Type: Seminar

Distribution: World

Expiry: 18 Sep 2012

Auth: zhangou@bari.maths.usyd.edu.au

Speaker: Prof. Claude Viallet (LPTHE) http://www.lpthe.jussieu.fr/~viallet/ Time: Friday, Aug. 24, 2:30--3:30PM Room: OMB-150, Old Main Building (UNSW) Lunch with speaker: we meet around 1PM at the entrance to the East Wing of Red Centre Building. One choice of commuting from Sydney: meet at Carslaw 620 around 12:25PM and share taxi to UNSW. The round trip is covered by school colloquium fund. ----------------------------------------------- Title: Discrete-time Dynamical System: Integrable Or Not Integrable? Abstract: to any discrete-time dynamical system with a rational evolution, it is possible to associate a number, the "algebraic entropy", which is a global index of complexity of the system. Its vanishing is the hallmark of integrability, and it proved to be an unsurpassed detector of integrability. But it is also an interesting object in itself. Its analysis links in particular to algebraic geometry and number theory. We will give the definition of the entropy, and show on simple examples how it can be calculated. We will then explain the link between this concept and the singularity structure of the system, relating it to the discrete Painlev\’e analysis (this is where algebraic geometry comes in). We finally describe, in a simple case, the set of values assumed by the entropy (this is where number theory enters, as it is conjectured to always be the logarithm of an algebraic integer), and introduce the notion of entropy gap. ----------------------------------------------- Joint Colloquium web site: http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html