Mathematics Colloquium 2010 - 2011

Programme

High-energy properties of the density of states of (almost)periodic Schrödinger operators

Date: 5th November 2010

Speaker: Sergey Morozov (University College London)

Abstract: In the last years, a number of results on the asymptotic behaviour of the density of states of multidimensional periodic and almost periodic Schrödinger operators have been obtained. We will review these results, and briefly discuss some of the modern techniques used in their proofs.

Number theory related to quantum chaos

Date: 12th November 2010

Speaker: Par Kurlberg (KTH Stockholm)

Abstract: Quantum chaos is concerned with properties of eigenvalues and eigenfunctions of "quantized Hamiltonians".  For instance, can classical chaos be detected by looking at the spacings between eigenvalues? Another problem is if classical ergodicity forces eigenfunctions to be equidistributed in a certain sense.  We will give a short introduction to quantized Hamiltonians, and then show that the study of the above mentioned questions for some simple dynamical systems gives rise to interesting problems in number theory.

The nodal set and its boundary intersections

Date: 17th December 2010

Speaker: Uzy Smilansky (Cardiff)

Abstract: Consider the  eigenfunctions (ordered so that the corresponding eigenvalues are non decreasing) of the Dirichlet Laplacian in a 2D domain with a sufficiently smooth boundary. The zero set of the n'th eigenfunction is the nodal set, and it intersects the domain boundary at a discrete number $\eta_n$ of points. The sequence
$\{ \eta_j\}_{j=1}^{\infty}$ of boundary intersection counts will be shown to encode information about the shape of the domain boundary. Moreover, the cumulant $Y(n)=\sum_{j=1}^n \eta_j$ allows an asymptotic expansion in terms of the closed geodesics in the domain similar in structure to the spectral trace formula. The  statements above are not proved rigorously. The purpose of the talk is to provide the available evidence for their validity and to intrigue further research.

On the power of linear dependences

Date: 9th March 2011

Speaker: Professor Imre Barany (UCL)

Abstract: I will talk about an old result, the so called vector-sum theorem and plan to explain how linear algebra can help solving various geometric problems. Several examples will illustrate the power of this technique, and some applications will also be presented.

Frobenius numbers, random graphs and ergodic theory

Date: 13th April 2011 - 15:10

Speaker: Jens Marklof (Bristol)

Abstract:My objective is to explain the connection between three seemingly unrelated topics:

(1) Frobenius numbers: Given d positive co-prime integers a_1,...,a_d, it is an old problem to find the largest integer--the Frobenius number-- that does not have a representation as a non-negative integer linear combination of the a_j. This puzzle is also known as the coin exchange problem. V.I. Arnold conjectured that the Frobenius number fluctuates wildly as a function of the a_j---I will report some recent progress on this question.

(2) Random graphs: The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. I will discuss an intermediate class of examples: Caley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. Here the diameter of a random circulant 2k-regular graph with n vertices scales as n^(1/k), and satisfies a non-standard limit theorem (joint work with A. Strombergsson, Uppsala).

(3) Ergodic theory: The key to both of the above problems is the dynamics of homogeneous flows on the space of lattices, and in particular an equidistribution theorem for Farey sequences on closed horospheres. The principal aim of my lecture is to give a gentle introduction to the basic ideas of this subject, and illuminate their relevance to (1) and (2).