### MMath Undergraduate Projects in Algebraic Geometry

For rules and further details of MA4K9 MMath Project, see

UGHandbook
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.