Phase Space Analysis of Partial Differential Equations

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These properties have been successfully studied so far in a number of special cases only under very restrictive conditions on the operator partly because they were not realised in the form of FIOs. However, recent research indicates that it should be possible to treat the general case of nondegenerate Fourier integral operators by combining recent developments in the local regularity theory with new approaches for establishing global estimates.

Phase Space Analysis of Partial Differential Equations

Global estimates for these operators are of crucial importance for nonlinear problems but were largely unapproachable in the past. It is expected that the new approach described in this proposal will allow me to deal with equations with variable coefficients which is nowadays one of the main challenges of the whole area.

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Present methods coming from spectral theory or from harmonic analysis generally fail when dealing with variable coefficients. At the same time the approach that I propose here is very well suited for it. In fact, already for some classes of equations it allowed to recover and improve most of the results that can be obtained with other approaches, and go far beyond!

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Another part of the project is to use all this as well as other recently discovered ideas and techniques to investigate dispersive, Strichartz, and smoothing estimates for dispersive equations with variable coefficients and lower order terms, and relations between them. The obtained results will be applied to local and global well-posedness questions of nonlinear hyperbolic, Schrodinger and other dispersive equations. It is important and challenging research with deep implications in theories of linear and nonlinear dispersive equations and their relation to geometry and other areas.

The research will be undertaken at the Mathematics Department of Imperial College, while collaboration with other mathematicians on some aspects of this project is expected. Phase Space Analysis of Evolution Equations. Is the information for this product incomplete, wrong or inappropriate?

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Send us a new image. Is this product missing categories? Add more categories. Review This Product. Welcome to Loot. In January we have a school for graduate students. It would be nice to include some topics from the conference into it.

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Another useful thing for our graduate students is a visit of some of the participant to UWI and present some results here. I hope that this has shown that in Jamaica we are also doing some serious things in mathematics. It can open the door for more visits to our country and extend our international collaboration. The conference 'Analysis and PDEs' hosted at Imperial College was an excellent opportunity to acquaint ourselves and discuss maths with scientists from around the world whom we otherwise would probably have no occasion to meet.

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Mathematics is very special in that it can be and is being done on a high level in every corner of the world irrespective of social-economic circumstances. We have seen impressive pieces of work accomplished both at leading UK universities and institutions from developing countries not very well known to the general mathematical society.

In fact, this event and the funding scheme behind it inspired me to discuss with an analysis research group in my country the possibility of establishing a collaboration with their colleagues in the UK, and I can report that they were more than interested. Nowadays, the number of scientific papers in mathematics have been growing rapidly and it is almost impossible for an individual researcher to follow all the new publications, even in a branch of mathematics, for example in analysis.

Moreover, particularly developing countries, such as Kazakhstan, have much difficulty accessing to new publications and lectures , which requires certain amount of funding in fundamental sciences. For these reason, exchange of new ideas as well as discussions on the conference have given lot of benefits especially to participants from developing countries. Here I also would like to thank the conference organizers, especially the chairman Prof. Michael Ruzhansky, for this amazing useful event. First I want to thank you for the opportunity to attend this conference.

Phase space analysis of partial differential equations

Initiatives like this make Imperial College to be internationally recognized as a leader and reference point for the world's researchers. I was very happy to attend this conference with researchers from several countries working on really interesting and relevant problems in analysis and PDE.

It was a happy combination of young and renowned researchers, which I would never have the opportunity to meet if it were not this conference. The talks addressed problems in active research lines and gave me a very broad overview of what has been done and what are the major problems of the area. I'm sure this was only a first step towards a stronger collaboration with the Imperial College, which will help us to improve the level of mathematics in my country and cooperate in the internationalization of my University.

This conference encouraged the participants to learn from each other's different experiences and mathematical approaches to a variety of recent problems arising in operator theory and its applications to the study of PDEs. It promoted the scientific information interchange between overseas participants. Several results and problems presented were new for me, and I learned about them and collected relevant bibliographic references.

The conferences were interesting, clear, concise and well-motivated, allowing all participants to be acknowledged with the main mathematical tools to solve the proposed problems. This conference gave me new ideas, mathematical and physical models, and problems to share with colleagues, undergraduate and graduate students in a permanent seminar on evolution PDEs held at the Department of Mathematics, Universidad del Valle, and further with colleagues in other academic institutions in Colombia. In particular, in this weekly seminar we study Sobolev spaces, semigroups , unbounded linear operators, pseudodifferential operators, and nonlinear positive operators defined in cones in a Frechet space, and in a Banach space, together with applications of this theory to the study of existence of solutions of boundary value problems for PDEs.