Details of Grant 

EPSRC Reference:	EP/F043074/1
Title:	Some Mathematical Aspects of the Calabi-Yau/Landau-Ginzburg Correspondence
Principal Investigator:	Segal, Dr EP
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	Imperial College London
Scheme:	Postdoc Research Fellowship
Starts:	01 July 2008	Ends:	30 June 2011	Value (£):	217,190
EPSRC Research Topic Classifications:	Algebra & Geometry Mathematical Physics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 13 Mar 2008 Mathematics Postdoctoral Interview Panel Announced 14 Feb 2008 Maths Postdoctoral Fellowships 2008 FinalDecisionYetToBeMade
Summary on Grant Application Form
String theory is an attempt to reconcile the two great theories of the early twentieth century physics, general relativity and quantum mechanics. It is based on the intuition that we should think of fundamental objects like electrons as being like little pieces of vibrating string, instead of the more traditional picture of little round balls or particles. Unfortunately the theory can only be made to work if we assume that the universe is ten-dimensional, and since we only observe four dimensions (three space plus time) physicists explain away the remaining six by assuming they are tightly curled up too small for us to see. If you only have one dimension to curl up then it's simple - you get a circle. If you have two dimensions however then there are lots of possibilities - a sphere and a torus (the surface of a ring doughnut) are two examples - and by the time you get to six dimensions it gets really complicated. This makes the physics very difficult, because you get different results depending on which particular six-dimensional space you choose to work with. This is where pure mathematics becomes useful, as mathematicians working over the past hundred years have developed lots of techniques for studying curved spaces in higher dimensions. Thus an important general goal is to take physical ideas and find mathematical tools (or if necessary, construct new ones) that correspond to them. This is very difficult, because string theory belongs to a branch of physics, called quantum field theory, which is still very poorly understood by mathematicians. However, in cases where it has been done it has proved amazingly useful, particularly to the mathematicians. This is because the physicists, using intuitions and calculations that mathematicians don't yet understand, can make predictions about the properties of these spaces that mathematicians would never have guessed. Many of these predictions have later been proved correct by other methods. The most interesting and impressive such tricks involve 'dualities', where the physics on two apparently unconnected spaces can be shown to be the same, which means that lots of deep mathematical properties of the spaces must also be the same. Sometimes one can even take the physics from one space and then change it continuously until it turns into the physics from another space, even though the two spaces are completely different and can't be continuously deformed into each other. My research is about the kind of strings which are 'open', which means they have end-points (as opposed to 'closed' strings which are loops). These end-points aren't allowed to roam freely in space, they have to stay on physical objects called 'branes'. When you consider all the branes in a space and all the strings between them (and then simplify things a bit) you get a mathematical object called a 'dg-category'. My aim is to understand examples of these dg-categories and the effects of the physical dualities that link them.
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