In this talk, we explore a family of polynomials whose roots are related to the Mandelbrot set. These roots correspond to the $k$-periodic points of the iteration defining the Mandelbrot set. We discuss some of a variety of approaches to compute the roots of these polynomials; classical iterative schemes, eigenvalues of companion matrices and a novel family of recursively defined matrices. Time permitting, we look at an experimentally-discovered asymptotic series for the largest-magnitude roots.

Joint work with Piers W. Lawrence and David J. Jeffrey.

20207115912

If rotating dust stars would exist in general relativity, this would represent an example of a complete balance between the attractive Newtonian force and the typically repulsive gravitomagnetic "force". Although such a balance is considered as improbable by all experts, till now all attempts for a non-existence proof for rotating dust stars failed.

I present a new attack for this problem by considering the level lines of the quasi-Newtonian potential U in the Bardeen metric form for stationary and axisymmetric systems. If rotating dust stars would exist, one U-level-line would run along the axis in the dust, would bifurcate at the poles of the star, would close in the exterior vacuum region, and would enclose a U-minimum there. Hereby the question of existence of rotating dust stars is reduced to the concrete but still open mathematical problem whether the potential U can have a minimum in the vacuum region.

20125121757

For a group G and a set of generators S of G, the Cayley graph Γ(G, S) is defined to have vertex set G and g, h ∈ G are adjacent if and only if gs = h or hs = g for some s ∈ S. The diameter of Γ(G, S) is the maximum distance among pairs of vertices; equivalently, the diameter is the minimum number d such that every group element can be written as a word of length at most d in terms of the elements of S and their inverses. The diameter problem may be interesting for a particular group and set of generators (how many turns do we need to solve Rubik’s cube?), but the mathematically most challenging questions are about estimating
diam(G) := max_S{diam(Γ(G, S))}
with the maximum taken over all sets of generators of G, and for G in an appropriate family of groups. The challenge driving most recent activities is Babai’s conjecture, which states that for all finite nonabelian simple groups, diam(G) < (log|G|)^c, for some absolute constant c. The conjecture was proven by Pyber, Szabó and Breillard, Green, Tao in 2011 for Lie-type groups of bounded rank, but the case of alternating groups cannot be handled by their machinery. For alternating groups A_n, Babai’s conjecture requires a polynomial, n^c, diameter bound. We can prove a slightly weaker quasi-polynomial result:
diam(A_n) < exp(O((log n)⁴ log log n)).

This is joint work with Harald Helfgott (ENS, Paris).

20131094635