The theory of webs was initiated by Blaschke and Bol in the 1930s. In its most basic form, it looks at the local analytic behavior of n intersecting foliations of complex 2-space by families of curves. For instance, you might fix three points and draw all the lines through each of them, or fix an algebraic curve and draw all the tangents to it. Then you look at the resulting configuration far from the curve or triple of points. Web geometry turns out to have numerous applications to differential equations, algebraic geometry and even physics.
Around the turn of the millenium, a group of French mathematicians made the very exciting discovery that so-called exceptional webs were intimately related to functional equations of polylogarithms. These are the functions you get by replacing the “k” in the denominator of the power-series expansion for the logarithm, by some power of k. Recent developments in algebraic K-theory have turned them from a curiosity into a major industry. (Capitalizing the k made it look more important.) One thing that, with some help, an undergraduate student might be able to do, is come up with a more algebro-geometric description of the (exceptional) Bol 5-web than I have seen in the literature.
There are related functions called Grassmanian polylogarithms, invented by A. Goncharov, which enjoy relatively simple functional equations. To try to relate these to webs, or to find a new (more geometric) approach to their functional equation, would also be interesting and potentially do-able.
One of the great collaborative success stories of the past two decades has been that between complex algebraic geometers and string theorists in the mirror symmetry program. The quest to produce Calabi-Yau 3-manifolds (three complex dimensions!) required mathematicians and physicists to confront the kinds of singularities – local failure of manifold structure – that arise from quotienting complex 3-space by a finite group action. Moreover, they had to figure out how to resolve them – the higher-dimensional analogue of lifting an (actual) string off itself.
While this story has only recently been thoroughly understood, it would be well within the powers of an interested undergraduate student to provide a down-to-earth account with basic examples. This is something I have not seen in the literature, and shouldn’t be thought of as an expository project – it would require some original thought. It would also acquaint you with toric geometry, an extremely useful tool which gives a dictionary between algebro-geometric concepts and the geometry of convex bodies (like polygons and polytopes) considered relative to a lattice. In working out examples, the latter boils down to some surprisingly entertaining 3-dimensional linear algebra which ultimately tells you how to draw a triangulation.
Mirror symmetry comes into this story in a number of ways. In one version, the resolutions of singularities you will construct are “mirror” to certain families of Riemann surfaces. An ambitious student might want to investigate this too.
(I will also accept non-amoebas as students on this project.) The classical theorems of Abel and Jacobi describe the divisors (configurations of zeroes and poles with multiplicity) of meromorphic functions on compact Riemann surfaces. Attempts to generalize these results to noncompact or singular settings, as well as to higher dimension, have motivated a lot of modern algebraic and differential geometry – like the Bloch-Beilinson and Hodge conjectures and the theory of webs. I don’t know of a good write-up of the one-dimensional generalizations, and you could already learn a lot by trying to trying to understand the situation for unions of lines, or for multiply connected regions.
In algebraic geometry, roughly speaking, we study solution sets of algebraic equations. Replace everywhere multiplication by addition and addition by “taking the maximum”, and you have an exciting new theory called tropical geometry – which even has its own version of Abel’s theorem! Amoebas are objects which provide a connection (via a limiting process) between algebraic curves and tropical curves, and it would be extremely interesting (though not necessary for an interesting project) to devise a connection whereby one Abel’s theorem becomes the limit of another.
For this project, all I really ask is that a student be familiar with basic complex analysis. It would also be useful to know what a Riemann surface is, but this could be dealt with in summer reading.
I would be happy to direct a reading course and subsequent write-up as well, on any of the following topics (or on appropriate student-proposed topics).