I have a variety of possible projects on offer related to modern
research in algebraic geometry, and to
MA4A5 Algebraic
Geometry
or MA4L7
Algebraic Curves.
I can also propose problems related to advanced topics in Galois theory,
commutative algebra and algebraic number theory for suitably motivated
students.
Finite subgroups of SL(2, CC) and
SL(3, CC)
Explicit generators, invariant theory. Work of Klein around 1870, but
recently developed in many directions. Start from my preliminary
chapter on
cyclic
quotient singularities. See for example my
Bourbaki
seminar or the references given on my McKay
correspondence
website.
Quaternions, octonions, special
geometric structures and
exceptional Lie groups
See any introductory text on quaternions (for example, Balazs Szendroi
and Miles Reid, Geometry and Topology, Chapter 8), followed by John
C Baez, The
Octonions, 56 pages, preprint available from uk.arXiv.org
as math.RA/0105155,
and John H. Conway and Derek A. Smith, On
quaternions and octonions: their geometry, arithmetic, and symmetry,
A K Peters, Ltd. 2003 IBSN 1568811349
(GBP25:50 from www.amazon.co.uk)
Topology, projective geometry and
enumerative geometry of
Grassmann varieties
and projective homogeneous spaces. (See me).
Applications of Riemann Roch on curves
and surfaces
Graded rings, computer algebra calculations. Computer enumeration
of graded rings of interest in algebraic geometry. Work of mine and
my
students. It contains lots of fairly simple minded
combinatorial problems
that lend themselves to computing. Start from my
preliminary chapter
on
graded
rings, then look at Gavin Brown's
GRDB website.