Project acronym SHPEF
Project Stability and hyperbolicity of polynomials and entire functions
Researcher (PI) Olga Holtz
Host Institution (HI) TECHNISCHE UNIVERSITAT BERLIN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The project is devoted to the theory, algorithms and applications of hyperbolic and stable multivariate polynomials. This line of research is meant to lead to new fundamental results in analysis, matrix and operator theory, combinatorics, and theoretical
computer science.
The central goal of the project is to develop a comprehensive, seamless, theory of hyperbolic and stable multivariate polynomials. The four areas and four objectives of the project are as follows:
Classical analysis: revisit and expand the theory of hyperbolic and stable polynomials and entire functions in both the univariate and the multivariate setting. Applications: apply the theory of hyperbolic and stable polynomials to problems of matrix theory,
combinatorics and theoretical computer science.
Operator theory: develop the theory of hypo- and hyperoscillating operators and apply it to problems of fluid dynamics. Algorithms: develop fast and accurate algorithms for testing hyperbolicity/stability and for related problems.
Summary
The project is devoted to the theory, algorithms and applications of hyperbolic and stable multivariate polynomials. This line of research is meant to lead to new fundamental results in analysis, matrix and operator theory, combinatorics, and theoretical
computer science.
The central goal of the project is to develop a comprehensive, seamless, theory of hyperbolic and stable multivariate polynomials. The four areas and four objectives of the project are as follows:
Classical analysis: revisit and expand the theory of hyperbolic and stable polynomials and entire functions in both the univariate and the multivariate setting. Applications: apply the theory of hyperbolic and stable polynomials to problems of matrix theory,
combinatorics and theoretical computer science.
Operator theory: develop the theory of hypo- and hyperoscillating operators and apply it to problems of fluid dynamics. Algorithms: develop fast and accurate algorithms for testing hyperbolicity/stability and for related problems.
Max ERC Funding
880 000 €
Duration
Start date: 2010-08-01, End date: 2015-07-31
Project acronym SPALORA
Project Sparse and Low Rank Recovery
Researcher (PI) Holger Rauhut
Host Institution (HI) RHEINISCH-WESTFAELISCHE TECHNISCHE HOCHSCHULE AACHEN
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary Compressive sensing is a novel field in signal processing at the interface of applied mathematics, electrical engineering and computer science, which caught significant interest over the past five years. It provides a fundamentally new approach to signal acquisition and processing that has large potential for many applications. Compressive sensing (sparse recovery) predicts the surprising phenomenon that many sparse signals (i.e. many real-world signals) can be recovered from what was previously believed to be highly incomplete measurements (information) using computationally efficient algorithms. In the past year exciting new developments emerged on the heels of compressive sensing: low rank matrix recovery (matrix completion); as well as a novel approach for the recovery of high-dimensional functions.
We plan to pursue the following research directions:
- Compressive Sensing (sparse recovery): We aim at a rigorous analysis of certain measurement matrices.
- Low rank matrix recovery: First results predict that low rank matrices can be recovered from incomplete linear information using convex optimization.
- Low rank tensor recovery: We plan to extend methods and mathematical results from low rank matrix recovery to tensors. This field is presently completely open.
- Recovery of high-dimensional functions: In order to reduce the huge computational burden usually observed in the computational treatment of high-dimensional functions, a recent novel approach assumes that the function of interest actually depends only on a small number of variables. Preliminary results suggest that compressive sensing
and low rank matrix recovery tools can be applied to the efficient recovery of such functions.
We plan to develop computational methods for all these topics and to derive rigorous mathematical results on their performance. With the experience I gained over the past
years, I strongly believe that I have the necessary competence to pursue this project.
Summary
Compressive sensing is a novel field in signal processing at the interface of applied mathematics, electrical engineering and computer science, which caught significant interest over the past five years. It provides a fundamentally new approach to signal acquisition and processing that has large potential for many applications. Compressive sensing (sparse recovery) predicts the surprising phenomenon that many sparse signals (i.e. many real-world signals) can be recovered from what was previously believed to be highly incomplete measurements (information) using computationally efficient algorithms. In the past year exciting new developments emerged on the heels of compressive sensing: low rank matrix recovery (matrix completion); as well as a novel approach for the recovery of high-dimensional functions.
We plan to pursue the following research directions:
- Compressive Sensing (sparse recovery): We aim at a rigorous analysis of certain measurement matrices.
- Low rank matrix recovery: First results predict that low rank matrices can be recovered from incomplete linear information using convex optimization.
- Low rank tensor recovery: We plan to extend methods and mathematical results from low rank matrix recovery to tensors. This field is presently completely open.
- Recovery of high-dimensional functions: In order to reduce the huge computational burden usually observed in the computational treatment of high-dimensional functions, a recent novel approach assumes that the function of interest actually depends only on a small number of variables. Preliminary results suggest that compressive sensing
and low rank matrix recovery tools can be applied to the efficient recovery of such functions.
We plan to develop computational methods for all these topics and to derive rigorous mathematical results on their performance. With the experience I gained over the past
years, I strongly believe that I have the necessary competence to pursue this project.
Max ERC Funding
1 010 220 €
Duration
Start date: 2011-01-01, End date: 2015-12-31
Project acronym SURFARI
Project Arithmetic of algebraic surfaces
Researcher (PI) Matthias Schütt
Host Institution (HI) GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER
Call Details Starting Grant (StG), PE1, ERC-2011-StG_20101014
Summary This research proposal concerns a fundamental problem in the theory of al-
gebraic surfaces which poses one of the most important challenges in order to
understand the inner structure of algebraic surfaces beyond the current state of
the art. Our research team will investigate in detail the structure of curves on
algebraic surfaces which is captured in the Neron-Severi group. In general, it
is a widely open problem to decide which shapes this group can take. By de-
signing innovative and unconventional approaches at the borderline of arithmetic
and geometry, we aim at groundbreaking results that will spark deep insights
into the inner structures of algebraic surfaces and lay cornerstones for future
investigations.
Summary
This research proposal concerns a fundamental problem in the theory of al-
gebraic surfaces which poses one of the most important challenges in order to
understand the inner structure of algebraic surfaces beyond the current state of
the art. Our research team will investigate in detail the structure of curves on
algebraic surfaces which is captured in the Neron-Severi group. In general, it
is a widely open problem to decide which shapes this group can take. By de-
signing innovative and unconventional approaches at the borderline of arithmetic
and geometry, we aim at groundbreaking results that will spark deep insights
into the inner structures of algebraic surfaces and lay cornerstones for future
investigations.
Max ERC Funding
899 847 €
Duration
Start date: 2011-10-01, End date: 2016-09-30
Project acronym TQFT
Project The geometry of topological quantum field theories
Researcher (PI) Katrin Wendland
Host Institution (HI) ALBERT-LUDWIGS-UNIVERSITAET FREIBURG
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary The predictive power of quantum field theory (QFT) is a perpetual driving force in geometry. Examples include the invention of Frobenius manifolds, mixed twistor structures, primitive forms, and harmonic bundles, up to the discovery of the McKay correspondence, mirror symmetry, and Gromov-Witten invariants. Still seemingly disparate, in fact these all are related to topological (T) QFT and thereby to the work by Cecotti, Vafa et al of more than 20 years ago. The broad aim of the proposed research is to pull the strands together which have evolved from TQFT, by implementing insights from mathematics and physics. The goal is a unified, conclusive picture of the geometry of TQFTs. Solving the fundamental questions on the underlying common structure will open new horizons for all disciplines built on TQFT. Hertling’s “TERP” structures, formally unifying the geometric ingredients, will be key. The work plan is textured into four independent strands which gain full power from their intricate interrelations. (1) To implement TQFT, a construction by Hitchin will be generalised to perform geometric quantisation for spaces with TERP structure. Quasi-classical limits and conformal blocks will be studied as well as TERP structures in the Barannikov-Kontsevich construction of Frobenius manifolds. (2) Relating to singularity theory, a complete picture is aspired, including matrix factorisation and allowing singularities of functions on complete intersections. A main new ingredient are QFT results by Martinec and Moore. (3) Incorporating D-branes, spaces of stability conditions in triangulated categories will be equipped with TERP structures. To use geometric quantisation is a novel approach which should solve the expected convergence issues. (4) For Borcherds automorphic forms and GKM algebras their as yet cryptic relation to “generalised indices” shall be demystified: In a geometric quantisation of TERP structures, generalised theta functions should appear naturally.
Summary
The predictive power of quantum field theory (QFT) is a perpetual driving force in geometry. Examples include the invention of Frobenius manifolds, mixed twistor structures, primitive forms, and harmonic bundles, up to the discovery of the McKay correspondence, mirror symmetry, and Gromov-Witten invariants. Still seemingly disparate, in fact these all are related to topological (T) QFT and thereby to the work by Cecotti, Vafa et al of more than 20 years ago. The broad aim of the proposed research is to pull the strands together which have evolved from TQFT, by implementing insights from mathematics and physics. The goal is a unified, conclusive picture of the geometry of TQFTs. Solving the fundamental questions on the underlying common structure will open new horizons for all disciplines built on TQFT. Hertling’s “TERP” structures, formally unifying the geometric ingredients, will be key. The work plan is textured into four independent strands which gain full power from their intricate interrelations. (1) To implement TQFT, a construction by Hitchin will be generalised to perform geometric quantisation for spaces with TERP structure. Quasi-classical limits and conformal blocks will be studied as well as TERP structures in the Barannikov-Kontsevich construction of Frobenius manifolds. (2) Relating to singularity theory, a complete picture is aspired, including matrix factorisation and allowing singularities of functions on complete intersections. A main new ingredient are QFT results by Martinec and Moore. (3) Incorporating D-branes, spaces of stability conditions in triangulated categories will be equipped with TERP structures. To use geometric quantisation is a novel approach which should solve the expected convergence issues. (4) For Borcherds automorphic forms and GKM algebras their as yet cryptic relation to “generalised indices” shall be demystified: In a geometric quantisation of TERP structures, generalised theta functions should appear naturally.
Max ERC Funding
750 000 €
Duration
Start date: 2009-01-01, End date: 2014-06-30
Project acronym UNICON
Project New Adaptive Computational Methods for Fluid-Structure Interaction using an Unified Continuum Formulation with Applications in Biology, Medicine and Industry
Researcher (PI) Johan Hoffman
Host Institution (HI) KUNGLIGA TEKNISKA HOEGSKOLAN
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary For many problems involving a fluid and a structure, decoupling of the two is not possible to accurately model the phenomenon at hand, instead the fluid-structure interaction (FSI) problem has to be solved as a coupled problem. This includes a multitude of important problems in biology, medicine and industry, such as the modeling of insect flight, the blood flow in our heart and arteries, human speech, acoustic noise generation in vehicles and wind induced vibrations in bridges and other structures. Major open challenges of computational FSI include; (i) robustness of the fluid-structure coupling, (ii) efficiency and reliability of the computations in the form of adaptivity and quantitative error estimates, and (iii) in the case of high Reynolds number flow the computation of turbulent flow. In this project we address (i)-(iii) by a novel approach which we refer to as a Unified continuum formulation (UCF), where we formulate the fundamental conservation laws for mass, momentum and energy for the combined FSI domain, which is treated as one single continuum, with the only difference being the constitutive relations for the fluid and the structure. The stability problems connected to FSI are related to the exchange of information (stresses and displacements) over the fluid-structure interface, but with UCF we achieve (i) by the global coupling of the conservation laws where the fluid-structure interface is just an interior surface. We achieve (ii)-(iii) by extending to FSI our technology for adaptive finite element methods for turbulent flow with a posteriori error estimation using duality. We typically discretize the equations using a Lagrangian coordinate system for the structure and Arbitrary Lagrangian-Eulerian (ALE) coordinates for the fluid. Preliminary results for the simulation of blood flow are very promising. The computational algorithms are implemented in the open source software FEniCS (www.fenics.org), of which our group is one of the main developers.
Summary
For many problems involving a fluid and a structure, decoupling of the two is not possible to accurately model the phenomenon at hand, instead the fluid-structure interaction (FSI) problem has to be solved as a coupled problem. This includes a multitude of important problems in biology, medicine and industry, such as the modeling of insect flight, the blood flow in our heart and arteries, human speech, acoustic noise generation in vehicles and wind induced vibrations in bridges and other structures. Major open challenges of computational FSI include; (i) robustness of the fluid-structure coupling, (ii) efficiency and reliability of the computations in the form of adaptivity and quantitative error estimates, and (iii) in the case of high Reynolds number flow the computation of turbulent flow. In this project we address (i)-(iii) by a novel approach which we refer to as a Unified continuum formulation (UCF), where we formulate the fundamental conservation laws for mass, momentum and energy for the combined FSI domain, which is treated as one single continuum, with the only difference being the constitutive relations for the fluid and the structure. The stability problems connected to FSI are related to the exchange of information (stresses and displacements) over the fluid-structure interface, but with UCF we achieve (i) by the global coupling of the conservation laws where the fluid-structure interface is just an interior surface. We achieve (ii)-(iii) by extending to FSI our technology for adaptive finite element methods for turbulent flow with a posteriori error estimation using duality. We typically discretize the equations using a Lagrangian coordinate system for the structure and Arbitrary Lagrangian-Eulerian (ALE) coordinates for the fluid. Preliminary results for the simulation of blood flow are very promising. The computational algorithms are implemented in the open source software FEniCS (www.fenics.org), of which our group is one of the main developers.
Max ERC Funding
500 000 €
Duration
Start date: 2008-06-01, End date: 2013-05-31
