| EPSRC Reference: |
EP/D037808/1 |
| Title: |
WORKSHOP: Quadratic Forms, Linear Algebraic Groups and Related Topics, 9-16 September 2005 |
| Principal Investigator: |
Hoffmann, Professor DW |
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| Department: |
Sch of Mathematical Sciences |
| Organisation: |
University of Nottingham |
| Scheme: |
Mathematics Small Grant PreFEC |
| Starts: |
13 September 2005 |
Ends: |
12 December 2005 |
Value (£): |
5,000
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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We all learn in school how to work with fractions: they can be added,subtracted, multiplied and one also can divide by a nonzero fraction. Fractions are an example of an algebraic object: we have elements which can be manipulated and composed in one or several ways following certain laws. Another example of an algebraic object would be the rotations of a wheel. One can rotate a wheel by a certain angle, perform several rotations by certain angles in succession and get again a rotation, one can undo a rotation by rotating the wheel by the same angle but in the opposite direction. A rotation by 360 degrees has the same effect as not rotating the wheel at all. Here, the elements of our algebraic object are rotations of the wheel, and composing two rotations just means performing one after the other. As an algebraic object, it doesn't matter whether the wheel is big or small or black or white. If we know how rotations work for one wheel, we know how they work for all wheels. In other words, we can identify the rotations of wheel A with the rotations of wheel B in an obvious way, and the laws of composition are preserved under this identification. In mathematical language, we have an isomorphism between two different algebraic objects (rotations of wheel A and rotations of wheel B). An isomorphism between algebraic objects is a bit like a perfect dictionary between two languages: It not only tells us unambiguously how one word from one language translates into a word from the other language, but it also tells us how the grammar in one language translates into the grammar of the other language. Algebra is the study of algebraic objects, and an important question is to find criteria that tell us when one object is isomorphic to another one. In 1970, an American mathematician, John Milnor, asked whether three particular algebraic objects which look quite different on the surface and which have their origins in different branches of algebra are in fact all isomorphic to each other. This has become known as the Milnor Conjecture and it has baffled algebraists ever since, and only partial results towards a solution could be shown until 1996, when a Russian mathematician, Vladimir Voevodsky, showed that these three algebraic objects are indeed isomorphic to each other by developing new powerful and very sophisticated mathematical tools often based on methods from other seemingly unrelated areas of mathematics like geometry. In 2002, Voevodsky received the Fields Medal, the most prestigious prize for mathematical achievement (incidentally, Milnor himself was a Fields Medal laureate in 1962).The topics of this workshop concern certain algebraic objects called quadratic forms and linear algebraic groups, and related ones. A simple example of a quadratic form is a map which takes, say, two real numbers, squares them and then adds them up (this particular quadratic form would then be called a 2-dimensional quadratic form over the real numbers). Quadratic forms are in a certain sense the building blocks of one of the three objects in the Milnor Conjecture, and their study has profited greatly from the new techniques which Voevodsky helped to develop. The algebraic object given by rotations of a wheel is a simple example of a linear algebraic group and it can be identified with an algebraic object attached to our quadratic form from before which is called the special orthogonal group of this quadratic form. At first glance it might not seem very obvious, but the theory of quadratic forms and that of linear algebraic groups are linked in many ways. In this workshop, some of the foremost experts in the study of these objects will come together to tell about the most recent advances of these theories and give an outlook on possible future developments. Many talented young researchers work in these areas of mathematics, and they will also be given an opportunity to present their contributions to these theories.
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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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| 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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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.nottingham.ac.uk |