F
ebruary 4 - 2000:Mathematical Aspects of Aperiodic Order.
14:15-16:00: Richard Gill (Utrecht University):
Asymptotically Optimal Quantum Statistics for Spin Half.
The meeting will take place in room 220 of the James Boswell Institute, Bijlhouwerstraat 6, Utrecht.
ABSTRACT MICHAEL BAAKE
Michael Baake is the main speaker of the Mark Kac seminar this year, and he will give three lectures in February, March and April. Here is the abstract for the series of three lectures:
The discovery of quasicrystals in the early eighties has triggered an intensive investigation of the various kinds of ordered states that are possible between periodic and random. In this series of lectures, I plan to describe the impact that quasicrystals had, focusing specifically of aspects of diffraction theory.
In the first lecture, I will start with the history of the field and summarize how and why quasicrystals challenged our understanding of the solid state. Some generalizations of crystallographic tools will be described that are suitable to cope with symmetry and equivalence concepts in this more general situation.
In the second lecture, I plan to survey mathematical diffraction theory, with special emphasis on the perfectly ordered systems. Of particular interest are point sets (representing atomic positions) which lead to pure point diffraction spectra, whose classification is far from complete.
In the third lecture, I will approach the diffraction of stochastic structures (such as random tilings) with methods from statistical physics. These systems show a variety of different spectral properties, including the possibility of practically relevant examples with singular continuous spectra.
ABSTRACT RICHARD GILL
"Quantum Statistics" means (for the speaker) the problem of determining the state of a quantum system from measurements on that system. A measurement on a quantum system yields a random outcome, whose distribution contains information about the state of the system. At the same time the measurement disturbs the original system, and therefore can prevent one from being able to get more information about the state. The class of all possible measurements (experiments) is not too difficult to describe mathematically, and one can therefore ask the question: which measurement is best.
I consider the simplest possible version of this problem, the case of a spin-half system, which is the quantum analogue of the problem of determining "p" from N tosses of a biased coin. for large N one can find in some special cases a fairly simple approximately optimal solution. However the general case contains challenging open problems.
Literature:
An introductory/survey paper, to appear in the van Zwet Festschrift:
http://www.math.uu.nl/people/gill/Preprints/paper.ps.gz
The real thing, joint with S. Massar, to appear in Phys. Rev. A
http://www.math.uu.nl/people/gill/Preprints/massar10.ps.gz