Fourth Year Projects

Room 802a, Department of Mathematics,

UCL, Gower Street, London, WC1E 6BT

j.d.evans • ucl.ac.uk

- My advice on mathematical reading.
- My suggestions for good stuff to read.

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

- Hypersurface singularities.
- Small exotic 4-manifolds.
- 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 H_{1},
H_{2}, ... H_{k}, one can look at the image of these
Hamiltonians as a subset of **R**^{k}. 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:

- multivariable calculus,
- complex analysis,
- algebraic topology.

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.

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:

**I expect you to go and read not only the things I tell you to go and read, but to follow up the references and explore the subject for yourself.**I recognise that this is difficult because it can take a long time to read mathematics at all, but it is the only way you will make the project your own and achieve the highest marks.**I expect you to think hard about questions before asking me.**It's very tempting to get your supervisor to explain every little detail, but you will learn more if you go through the pain of thinking before asking.**Of course, I also expect you to ask me questions!**Even ones which you think are silly (they're often the most important ones).**I expect to see you for about an hour a week during termtime to check you're on track and hear about what you've been reading.****I expect you to start reading (and emailing me with questions) quite intensively over the summer**: that will give you about nine months to do your project. This early stage is important because there will be a lot of background for you to fill in and it is best to get this secure as soon as possible. Bear in mind that much of what you learn will also be useful in preparing you for fourth year courses (in particular in geometry and topology). The guidelines say you should spend approximately 200-250 hours on your project: that's probably an underestimate and the holidays are an excellent chance to make up for time you will not have during term.**I expect you to try to**Gleaning understanding from a mathematical proof is much harder than following it step-by-step - it helps to work out the proof in simple cases and to try explaining the proof to other people (e.g. me or your fellow fourth year students).*understand*what you read, not just to*follow*it.**See my advice on mathematical reading.****I expect you to write up as you go along.**Leaving writing till the last minute will cause everybody stress and pain. Writing as you go will mean that you (a) remember stuff five months down the line, (b) learn LaTeX more effectively, (c) will get to February and be able to piece together your project in a leisurely way from the material you've written already instead of writing up in a mad panic.**I expect your project to be your own take on the material you are reading**, illustrated with examples you have worked out and proofs you have explained in your own words, or even adapted and simplified: I am more interested in your ability to explain and show understanding of a proof in a simple case or a particular example than just copying out a chunk of textbook.**I expect you to reference properly.**Your references should (a) give historical context to the mathematics you're describing, (b) point to sources containing proofs you have omitted (because they were too long or tangential to the point you were trying to make) and (c) acknowledge where you found the ideas for the proofs you have included.

- 2016-2017 Christopher Evans, "Quotients in algebraic and symplectic geometry"
- 2015-2016 Brunella Torricelli (visiting from ETH
Zürich), "A survey on the Lagrangian SU(n)/(
**Z**/n) in**CP**^{n2-1}" - 2015-2016 Cheng Zhi Lim, "Morse theory"
- 2014-2015 Adam Paxton, "The de Rham cohomology, cobordism and characteristic classes"
- 2013-2014 Grace Oswin, "Mapping class groups of surfaces"

Last updated 23rd March 2017.