Polynomial automorphisms over finite fields
Let k be a field. We want to find a good description of the group
of polynomial automorphisms of kn, Autn(k).
If n = 1, this is trivial, because the only polynomial automorphisms are the affine maps. For n = 2 Jung in 1942 showed that if char(k) = 0, then the automorphism group is generated by the tame maps. In 1953 Van der Kulk generalized this to any field. For n ≥ 3 it is still open, but in 2004 Shestakov and Umirbaev showed that in case n = 3,
N = (x - 2(xz + y2 )y - (xz + y2)2 z, y + (xz + y2 )z, z)
Nagata's map (1972) in char(k) = 0 is not tame. If char(k) = 0 the problem is still open for n ≥ 3. This talk will be about some results and some open problems in the case where k = Fq a finite field.
Topological quantum computers
It is well-known that quantum computers can perform certain quantum algorithms, whose (algorithmic) complexity is lower than their classical counterparts. Freedman and Kitaev independently proposed the concept of a topological quantum computer. Their proposals try to exploit topological properties to tackle an important problem with quantum computers: the computations are influenced by interactions with the environment. This is called decoherence. Even though the proposals are different, the underlying mathematical structure is the same. Namely, of central importance is the concept of a modular category, a certain type of braided tensor category. In physics terms, what is important is the existence of a certain type of (quasi)particles called anyons.
Despite the fact that there is considerable interest in systems with anyons in two dimensions, not much work has been done on their mathematically rigorous study. In this talk, I will outline the essential mathematical features of topological quantum computers. After this introduction, I will discuss our approach to a rigorous description of systems with the required properties. This approach is influenced by results in algebraic quantum field theory, a theory that has been developed in close interaction with the theory of operator algebras. Besides the theory of modular categories, this is the mathematical setting of the project.