MMath Undergraduate Projects in Algebraic Geometry

For rules and further details of MA469 MMath Project and
MA497 Project Extension, see Mauve PYDC

I have a variety of possible projects on offer related to modern
research in algebraic geometry, and to MA4A5 Algebraic Geometry
or MA426 Elliptic 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.