Details of Grant 

EPSRC Reference:	EP/C013271/1
Title:	Coherent Sheaves in Mirror Symmetry
Principal Investigator:	Teleman, Professor C
Other Investigators:	Corti, Professor A
Researcher Co-Investigators:
Project Partners:
Department:	Pure Maths and Mathematical Statistics
Organisation:	University of Cambridge
Scheme:	Mathematics Small Grant PreFEC
Starts:	01 May 2005	Ends:	31 July 2005	Value (£):	9,934
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
This project aims to bring together researchers in different areas of maths to work on problems arising from Mirror Symmetry. Mirror Symmetry is a mathematical theory started by physicists working on strings. The idea of string theory is that we should think of an elementary particle as a tiny closed string rather than a point-like object.Quantities of physical interest in string theory can be computed using two different mathematical theories. The reason is that the physical *quantum* theory has more symmetry than is visible in the space where the string can move; from the physical perspective, this space describes a *classical* theory. In one picture, one is led to enumerate one-dimensional solutions ofalgebraic equations over the complex numbers. In the other picture, one is led to solve differential equations.Teleman will study a very special case in which MS is computationally more tractable because of the large amount of symmetry involved. The building blocks for the underlying spaces, called Lie groups, are homogeneous, in the sense that the geometry at all their points looks the same. They have been much studied in mathematics, sincethey themselves describe the symmetries of other geometric objects;and it turns out that the mirror symmetry is related here to another symmetry for Lie groups (known to mathematicians as Langlands duality)although the precise formulation of the connection is still unclear.Corti is interested in the classification of the possible structure of the geometry of spacetime. In some models the spacetime is a mathematical object called a 3-dimensional Fano variety. One of the proposed visitors, Golyshev, has used the two pictures described above to establish a correspondence between Fano varieties and modular forms (a mathematical concept coming from number theory). This result is surprising because it relates concepts coming from different parts of mathematics. Corti and Golyshev will use this opportunity to work on a joint project to generalize this correspondence to more general kinds of Fano varieties.in different areas
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Organisation Website:	http://www.cam.ac.uk