Plato's Cave: What we still don't know about linear projections in Algebraic Geometry

In the nineteenth century people studied algebraic curves by projecting them into the plane. It was important to know that the generic projection of a smooth curve had only ordinary nodes. This result generalizes easily to surfaces, and a little further, but there are many mysteries for in higher dimension. I'll survey what's (not) known and what's conjectured, and explain some of my recent joint work with Joe Harris and Roya Behesht on the problems raised.

Quotients of algebraic varieties by group actions

Many moduli spaces in algebraic geometry arise naturally as quotients of linear algebraic group actions. Mumford's geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of reductive groups on projective varieties. This talk will review the main ideas of GIT, discuss some ways to extend them to actions of linear algebraic groups which are not necessarily reductive, and (if time permits) describe an application of non-reductive GIT to the study of entire holomorphic curves in projective hypersurfaces, using Demailly's theory of jet differentials.

The combinatorics of solving linear equations

A major branch of modern combinatorics, usually called Ramsey theory, studies properties of structures which are preserved under partitions. Its guiding philosophy can be neatly summarized by the statement, ``Complete disorder is impossible". In this talk I will survey what is known and what is still unknown from this perspective for solution sets of linear equations over the integers.

11:45 - 12:45 *Kirwan Lecture Theatre*

Approximate structure in additive combinatorics

I will introduce notions of approximate group, approximate homomorphism and approximate polynomial. Much of the subject of additive combinatorics can be reduced to the study of such objects, and in particular to the question of how much they resemble more ``rigid'' algebraic objects. The aim of this talk will be to say a little about what is known and what is conjectured, and about what this has to do with counting arithmetic progressions of prime numbers. Strenuous efforts will be made to keep the talk accessible to a general mathematical audience.

Hanoi Tower game and self-similar groups

We are going to explain the relation between the famous combinatorial problem known as the Hanoi Towers Game on k>2 pegs and group theory. The corresponding groups H^k are called Hanoi Tower groups and happen to be self-similar (weakly) branch groups. Moreover, H^3 and some of its subgroups serve as the iterated monodromy groups of some rational functions of degree 3 studied by Julia and Devaney. The groups H^k posses interesting algebraic and asymptotic properties. At the same time the associated Schreier graphs also have interesting combinatorial and asymptotic properties some of which (including spectral) will be discussed at the end of the talk.