Term 1, 2012-13
The mini-courses are intended to last two hours, and since the number of lectures in each series cannot be determined in advance (or, since we don't want to do this in advance), the schedule may be stretched over time. So, please check back here if you are interested in a particular series.
The schedule of the seminars can be found here.
- Thursday, October 18, 1-3pm
Stochastic behaviour in deterministic dynamical systems I: Weak chaos sounds clearer in odd dimensions
Abstract: The serious background to this title is the classical (linear) Huygen's principle that sounds propagates in odd but not even dimensions. An analogous nonlinear dynamics statement is in the context of dynamical systems with Euclidean symmetry (rotations and translations of R^d). If the dynamics is simple, then it can be shown that typically solutions propagate linearly (growth rate a=1) if d is odd, and are bounded (a=0) if d is even. The dichotomy disappears for chaotic dynamics. At least, "strong chaos" leads to Brownian motion with the normal diffusion rate a=1/2 in all dimensions. But the dichotomy reappears for "weak chaos". It is not all or nothing, hence the title. "Weakly chaotic" dynamics leads to Levy flights and superdiffusive growth rate a>1/2, but only if d is odd. If d is even, typically the anomalous diffusion is suppressed and Brownian motion with growth rate a=1/2 is restored.
In the first part of this mini-course, I will introduce enough background to give a flavour of the ideas behind (and meaning of) these statements. Then, I will give more details about how to prove central limit theorems(and convergence to Brownian motion) for large classes of deterministic dynamical systems encompassing those occurring above.
Prior knowledge of the underlying ergodic theory/probability theory/pattern formation will not be assumed.
- Thursday, October 25, 1-3pm
Stochastic behaviour in deterministic dynamical systems II
- Thursday, November 1, 1-3pm, Room B1.12
Stochastic behaviour in deterministic dynamical systems III
- Thursday, November 8, 1-3pm, Room B1.12
Noncommutative geometry and dynamics I
Abstract: In this series of lectures, we will present an overview of the core ideas in noncommutative geometry and illustrate them with examples coming from dynamics. We will discuss spectral triples, which serve as the analogue of a Riemannian manifold in noncommutative geometry, by looking at the noncommutative 2-torus. This object arises naturally by considering rotation actions on the circle. The class of Cuntz-Krieger algebras, a type of universal C*-algebra, is introduced via their close relation to subshifts of finite type. In certain instances, these can be related to crossed product algebras associated to the action of a Kleinan group on its limit set. We will introduce K-theory, and its relation to spectral triples, via the analogy with classical algebraic topology. For the noncommutative torus, the Cuntz-Krieger algebras, as well as for the action of a Fuchsian group on its limit set , we will do concrete calculations of K-groups. This will shed some light on the meaning and significance of K-theory. If time allows, there will be discussion of Chern characters, cyclic homology and noncommutative index theory in the context of dynamics.
This lecture series is aimed at non specialists (in both noncommutative geometry as well as in dynamics) who would like to learn a bit about the subject. Everyone is welcome to attend.
- Thursday, November 15, 1-3pm, Room B1.12
Noncommutative geometry and dynamics II