[CvGmt News] Colloquium and Mini-Course on Arnol'd diffusion (Vadim Kaloshin)

[Alfonso Sorrentino]

COLLOQUIUM & MINI-COURSE (2 Lectures) ON ARNOL'D DIFFUSION

Prof. Vadim Kaloshin (University of Maryland) will be visiting "Roma
Tre" for one week in April and he will deliver a series of lectures on
his recent works on Arnol'd diffusion. More precisely, here is the
schedule of the colloquium + two lectures (see also the abstracts
below):

I. Wednesday 16/4/2014, 16:00 (room: F), Departmental Colloquium:
"Arnold Diffusion via Invariant Cylinders and Mather Variational
Method";

II. Thursday  17/4/2014, 11:30 - 13:00 (room: 211), Lecture 1: "Single
and double resonant normal forms and Conley's isolating block".

III.  Thursday  17/4/2014, 16:00 - 17:30 (room: 211), Lecture 2:
"Normally hyperbolic invariant cylinders for single and double
resonant systems.Variational lambda lemma."


If you need any additional information, please feel free to contact
sorrentino at mat.uniroma3.it.

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Abstracts:

I. Colloquium: "Arnold Diffusion via Invariant Cylinders and Mather
Variational Method"
The famous ergodic hypothesis claims that a typical Hamiltonian
dynamics on a typical energy
surface is ergodic. However, KAM theory disproves this. It establishes
a persistent set of positive measure of invariant KAM tori. The
(weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff,
says that a typical Hamiltonian dynamics on a typical energy surface
has a dense orbit. This question is wide open. In early 60th Arnold
constructed an example of instabilities for a nearly integrable
Hamiltonian of dimension n>2 and conjectured that this is a generic
phenomenon, nowadays, called Arnold diffusion.  In the last two
decades a variety of powerful techniques to attack this problem were
developed.  In particular, Mather discovered a large class of
invariant sets and a delicate variational technique to shadow them. In
a series of preprints: one joint with P. Bernard, K. Zhang and one
with K. Zhang and one with M. Guardia we prove strong form of Arnold's
conjecture in dimension n=3.

II. Lecture 1: "Single and double resonant normal forms and Conley's
isolating block".
In the lecture I will describe normal forms for nearly integrable
systems near resonance.Then I present a condition on a vector field to
guarantee existence of a normally hyperbolic invariant cylinder.

III. Lecture 2: "Normally hyperbolic invariant cylinders for single
and double resonant systems.
Variational lambda lemma."
Using resonant normal forms I show construction of normally hyperbolic
invariant cylinder
at single and double resonances for nearly integrable systems of two
and a half degrees of
freedom. Time permitting I will describe variational lambda  lemma and
notion of c-equivalence
proposed by Bernard.