Pseudorandomness and explicit constructions in discrete mathematics
Programme type: PEOPLE, MARIE CURIE ACTIONS, INTRA-EUROPEAN FELLOWSHIPS (IEF)
Objectives:
The project concerns research on the frontier between discrete mathematics and theoretical computer science. Discrete mathematics is an established mathematical discipline and is playing an increasing role in various fields of mathematics. Many real-life problems can be formulated using the language of discrete mathematics. Deep mathematics is hidden behind practical problems such as devising optimal schedules, or efficient routing of data packets through the internet. These problems also suggest that besides problems typical to pure mathematics (such as existence of a solution) one often seeks their efficient algorithmic counterparts (algorithm design).
The project goals are the following:
To explore constructions of specialized families of expanders, in particular of monotone expanders
To study the applications of dimension expanders (and related) in the area of explicit Ramsey graphs
To study in detail the new concept of partition expanders
To study the combinatorial construction of Mendel and Naor can yield explicit constructions of partition expanders, and to look for applications, especially in derandomization and communication complexity
To understand the role of extractors in theoretical computer science
To understand, extend, and simplify constructions of Ramsey graph of Barak et al
To find a purely combinatorial construction of Ramsey graphs
Grant: FEALORA(339691)
from 01/01/2014
to 31/12/2018
Grantor: European Research Council Executive Agency
Feasibility, Logic and Randomness in computational complexity
Programme type: FP7 ERC Advanced Grant
Objectives:
This project aims at making progress in the study of basic open problems in computational complexity, such as the P versus NP problem. There are several approaches to these difficult problems, one of which is proof complexity. In proof complexity we not only study the lengths of proofs in various proof systems, but also first order theories associated with complexity classes, collectively called bounded arithmetic. Proving separations between proof systems or theories in bounded arithmetic, however, seems as difficult as separating the corresponding complexity classes.
Our group has been working in proof complexity for more than two decades and has played an important role in the development of the field. The novelty of this project is its focus on the role of the following two concepts in proof complexity: incompleteness and pseudorandomness. The incompleteness phenomenon is well understood in the context of stronger arithmetical theories, but little is known in bounded arithmetic. As it may be extremely difficult to solve the problems about incompleteness in the polynomial time domain, one of the approaches we will try is proposing axioms that will decide these questions. Similarly, pseudorandomness has been intensively researched in computational complexity, but its role in proof complexity still needs more research.
Grant: MATHEF(320078)
from 01/05/2013
to 30/04/2018
Grantor: European Research Council Executive Agency
Mathematical Thermodynamics of Fluids
Programme type: FP7 ERC Advanced Grant
Objectives:
The main goal of the present research proposal is to build up a general mathematical theory describing the motion of a compressible, viscous, and heat conductive fluid. Our approach is based on the concept of generalized (weak) solutions satisfying the basic physical principles of balance of mass, momentum, and energy. The energy balance is expressed in terms of a variant of entropy inequality supplemented with an integral identity for the total energy balance.
We propose to identify a class of suitable weak solutions, where admissibility is based on a direct application of the principle of maximal entropy production compatible with Second law of thermodynamics. Stability of the solution family will be investigated by the method of relative entropies constructed on the basis of certain thermodynamic potentials as ballistic free energy.
The new solution framework will be applied to multiscale problems, where several characteristic scales become small or extremely large. We focus on mutual interaction of scales during this process and identify the asymptotic behavior of the quantities that are filtered out in the singular limits. We also propose to study the influence of the geometry of the underlying physical space that may change in the course of the limit process. In particular, problems arising in homogenization and optimal shape design in combination with various singular limits are taken into account.
The abstract approximate scheme used in the existence theory will be adapted in order to develop adequate numerical methods. We study stability and convergence of these methods using the tools developed in the abstract part, in particular, the relative entropies.
Stochastic and deterministic modelling of biological and biochemical phenomena with applications to circadian rhythms and pattern formation
Marie Curie Intra European Fellowship for Tomáš Vejchodsky at the University of Oxford. Grant Agreement Number: PIEF-GA-2012-328008.
Grant: DP130101172
from 01/01/2013
to 31/12/2016
Grantor: Australian Research Council
Enriched higher category theory
Objectives:
Higher category theory is a very young branch of mathematics (less then 20 years old), which has already became a vital tool in many areas of mathematics and theoretical physics such as algebra, geometry, topology, mathematical logic, quantum field theory and computer science. The impact of higher category theory for the future development of mathematics and physics will be immense. In its present shape, however, this theory is technically very difficult. The challenge is to find an approach to this theory which would allow to make it transparent and accessible for the wider scientific community. In our project we are going to propose such an approach and we will study its application to important open problems in geometry and topology.
Grant: AOS(318910)
from 01/11/2012
to 31/10/2016
Grantor: European Commission
Asymptotics of Operator Semigroups
Programme type: FP7 Marie Curie Action "International Research Staff Exchange Scheme"
Objectives:
The theory of asymptotic behaviour of operator semigroups is a comparatively new field serving as a common denominator for many other areas of mathematics, such as for instance the theory of partial differential equations, complex analysis, harmonic analysis and topology.
The primary interest in the study of asymptotic properties of strongly continuous operator semi-groups comes from the fact that such semigroups solve abstract Cauchy problems which are often models for various phenomena arising in natural sciences, engineering and economics.
Knowledge of the asymptotics of semigroups allows one to determine the character of long-time evolution of these phenomena.
Despite an obvious importance, the asymptotic theory of one-parameter strongly continuous operator semigroups was for a very long time a collection of scattered facts rather than an organized area of research. The interest increased in the 1980s and the theory has witnessed a dramatic development over the past thirty years. Still there is a number of notorious open problems that have been left open. These missing blocks prevent the theory from being complete, slow down the development of the theory and discourage specialists from related fields to engage into the theory.
The goal of the project is to give new impetus to the theory of asymptotic behaviour of operator semigroups. To this aim we plan to extend and unify various aspects of the asymptotic theory of operator semigroups: stability, hyperbolicity, rigidity, boundedness, relations to Fredholm property, to work out new methods and to solve several long-standing open problems thus giving the theory its final shape.
We intend to create an international forum that enables and promotes a multi- and cross- disciplinary exchange of ideas, methods and tools under the common umbrella of asymptotic theory of operator semigroups. Thus we expect that, moreover, a wide range of modern analysis will benefit from the project.
Category-theoretic framework for the Fraisse-Jonsson construction
Objectives:
The main probject objective is to develop the general and applicable theory of Fraissé-Jónsson limits, based on category theory. The project is divided into the following 4 intermediate goals:
Developing the theory of Fraissé-Jónsson limits of singular length.
Finding a suitable theory of categories with measures, capturing the case of epsilon-isometries of metric or Banach spaces.
Investigating automorphism groups of category-theoretic Fraissé limits.
Studying Fraissé-Jónsson limits of projection-embedding pairs.
Goal 1 aims at better understanding of Fraissé-Jónsson limits induced by ``pushout generated arrows", in particular, when the construction has a singular length. Successful results of Goal 2 will shed more light at objects like the Gurarii space, whose structure and properties are still not well understood. Goal 3 aims at deeper understanding of combinatorial properties of categories related to topological dynamics. Finally, Goal 4 is devoted to the study of important examples of categories related to projections. Particular examples come from domain theory.
Grant: TraFlu(SCIEX 11.152)
from 01/07/2012
to 31/12/2013
Grantor: Scientific Exchange Programme NMS.CH (Switzerland)
Transport phenomena in continuum fluid dynamics
Sciex Postdoctoral Fellowship for Ondřej Kreml at University of Zurich (Host institution).
Objectives:
Ondřej Kreml will study the results of Camillo De Lellis and László Székelyhidi about ill-posedness of bounded weak solutions for the incompressible Euler equations and bounded admissible solutions for the compressible isentropic Euler system in multiple space dimensions. The objectives of the project are to generalize the ill-posedness results for compressible isentropic Euler system and to study the Riemann problem for this system. Another objective is to modify the method of De Lellis and Székelyhidi to be applicable in other systems of partial differential equations describing inviscid fluid flows.
Programme type: FP7 Competitiveness and Innovation framework Programme (CIP)
Objectives:
In the light of mathematicians reliance on their discipline's rich published heritage and the key role of mathematics in enabling other scientific disciplines, the European Digital Mathematics Library strives to make the significant corpus of mathematics scholarship published in Europe available online, in the form of an authoritative and enduring digital collection, developed and curated by a network of institutions.
National efforts have led to the digitisation of large quantities of mathematical literature, primarily by partners in this project. Publishers produce new material that needs to be archived safely over the long term, made more visible, usable, and interoperable with the legacy corpus on which it settles. In EuDML, these partners will join together with leading technology providers in constructing the Europe-wide interconnections between their collections to create a document network as integrated and trans-national as the discipline of mathematics itself. They will future-proof their work by providing the organisational and technical infrastructure to accommodate new collections and mathematically rich metadata formats, and will work towards truly open access for the whole European Community to this foundational resource, thereby retaining Europe's leadership in the provision, accessibility and exploitation of electronic mathematical content.
EuDML will design and build a collaborative digital library service that will collate the currently distributed content by the diversity of providers. This will be achieved by implementing a single access platform for heterogeneous and multilingual collections. The network of documents will be constructed by merging and augmenting the information available about each document from each collection, and matching documents and references across the entire combined library. In return for this added value, the rights holders agree to a moving wall policy to secure eventual open access to their full texts.
Grant: NaDiMa
from 01/10/2008
to 30/09/2010
Grantor: The Education, Audiovisual and Culture Executive Agency
Motivation via Natural Differentiation in Mathematics
Objectives:
The European materials on educational policy stress the importance of formation of pupils´ competences as: competence to learn, to communicate, to solve problems, to make conjectures, etc. This process should start in early age and on the primary school level it is supported mainly by the teacher.
On the general level this project aims at the development of primary school pupils:
to support the development of their learning competences;
to support the consciousness of the meaning of mathematics as a part of human culture;
to encourage pupils' motivation to learn mathematics;
to realize pupils' individual (cognitive) potentials;
to create the possibility for students to experience success in the process of problem solving.
These aims should be achieved by means of support and enhancement of teachers directed on:
to get to know examples for substantial learning environments and by that experience the nature of mathematics and recognize and use the potential for teaching and learning;
to strengthen teacher’s mathematical content knowledge in relation to and requested for these specific learning environments;
how to cope with the heterogeneity and realize natural differentiation in mathematics classrooms;
to support the change of belief on the substance and importance of mathematics for primary school level.
In concrete terms each local team in cooperation with teachers from partners and associated schools will prepare elaborated materials and examples of substantial learning environments:
its mathematical background (importance for the development of mathematical thinking, connections to crucial mathematical ideas, different ways of solutions);
different possibilities how to put it into practice in the classrooms;
our experience from field experiment (concentrating mostly on the pupils´ motivation and development of their different competences);
possibilities to develop, cultivate, and enrich mathematical ideas.
Programme type: FP7 Information and Communication Technologies
Objectives:
The objective of DISC is the design of supervisors and fault detectors exploiting the concurrency and the modularity of the plant model. Coordinated controllers should preferably be designed using only local plant behaviour models, and requiring only limited information exchange between the different local controllers.
We plan to use several techniques to reduce the computational complexity of solving the above mentioned problem for distributed plants: modularity in the modelling and control design phases;
decentralized control with communicating controllers;
modular state identification, distributed diagnosis and modular fault detection based on the design of partially decentralized observers;
fluidisation of some discrete event dynamics to reduce state space cardinality.
The expected outcome of this project are: new methodologies for applying the above described techniques for embedded controllers to distributed plants; new tools for the modelling, simulation and supervisory control design that will be part of an integrated software platform; the application of these methodologies to a few cases of industrial relevance using the developed tools; the dissemination of the results.