Details of Grant 

EPSRC Reference:	EP/L013037/1
Title:	Geometry and Invariant Theory in Modular Lie Theory
Principal Investigator:	Tange, Dr R
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	University of Leeds
Scheme:	First Grant - Revised 2009
Starts:	17 October 2014	Ends:	16 October 2016	Value (£):	93,256
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 27 Nov 2013 Mathematics Prioritisation Panel Meeting Nov 2013 Announced
Summary on Grant Application Form
Lie Theory is a branch of pure mathematics which has its roots in the work of the Norwegian mathematician Sophus Lie. He invented the infinitesimal analogue of the notion of a group: the notion of a Lie algebra. In fact there is a class of groups to which we can canonically associate such a Lie algebra, these are called Lie groups. Basically, these are groups endowed with the structure of a differentiable manifold which is compatible with the group structure. If one replaces "differentiable manifold" by "algebraic variety", then one obtains the notion of an algebraic group. This notion makes sense over an algebraically closed field of arbitrary characteristic. Lie theory can be described as the area of mathematics which studies Lie groups, algebraic groups, Lie algebras and their actions. These actions reveal something of the structure of the acting Lie algebra, Lie group or algebraic group, but also something of the structure of the object on which they act. In the latter case one can say that the action is a mathematically precise way to take the symmetry of the object into account. Lie theory has established itself as one of the most central branches of mathematics with strong links to mathematical physics, differential equations, representation theory, ring theory, algebraic and differential geometry, combinatorics and number theory. It is a very active area of research to which many big mathematicians have contributed. This proposal focuses on certain problems concerning actions of reductive groups, mainly over fields of prime characteristic (that is what the word "modular" in the title refers to). Reductive groups are a very important class of algebraic groups. They have been classified by means of root systems. The most basic example which is also very important in this proposal is the general linear group GL_n which consists of the invertible nxn matrices. Modular Lie theory has its own intrinsic beauty, but is also important because of its relation with ordinary Lie theory via reduction mod p. Several results in characteristic 0 are proved via reduction mod p. It is also worth noting that fields of prime characteristic have important applications in coding theory and cryptography and that they are easier for the computer to handle. One of the biggest problems in modular Lie theory is Lusztig's Conjecture which predicts what the irreducible constituents are of the linear actions of reductive groups. There are many other related fundamental problems, some of which are addressed in the proposal. One of the main ideas of the proposal is that in the presence of problems like Lusztig's Conjecture many related typical characteristic p phenomena should be studied, especially the most elementary ones. In the proposed research I plan to investigate three problems which should give new insights in the the geometry and invariant theory and also in the representation theory of reductive groups in prime characteristic.
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Organisation Website:	http://www.leeds.ac.uk