#### Event Title

### Multiple singular values of Hankel operators and weak turbulence in the cubic Szeg\H{o} equation

#### Location

Brown Hall 100

#### Start Date

7-20-2016 11:40 AM

#### End Date

20-7-2016 12:30 PM

#### Description

We establish an inverse spectral result on compact Hankel operators on the unit sphere. Namely, we describe the set of symbols of compact Hankel operators having a prescribed sequence of singular values. It is done by constructing a one-to-one correspondence between a symbol of a compact Hankel operator and its sequence of singular values as well as some additional spectral parameters. \\ This one-to-one correspondence plays the role of a non linear Fourier transform for some hamiltonian equation: the cubic Szeg\H{o} equation. It allows to obtain explicit formulae of the solutions and to prove a wave turbulence phenomenon: for a dense $G_\delta$ of initial data, solutions develop large oscillations on small space scales. It is from joint works with Patrick G\'erard.

Multiple singular values of Hankel operators and weak turbulence in the cubic Szeg\H{o} equation

Brown Hall 100

We establish an inverse spectral result on compact Hankel operators on the unit sphere. Namely, we describe the set of symbols of compact Hankel operators having a prescribed sequence of singular values. It is done by constructing a one-to-one correspondence between a symbol of a compact Hankel operator and its sequence of singular values as well as some additional spectral parameters. \\ This one-to-one correspondence plays the role of a non linear Fourier transform for some hamiltonian equation: the cubic Szeg\H{o} equation. It allows to obtain explicit formulae of the solutions and to prove a wave turbulence phenomenon: for a dense $G_\delta$ of initial data, solutions develop large oscillations on small space scales. It is from joint works with Patrick G\'erard.