Project acronym FIELDS-KNOTS
Project Quantum fields and knot homologies
Researcher (PI) Piotr Sulkowski
Host Institution (HI) UNIWERSYTET WARSZAWSKI
Country Poland
Call Details Starting Grant (StG), PE2, ERC-2013-StG
Summary This project is concerned with fundamental problems arising at the interface of quantum field theory, knot theory, and the theory of random matrices. The main aim of the project is to understand two of the most profound phenomena in physics and mathematics, namely quantization and categorification, and to establish an explicit and rigorous framework where they come into play in an interrelated fashion. The project and its aims focus on the following areas:
- Knot homologies and superpolynomials. The aim of the project in this area is to determine homological knot invariants and to derive an explicit form of colored superpolynomials for a large class of knots and links.
- Super-A-polynomial. The aim of the project in this area is to develop a theory of the super-A-polynomial, to find an explicit form of the super-A-polynomial for a large class of knots, and to understand its properties.
- Three-dimensional supersymmetric N=2 theories. This project aims to find and understand dualities between theories in this class, in particular theories related to knots by 3d-3d duality, and to generalize this duality to the level of homological knot invariants.
- Topological recursion and quantization. The project aims to develop a quantization procedure based on the topological recursion, to demonstrate its consistency with knot-theoretic quantization of A-polynomials, and to generalize this quantization scheme to super-A-polynomials.
All these research areas are connected via remarkable dualities unraveled very recently by physicists and mathematicians. The project is interdisciplinary and aims to reach the above goals by taking advantage of these dualities, and through simultaneous and complementary development in quantum field theory, knot theory, and random matrix theory, in collaboration with renowned experts in each of those fields.
Summary
This project is concerned with fundamental problems arising at the interface of quantum field theory, knot theory, and the theory of random matrices. The main aim of the project is to understand two of the most profound phenomena in physics and mathematics, namely quantization and categorification, and to establish an explicit and rigorous framework where they come into play in an interrelated fashion. The project and its aims focus on the following areas:
- Knot homologies and superpolynomials. The aim of the project in this area is to determine homological knot invariants and to derive an explicit form of colored superpolynomials for a large class of knots and links.
- Super-A-polynomial. The aim of the project in this area is to develop a theory of the super-A-polynomial, to find an explicit form of the super-A-polynomial for a large class of knots, and to understand its properties.
- Three-dimensional supersymmetric N=2 theories. This project aims to find and understand dualities between theories in this class, in particular theories related to knots by 3d-3d duality, and to generalize this duality to the level of homological knot invariants.
- Topological recursion and quantization. The project aims to develop a quantization procedure based on the topological recursion, to demonstrate its consistency with knot-theoretic quantization of A-polynomials, and to generalize this quantization scheme to super-A-polynomials.
All these research areas are connected via remarkable dualities unraveled very recently by physicists and mathematicians. The project is interdisciplinary and aims to reach the above goals by taking advantage of these dualities, and through simultaneous and complementary development in quantum field theory, knot theory, and random matrix theory, in collaboration with renowned experts in each of those fields.
Max ERC Funding
1 345 080 €
Duration
Start date: 2013-12-01, End date: 2018-11-30
Project acronym QOLAPS
Project Quantum resources: conceptuals and applications
Researcher (PI) Ryszard Horodecki
Host Institution (HI) UNIWERSYTET GDANSKI
Country Poland
Call Details Advanced Grant (AdG), PE2, ERC-2011-ADG_20110209
Summary "The studies of quantum resources - entanglement (E) and non-locality (NL) carried out over the last decade have broadened horizons of our conceptual understanding of Nature and at the same time opened unprecedented possibilities for practical applications.
The project aims at taking advantage of the most recent discoveries to understand the ultimate power and find novel applications of these resources. The main objectives are: E) to study novel entanglement-induced non-additivity effects in quantum communication and application of mixed state entanglement to quantum metrology NL) to recognize the influence of information causality on the power of quantum non-locality and verify the power of non-locality, and more generally – contextuality – for quantum computational speed-up. In particular, it is planned: E) to find new non-additivities by providing explicit constructions of bipartite channels, broadcast channels and quantum networks; to demonstrate experimentally non-additivity effects; to provide experimentally friendly entanglement measures in quantum networks; to analyse entanglement-enhanced metrology in presence of decoherence NL) to determine to what extent information-causality reproduces quantum mechanics; to generalize information causality to multipartite systems; to provide new fundamental information-theoretical principles behind quantum mechanics; to quantify and classify contextuality; to design and analyse multiparty non-local systems independently of quantum mechanics; to verify their usefulness for communication and computational tasks.
We shall extensively exploit multiple interrelations between these two aspects of quantum physics. The results of theoretical investigations will be implemented in labs by experimental partners. In particular, we plan pioneering implementations of quantum channel non-additivity effects. The proposed research lines will bring ground-breaking results for quantum information processing."
Summary
"The studies of quantum resources - entanglement (E) and non-locality (NL) carried out over the last decade have broadened horizons of our conceptual understanding of Nature and at the same time opened unprecedented possibilities for practical applications.
The project aims at taking advantage of the most recent discoveries to understand the ultimate power and find novel applications of these resources. The main objectives are: E) to study novel entanglement-induced non-additivity effects in quantum communication and application of mixed state entanglement to quantum metrology NL) to recognize the influence of information causality on the power of quantum non-locality and verify the power of non-locality, and more generally – contextuality – for quantum computational speed-up. In particular, it is planned: E) to find new non-additivities by providing explicit constructions of bipartite channels, broadcast channels and quantum networks; to demonstrate experimentally non-additivity effects; to provide experimentally friendly entanglement measures in quantum networks; to analyse entanglement-enhanced metrology in presence of decoherence NL) to determine to what extent information-causality reproduces quantum mechanics; to generalize information causality to multipartite systems; to provide new fundamental information-theoretical principles behind quantum mechanics; to quantify and classify contextuality; to design and analyse multiparty non-local systems independently of quantum mechanics; to verify their usefulness for communication and computational tasks.
We shall extensively exploit multiple interrelations between these two aspects of quantum physics. The results of theoretical investigations will be implemented in labs by experimental partners. In particular, we plan pioneering implementations of quantum channel non-additivity effects. The proposed research lines will bring ground-breaking results for quantum information processing."
Max ERC Funding
1 970 380 €
Duration
Start date: 2012-01-01, End date: 2016-12-31
