| EPSRC Reference: |
EP/M000079/1 |
| Title: |
Microlocal analysis of first order systems of partial differential equations |
| Principal Investigator: |
Vassiliev, Professor D |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
UCL |
| Scheme: |
Standard Research |
| Starts: |
30 November 2014 |
Ends: |
24 March 2018 |
Value (£): |
364,832
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| EPSRC Research Topic Classifications: |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
| Panel Date | Panel Name | Outcome |
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11 Jun 2014
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EPSRC Mathematics Prioritisation Meeting June 2014
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Announced
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Summary on Grant Application Form |
Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space: if one moves along a straight line (geodesic) in this universe one eventually comes back to the starting point. And suppose that there is only one particle living in the universe, a massless neutrino. The study of the behaviour of this single particle is a typical example of the class of mathematical problems that we propose to examine.
It may seem that there is little practical value in studying the highly idealised situation described above. However, it may in the long run help us understand the internal structure of the neutrino, tell us why neutrinos propagate with the speed of light and why they carry spin, explain the difference between the neutrino and antineutrino, give us a clearer picture of how neutrinos interact with gravity etc.
The massless neutrino is described mathematically by a system of two equations which, taken together, are called "massless Dirac equation". The massless Dirac equation is the accepted mathematical model for the neutrino and the corresponding antiparticle, antineutrino. This equation models the neutrino as a wave running through our universe. The aim of the research project is to perform a comprehensive analysis of a class of systems of equations, of which the massless Dirac equation is a characteristic representative.
The main difficulty here is that our wave-like solution has a complicated structure because the universe over which this wave propagates is curved. The plan is to tackle this difficulty by means of a mathematical technique called "microlocal analysis". Microlocal analysis is a powerful mathematical technique developed by Lars Hormander (Fields Medal 1962). Though the subject of microlocal analysis is highly technical (Hormander's book runs into 4 volumes), the underlying idea is quite straightforward: we assume, in the first approximation, that our universe is flat, construct our wave-like solution explicitly and then correct it by taking account of the curvature of our universe.
Microlocal analysis has been applied with great success to the study of scalar (single) equations. As explained above, the massless Dirac equation is actually a system of two equations and the fact that it is a system makes the structure of the wave-like solution particularly complicated and its study fascinating.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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