Icosahedral Fibres of the Symmetric Cube and Algebraicity

For any number field F, call a cusp form π = π_∞⊗πf on GL(2)/F special icosahedral, or just s-icosahedral for short, if π is not solvable polyhedral, and for a suitable “conjugate” cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of π equals the Rankin-Selberg L-function L(s, sym^2(π') × π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let π be s-icosahedral (of trivial central character). Then π f is algebraic without local components of Steinberg type, π ∞ is of Galois type, and π_v is tempered every-where. Moreover, if π' is also of trivial central character, it is s-icosahedral, and the field of rationality Q(πf) (of πf) is K := Q[√5], with π' _f being the Galois conjugate of πf under the non-trivial automorphism of K. There is an analogue in the case of non-trivial central character ω, with the conclusion that π is algebraic when ω is, and when ω has finite order, Q(πf) is contained in a cyclotomic field

A categorification of the Jones polynomial

We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.Comment: latex, 51 pages, 72 eps figures, to appear in Duke Mathematical Journa

Orbit Parametrizations for K3 Surfaces

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose N\'eron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm

Groups of PL homeomorphisms of cubes

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted.Comment: 23 pages, 3 figure