| EPSRC Reference: |
EP/D054214/1 |
| Title: |
Noncommutative geometry of groups and spaces. |
| Principal Investigator: |
Brodzki, Professor J |
| Other Investigators: |
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| Department: |
School of Mathematics |
| Organisation: |
University of Southampton |
| Scheme: |
Mathematics Small Grant PreFEC |
| Starts: |
11 November 2005 |
Ends: |
10 April 2006 |
Value (£): |
7,800
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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Analysis is interested in how spaces look on a small scale. Imagine that you try to find your way down a rocky mountain ridge in complete darkness: to do this, you would need to discern by feel the shape of the terain in a small (depending how far you can reach) disc around you. Using this local information, you would then need to imagine a likely shape of the ridge and then decide which is the safe direction to follow. Groups, unlike mountain ranges, do not have any shape, it seems. A group is simply a set of elements with a rule on how to multiply any two of them. It has no more shape than a cloud of dust. However, there is a very natural way of measuring the distance between any two points in the group. Typically, any element in a (disrete) group can be obtained by multiplying out (in a certain way) a small number of chosen elements, called the generators. This is just like writing words in a language with an alphabet: any word in the English language can be written using a selection from a fixed set of twenty six letters. Just like words, which have diferent lenghts depending on how many letters are needed to write them, elements of a group can be given a length, which depends on the number of generators one needs to multiply to get a particular group element. This notion of length immediately allows one to define a metric on the group, that is a way of measuring distances between group elements. This gives shape to discrete groups. However, compared to spaces encountered in analysis, where points tend to be `densely packed', points in groups seem to be fairly isolated from one another. A very ingenious way to get around this problem has been proposed Gromov. Imagine that you watch the sky on a clear night: you would see the stars as single points; locally they might trace out some recognizable figures (as was noticed by the ancients who gave names to the Zodiac signs), but there would be no global structure to what you see, no overal shape that contains all the stars. Imagine now that you move back by a huge distance; single stars will no longer be visible, but all of a sudden you will see shapes: for instance, you would be able to see that the stars in our sky form a beautiful spiral galaxy. This galaxy will have a definite structure and will resemble spaces that can be studied by analytic means. This is how a contact can be made between analysis and group theory. This approach has been very successful in the past, and now there is a thriving area of mathematics, called noncommutative geometry, centered around these ideas. This proposal will use methods from analysis and geometry to study geometric properties of spaces that emerge from the study of groups. Some of the spaces of interest to us arise in mathematical physics. There one is interested in describing what a particular physical system (e.g. gas in a box) `can do', for instance by measuring all possible positions and velocities of particles that comprise it. This space of permitted configurations of the system can have a very complicated structure (this happens for instance in string theory, which studies the elementary forces that hold all matter together). Noncommutative geometry is perfectly suited to the study and understanding such complicated objects.
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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 |
| Summary |
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| Date Materialised |
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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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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.soton.ac.uk |