant mathematical results in order to
get further insight into new models for phase transitions and the
corresponding evolution PDE systems. The new approach presented here turns
out to be particularly helpful within the investigation of issues like as existence, uniqueness,
control, and long-time behavior of the solutions for such evolutionary PDEs.
Moreover, the importance of the opportunity to apply such new theory to phase transitions lies
in the fact that such phenomena arise in a variety of applied problems like, e.g.,
melting and freezing in solid-liquid mixtures, phase changes in solids, crystal growth, soil freezing,
damage in elastic materials, plasticity, food conservation, collisions, and so on. From
the practical viewpoint, the possibility to describe these phenomena in a quantitative way
has deeply influenced the technological
development of our society, stimulating the related mathematical interest.
Max ERC Funding
659 785 €
Duration
Start date: 2011-04-01, End date: 2017-03-31
Project acronym GeoMeG
Project Geometry of Metric groups
Researcher (PI) Enrico LE DONNE
Host Institution (HI) UNIVERSITA DI PISA
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary What are the best trajectories to park a truck with several trailers?
How fast can a lattice grow? These are some of the questions studied in this project because both the infinitesimal control structure of movement of a truck and the asymptotic geometry of a (nilpotent) lattice are examples of metric groups: Lie groups with homogeneous distances.
The PI plans to study geometric properties of metric groups and their implications to control systems and nilpotent groups. In particular, the plan is to exploit the relation between the regularity of distinguished curves, sets, and maps in subRiemannian groups, volume asymptotics in nilpotent groups, and embedding results.
The general goal is to develop an adapted geometric measure theory.
SubRiemannian spaces, and in particular Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, asymptotic geometry, subelliptic PDE's and geometric group theory. The results in this project will provide more links between such areas.
The PI has developed a net of high-level international collaborations and obtained several results via a combination of analysis on metric spaces (differentiation of Lipschitz maps, tangents of measures, and Gromov-Hausdorff limits) and the theory of locally compact groups (Lie group techniques and the solutions of the Hilbert 5th problem). This allowed the PI to solve a number of open problems in the field, such as the analogue of Myers-Steenrod theorem on the smoothness of isometries, the analogue of Nash isometric embedding and the non-minimality of curves with corners.
Some of the next aims are to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problem.
The goal of this project is to tackle them, together with many more related questions.
The PI received his first degree at SNS Pisa (advisor: M.Abate) and his PhD from Yale University (advisor: B.Kleiner). Before obtaining a permanent position only three years after graduation, he was at ETH, Orsay, and MSRI. He received the prestigious position of research fellow of the Academy of Finland.
Summary
What are the best trajectories to park a truck with several trailers?
How fast can a lattice grow? These are some of the questions studied in this project because both the infinitesimal control structure of movement of a truck and the asymptotic geometry of a (nilpotent) lattice are examples of metric groups: Lie groups with homogeneous distances.
The PI plans to study geometric properties of metric groups and their implications to control systems and nilpotent groups. In particular, the plan is to exploit the relation between the regularity of distinguished curves, sets, and maps in subRiemannian groups, volume asymptotics in nilpotent groups, and embedding results.
The general goal is to develop an adapted geometric measure theory.
SubRiemannian spaces, and in particular Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, asymptotic geometry, subelliptic PDE's and geometric group theory. The results in this project will provide more links between such areas.
The PI has developed a net of high-level international collaborations and obtained several results via a combination of analysis on metric spaces (differentiation of Lipschitz maps, tangents of measures, and Gromov-Hausdorff limits) and the theory of locally compact groups (Lie group techniques and the solutions of the Hilbert 5th problem). This allowed the PI to solve a number of open problems in the field, such as the analogue of Myers-Steenrod theorem on the smoothness of isometries, the analogue of Nash isometric embedding and the non-minimality of curves with corners.
Some of the next aims are to establish an analogue of the De Giorgi's rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problem.
The goal of this project is to tackle them, together with many more related questions.
The PI received his first degree at SNS Pisa (advisor: M.Abate) and his PhD from Yale University (advisor: B.Kleiner). Before obtaining a permanent position only three years after graduation, he was at ETH, Orsay, and MSRI. He received the prestigious position of research fellow of the Academy of Finland.
Max ERC Funding
1 248 560 €
Duration
Start date: 2017-08-01, End date: 2022-07-31
Project acronym HHNCDMIR
Project Hochschild cohomology, non-commutative deformations and mirror symmetry
Researcher (PI) Wendy Lowen
Host Institution (HI) UNIVERSITEIT ANTWERPEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary "Our research programme addresses several interesting current issues in non-commutative algebraic geometry, and important links with symplectic geometry and algebraic topology. Non-commutative algebraic geometry is concerned with the study of algebraic objects in geometric ways. One of the basic philosophies is that, in analogy with (derived) categories of (quasi-)coherent sheaves over schemes and (derived) module categories, non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful in non-commutative algebra, algebraic geometry and more recently in string theory thanks to the Homological Mirror Symmetry conjecture. One of our main aims is to set up a deformation framework for non-commutative spaces represented by ""enhanced"" triangulated categories, encompassing both the non-commutative schemes represented by derived abelian categories and the derived-affine spaces, represented by dg algebras. This framework should clarify and resolve some of the important problems known to exist in the deformation theory of derived-affine spaces. It should moreover be applicable to Fukaya-type categories, and yield a new way of proving and interpreting instances of ""deformed mirror symmetry"". This theory will be developed in interaction with concrete applications of the abelian deformation theory developed in our earlier work, and with the development of new decomposition and comparison techniques for Hochschild cohomology. By understanding the links between the different theories and fields of application, we aim to achieve an interdisciplinary understanding of non-commutative spaces using abelian and triangulated structures."
Summary
"Our research programme addresses several interesting current issues in non-commutative algebraic geometry, and important links with symplectic geometry and algebraic topology. Non-commutative algebraic geometry is concerned with the study of algebraic objects in geometric ways. One of the basic philosophies is that, in analogy with (derived) categories of (quasi-)coherent sheaves over schemes and (derived) module categories, non-commutative spaces can be represented by suitable abelian or triangulated categories. This point of view has proven extremely useful in non-commutative algebra, algebraic geometry and more recently in string theory thanks to the Homological Mirror Symmetry conjecture. One of our main aims is to set up a deformation framework for non-commutative spaces represented by ""enhanced"" triangulated categories, encompassing both the non-commutative schemes represented by derived abelian categories and the derived-affine spaces, represented by dg algebras. This framework should clarify and resolve some of the important problems known to exist in the deformation theory of derived-affine spaces. It should moreover be applicable to Fukaya-type categories, and yield a new way of proving and interpreting instances of ""deformed mirror symmetry"". This theory will be developed in interaction with concrete applications of the abelian deformation theory developed in our earlier work, and with the development of new decomposition and comparison techniques for Hochschild cohomology. By understanding the links between the different theories and fields of application, we aim to achieve an interdisciplinary understanding of non-commutative spaces using abelian and triangulated structures."
Max ERC Funding
703 080 €
Duration
Start date: 2010-10-01, End date: 2016-09-30
Project acronym iHEART
Project An Integrated Heart Model for the simulation of the cardiac function
Researcher (PI) Alfio Maria QUARTERONI
Host Institution (HI) POLITECNICO DI MILANO
Call Details Advanced Grant (AdG), PE1, ERC-2016-ADG
Summary The goal of this project is to construct, mathematically analyze, numerically approximate, computationally solve, and validate on clinically relevant cases a mathematically-based integrated heart model (IHM) for the human cardiac function. The IHM comprises several core cardiac models – electrophysiology, solid and fluid mechanics, microscopic cellular force generation, and valve dynamics – which are then coupled and finally embedded into the systemic and pulmonary blood circulations. It is a multiscale system of Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs) featuring multiphysics interactions among the core models.
The physical and mathematical properties of each core model and those of the even more complex integrated heart model (IHM) will be analyzed. The numerical approximation of IHM develops along several steps: introduce new high order methods for the core models, carry out their stability and convergence analysis, devise new paradigms for their numerical coupling, and construct optimal, scalable, and adaptive preconditioners for the efficient solution of the resulting large-scale discrete problems. To address data variability in clinically relevant cases, new reduced order models and efficient computational techniques will be developed also for forward and inverse uncertainty quantification problems. Two software libraries, LifeHEART and RedHEART, will be built and made available to the scientific community.
The project is original, very ambitious, mathematically inspired and rigorous, tremendously challenging, and groundbreaking. If successful, it will provide researchers from applied mathematics and life sciences, cardiologists, and cardiac surgeons with a powerful tool for both the qualitative and quantitative study of cardiac function and dysfunction. iHEART has the potential to drive improvements in diagnosis and treatment for cardiovascular pathologies that are responsible for more than 45% of deaths in Europe.
Summary
The goal of this project is to construct, mathematically analyze, numerically approximate, computationally solve, and validate on clinically relevant cases a mathematically-based integrated heart model (IHM) for the human cardiac function. The IHM comprises several core cardiac models – electrophysiology, solid and fluid mechanics, microscopic cellular force generation, and valve dynamics – which are then coupled and finally embedded into the systemic and pulmonary blood circulations. It is a multiscale system of Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs) featuring multiphysics interactions among the core models.
The physical and mathematical properties of each core model and those of the even more complex integrated heart model (IHM) will be analyzed. The numerical approximation of IHM develops along several steps: introduce new high order methods for the core models, carry out their stability and convergence analysis, devise new paradigms for their numerical coupling, and construct optimal, scalable, and adaptive preconditioners for the efficient solution of the resulting large-scale discrete problems. To address data variability in clinically relevant cases, new reduced order models and efficient computational techniques will be developed also for forward and inverse uncertainty quantification problems. Two software libraries, LifeHEART and RedHEART, will be built and made available to the scientific community.
The project is original, very ambitious, mathematically inspired and rigorous, tremendously challenging, and groundbreaking. If successful, it will provide researchers from applied mathematics and life sciences, cardiologists, and cardiac surgeons with a powerful tool for both the qualitative and quantitative study of cardiac function and dysfunction. iHEART has the potential to drive improvements in diagnosis and treatment for cardiovascular pathologies that are responsible for more than 45% of deaths in Europe.
Max ERC Funding
2 351 544 €
Duration
Start date: 2017-12-01, End date: 2022-11-30
Project acronym UniCoSM
Project Universality in Condensed Matter and Statistical Mechanics
Researcher (PI) Alessandro GIULIANI
Host Institution (HI) UNIVERSITA DEGLI STUDI ROMA TRE
Call Details Consolidator Grant (CoG), PE1, ERC-2016-COG
Summary Universality is a central concept in several branches of mathematics and physics. In the broad context of statistical mechanics and condensed matter, it refers to the independence of certain key observables from the microscopic details of the system. Remarkable examples of this phenomenon are: the universality of the scaling theory at a second order phase transition, at a quantum critical point, or in a phase with broken continuous symmetry; the quantization of the conductivity in interacting or disordered quantum many-body systems; the equivalence between bulk and edge transport coefficients. Notwithstanding the striking evidence for the validity of the universality hypothesis in these and many other settings, a fundamental understanding of these phenomena is still lacking, particularly in the case of interacting systems.
This project will investigate several key problems, representative of different instances of universality. It will develop along three inter-connected research lines: scaling limits in Ising and dimer models, quantum transport in interacting Fermi systems, continuous symmetry breaking in spin systems and in models for pattern formation or nematic order. Progresses on these problems will come from an effective combination of the complementary techniques that are currently used in the mathematical theory of universality, such as: constructive renormalization group, reflection positivity, functional inequalities, discrete harmonic analysis, rigidity estimates. We will pay particular attention to the study of some poorly understood aspects of the theory, such as the role of boundary corrections, the loss of translational invariance in multiscale analysis, and the phenomenon of continuous non-abelian symmetry breaking. The final goal of the project is the development of new tools for the mathematical analysis of strongly interacting systems. Its impact will be an improved fundamental understanding of universality phenomena in condensed matter.
Summary
Universality is a central concept in several branches of mathematics and physics. In the broad context of statistical mechanics and condensed matter, it refers to the independence of certain key observables from the microscopic details of the system. Remarkable examples of this phenomenon are: the universality of the scaling theory at a second order phase transition, at a quantum critical point, or in a phase with broken continuous symmetry; the quantization of the conductivity in interacting or disordered quantum many-body systems; the equivalence between bulk and edge transport coefficients. Notwithstanding the striking evidence for the validity of the universality hypothesis in these and many other settings, a fundamental understanding of these phenomena is still lacking, particularly in the case of interacting systems.
This project will investigate several key problems, representative of different instances of universality. It will develop along three inter-connected research lines: scaling limits in Ising and dimer models, quantum transport in interacting Fermi systems, continuous symmetry breaking in spin systems and in models for pattern formation or nematic order. Progresses on these problems will come from an effective combination of the complementary techniques that are currently used in the mathematical theory of universality, such as: constructive renormalization group, reflection positivity, functional inequalities, discrete harmonic analysis, rigidity estimates. We will pay particular attention to the study of some poorly understood aspects of the theory, such as the role of boundary corrections, the loss of translational invariance in multiscale analysis, and the phenomenon of continuous non-abelian symmetry breaking. The final goal of the project is the development of new tools for the mathematical analysis of strongly interacting systems. Its impact will be an improved fundamental understanding of universality phenomena in condensed matter.
Max ERC Funding
1 235 875 €
Duration
Start date: 2017-03-01, End date: 2022-02-28
