# Le jour d’analyse 2007

*Analysis day 2007*

### May 2 mai 2007

### Organisateur/Organizer: Dmitry Jakobson (McGill University)

Toutes les conférences auront lieu au local 5340

*All talks will be held in Room 5340 *
Centre de Recherches Mathematiques,

Université de Montréal, Pavillon André-Aisenstadt,

2920 Chemin de la tour, Montréal

### Mercredi / Wednesday, May 2 mai 2007

## Horaire/Schedule

**11:00-12:00, Igor Wigman (McGill/CRM)**

*Title*: *Nodal lines for random eigenfunctions of the Laplacian on the torus*.

**Abstract:**

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4π^{2}2λ with growing multiplicity `N` → ∞, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is const λ^{1/2}. Our main result is that the variance of the volume normalized by λ^{1/2} is bounded by `O`(1/`N`^{1/2})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

**12:00-13:30 Lunch**

*13:30-14:30 Francis Clarke (Institut universitaire de France et Université de Lyon)*

*Title: Regularity of solutions in the calculus of variations*

**Abstract: **

An overview of some classical and recent results on the regularity of solutions in the (single and multiple integral) calculus of variations. Along the way are reviewed the celebrated theorems of Hilbert-Haar and De Giorgi.

**14:40-15:40 Francois Germinet (University of Cergy-Pontoise, Paris)**

*Title: Single energy multiscale analysis and Anderson localization*

**Abstract: **

In the talk, I will show how to extract exponentially decaying eigenfunctions out of the basic and standard single energy multiscale analysis introduced by Fröhlich and Spencer in 1983.