# EP/H028803/1 - New approaches for isospectrality and nodal domains

Research Perspectives grant details from EPSRC portfolio

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Research Perspectives grant details from EPSRC portfolio

Principal Investigator - Mathematics, University of Bristol

Start Date

08/2010

End Date

08/2013

Value

£218,282

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Summary and Description of the grant

The physical systems around us present the challenge of inverse problems. Very often we wish to determine the properties of a physical object, but we are not capable of measuring them directly. This raises the question of what is possible to learn about a system, looking through the glasses that nature supplies. Inverse problems, appearing already in the Allegory of the cave by Plato, affect everyday life, since their solutions allow medical imaging, oil and gas exploration, land-mine detection, navigation and much more. This is the perspective of my research which concerns two main themes: isospectrality and nodal domains. For physicists and mathematicians, a drum is a solid planar frame to which an elastic membrane is attached. The spectrum of frequencies which can be produced by such a drum when it vibrates depends on the shape of the frame. In 1966, Marc Kac asked his famous question 'Can one hear the shape of a drum?', namely, can one deduce the shape of the frame from the knowledge of the spectrum of vibrations. Since Kac asked his question much research was invested in trying to find two systems, in particular two different planar drums, which have the same spectrum - an isospectral pair. The existence of such a pair would show that Kac's question has to be answered by a simple no . The first example was discovered in 1992 by the mathematicians Gordon, Webb and Wolpert. They came up with two drums that have equal areas and perimeters but different geometric shapes. One aspect of my research is developing tools which enable to construct isospectral pairs.The existence of isospectral examples leads to the next natural question: 'Can we find other observable properties related to the vibration of the drum that can distinguish the isospectral drums?'This can be answered by the next theme of my research, nodal domains. Nodal domains were first presented in full glory by Chladni's Sound figures. By the end of the 18th century Chladni was performing the following demonstration: he spread sand on a brass plate and stroke it with a violin bow. This caused the sand to accumulate in intricate patterns of nodal lines - the lines where the vibration amplitude vanishes. The areas bounded by the nodal lines are called nodal domains. The impact of the nodal domains research on present day science is immense and there are many physical systems for which the study of nodal domains and nodal lines form an important role. Few such examples are density fluctuations of the cosmic matter, earthquakes, turbulent flows, microwave cavities and models of the visual cortex. A specific question which I currently investigate concerns the information that one can deduce about a system just from counting the numbers of its nodal domains.

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Grant Event Details:

Name: New approaches for isospectrality and nodal domains - EP/H028803/1

Start Date: 2010-08-01T00:00:00+00:00

End Date: 2013-08-09T00:00:00+00:00

Organization: University of Bristol

Description: The physical systems around us present the challenge of inverse problems. Very often we wish to determine the properties of a physical object, but we are not capable of measuring them directly. This raises the question of what is possible to learn about a ...