Details of Grant 

EPSRC Reference:	EP/F043570/1
Title:	Topological methods in algebraic geometry
Principal Investigator:	Pridham, Dr JP
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Pure Maths and Mathematical Statistics
Organisation:	University of Cambridge
Scheme:	Postdoc Research Fellowship
Starts:	01 October 2008	Ends:	30 March 2011	Value (£):	197,159
EPSRC Research Topic Classifications:	Algebra & Geometry
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 InvitedForInterview
Summary on Grant Application Form
In homotopy theory, topological spaces (i.e. shapes) are regarded as being the same if we can deform continuously from one to the other. Algebraic varieties are spaces defined by polynomial equations; studying their homotopy theory means trying to tell which topological spaces can be deformed continuously to get algebraic varieties, or when a continuous map between algebraic varieties can be continuously deformed to a map defined by polynomials.If the polynomials defining a variety are rational numbers (i.e. fractions), this automatically gives the variety a group of symmetries, called the Galois group. Although these symmetries are not continuous, they behave well enough to preserve many of the topological features of the variety. Much of my work involves investigating how the Galois group interacts with the topology. I also study algebraic varieties in finite characteristics. These latter are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. For instance, in characteristic 3 the equation 1+1+1=0 holds. Topology and geometry of varieties still make sense in finite characteristics, where we gain much valuable insight into the behaviour of the Galois group.Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry. To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can have symmetries and so on, giving rise to infinity stacks.Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities (points where the space is not smooth), and regions of different dimensions. Partially inspired by ideas from theoretical physics, it has been conjectured that moduli stacks should be extended to derived moduli stacks, which have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness''. Different classification problems can have the same moduli stack, but different derived moduli stacks. Part of my work will be to try to construct derived moduli stacks for a large class of problems.
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Organisation Website:	http://www.cam.ac.uk