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EPSRC Reference: EP/C538803/1
Title: Visits to discuss the use of Frobenius skew polynomial ring in tight closure theory, and related topics
Principal Investigator: Sharp, Professor RY
Other Investigators:
Researcher Co-Investigators:
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Department: Pure Mathematics
Organisation: University of Sheffield
Scheme: Mathematics Small Grant PreFEC
Starts: 01 June 2005 Ends: 30 November 2006 Value (£): 5,097
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Panel History:  
Summary on Grant Application Form
It is often the case that an effective way to study objects in algebra is to investigate slighter larger objects, that are 'closed' in some sense. This is the principle behind the theory of tight closure, which was developed in the late 1980s by M. Hochster and C. Huneke, two of the world's leaders in commutative algebra. Their theory associates to many algebraic objects their 'tight closures'; the tight closure can be expected to be a little bigger, in some sense, than the original object, but 'tightly linked' to it. This theory of tight closure has had astonishing success in providing short proofs of major theorems in commutative algebra, and providing routes to approach long-standing unsolved problems. Moreover, there have recently been important applications of tight closure theory to algebraic geometry.However, tight closure theory itself has thrown up its own irritating problems, one of which concerns expected 'good behaviour' of a 'closure' operation. We would like very much to know that it does not matter whether we perform this 'tight closure' operation before or after another absolutely standard operation in commutative algebra, called 'fraction formation', or 'localization'. (The latter operation is concerned with the introduction of denominators, in a similar but slightly more complicated way to that in which rational numbers ('fractions') are produced from the integers ('whole numbers').) The fact that no-one has so far managed to settle this question is referred to as 'the localization problem in tight closure'. The development of the theory has had to be more complicated than it would have been if this problem had been solved (with the hoped-for outcome) in the early 1990s. This localization problem is still unsolved.The research to which this grant application applies concerns cross-fertilization of ideas from 'non-commutative' algebra into this theory of tight closure, with the principal objective being the discovery of new information about the difficulties surrounding the localization problem in tight closure. Many of the objects studied in tight closure theory can be considered, quite naturally, as operated upon by a large structure that has a 'twist' in its multiplication (and this explains the epithet 'non-commutative'). A few mathematicians have pursued this; the Principal Investigator had some success with this approach in 2002 and 2003, in research partially supported by an EPSRC Overseas Travel Grant (the reference number is given in the attached Case for Support). Now the PI wishes to build on that success during Study Leave (from October 2004 until 11 June 2005) following a period of six years as Head of the Department of Pure Mathematics at Sheffield University, when the heavy administrative load associated with that post provided him with only limited opportunities for research. The Overseas Travel Grant requested in this application would enable the PI to make one short visit to the US in April 2005 to attend a lecture series entitled 'Recent developments in tight closure' that would clearly benefit this research, and three further visits in summer 2005, all following the end of his Study Leave, that would give good opportunities for discussion of his ideas and results, and possible directions for his future research, with experts in tight closure theory and related topics.
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Organisation Website: http://www.shef.ac.uk