Fourth Year Projects
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Room 802a, Department of Mathematics,
UCL, Gower Street, London, WC1E 6BT
j.d.evans •


Useful links:

What's on offer?

I will take on at most one student for 2017–2018.

  1. Hypersurface singularities.
  2. Small exotic 4-manifolds.
  3. Convexity and commuting Hamiltonians.

1. Consider the family of plane curves xy=c (as c varies this gives you a family of hyperbolae). If you work over the complex numbers, each hyperbola is (topologically) an annulus except the curve xy=0, which is a pair of lines that intersect at a point. We say that this special hyperbola is singular. You can imagine that the singular fibre is obtained from the nonsingular fibres by crushing a circle in the annulus down to a point. If you look at more complicated polynomials p(x,y,...)=0 which cut out singular spaces, then, in the smoothing, you will always find a configuration of spheres called vanishing cycles which get crushed down to the singularity. The purpose of this project would be to understand Milnor's proof that the smoothing retracts onto the vanishing cycles, and to study some explicit polynomials to see what kinds of vanishing cycle configurations you get. In particular, there is a nice classification of "simple singularities" and their vanishing cycle configurations are the ADE Dynkin diagrams which occur in the classification of semisimple Lie groups/Platonic solids.

2. One of the biggest open problems in topology is the 4-dimensional smooth Poincaré conjecture: the question of whether there is a simply-connected 4-manifold with the cohomology of the 4-sphere but which isn't actually diffeomorphic to the 4-sphere. Although we don't know any counterexamples, there are many examples of "small" manifolds (i.e. with low second Betti number) which have "exotic" cousins (i.e. nondiffeomorphic simply-connected manifolds with the same cohomology ring). The goal of this project would be to understand some of these examples. As a start, you could learn about the Kirby calculus (a way of describing any 4-manifold just by drawing pictures of knots) and later you could learn about some of the more advanced tools like Seiberg-Witten invariants which are used to distinguish nondiffeomorphic 4-manifolds where the classical invariants like homology are not enough to tell them apart.

3. In Hamilton's formulation of classical dynamics, the evolution of a system is specified by a function called the "Hamiltonian" of the system. Two Hamiltonians F and G commute if you evolve the system using F and then G then you have the same effect as evolving using G and then F. Given a system of commuting Hamiltonians H1, H2, ... Hk, one can look at the image of these Hamiltonians as a subset of Rk. It turns out this is always a convex polytope! This is responsible for some classical theorems on eigenvalues of matrices (Schur-Horn convexity) and gives a very fruitful link between classical mechanics and combinatorial algebraic geometry (toric geometry). The proof of the convexity theorem is a very pretty application of Morse theory and the aim of this project would be to understand this proof.

These projects will be best enjoyed if you have some familiarity with the following courses:

On a related note, I would advise anyone doing a project with me to take a subset of {Topology and Groups, Lie Groups and Lie Algebras, Algebraic Geometry, Riemannian Geometry}. It's likely that these will be very relevant for your project.

My expectations

The purpose of doing this project is partly to learn some beautiful mathematics and partly to develop your independence and persistence as a learner and researcher, skills which will be useful whether you end up a mathematician or not. So here is what I expect of you if you take a project with me:

Past/current projects

Last updated 23rd March 2017.