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November 7 - 2003

11:15-13:00: Antal Jarai
        The Abelian sandpile model and the uniform spanning tree.


14:15-16:00: Madalin Guta      
        Estimation of quantum states through quantum tomography.

Abstracts:

Antal Jarai
The Abelian sandpile model and the uniform spanning tree.

The Abelian sandpile model (ASM) was introduced by Bak, Tang and Wiesenfeld as a model exhibiting so called self-organised critical behaviour. Roughly speaking, self-organised criticality arises when a stochastic dynamics drives a system towards a state that is characterised by power laws. The mathematical study of the ASM, initiated by Dhar, has revealed a rich structure, making the analysis of the ASM more tractable than other models, and mathematically interesting in itself. By a deep observation of Dhar and Majumdar, the ASM can be mapped onto the uniform spanning tree, that, at least heuristically, explains the appearance of power laws.

In this talk I will describe the correspondance between the ASM and spanning trees, and discuss the existence of the infinite volume limit for the stationary measure of the model. This is joint work with S. R. Athreya. I will also mention ongoing research joint with F. Redig.


Madalin Guta
Estimation of quantum states through quantum tomography.

Abstract: Quantum mechanics ascribes to each quantum system a state represented by a density matrix on a Hilbert space. By performing a measurement on the quantum system, we obtain a random result, whose distribution depends on the initial state of the system. The inverse route is to estimate the initial (unknown) state, based on the results of the measurements. Curiously, this has been explored experimentally only in 1993 through a measurement technique called quantum homodyne tomography. From the point of view of theoretical statistics, we deal with a new class of non-parametric estimation problems. We will describe two approaches based on linear estimators and respectively maximum likelihood, the latter using entropy techniques from the theory of empirical processes.