**
Ugo Boscain's ERC Starting Grant 2009 GeCoMethods
**

I am winner of an ERC-starting grant 2009 whose title is

"**Geometric Control Methods for the
Heat
and
Schroedinger Equations (GeCoMethods)" **

**Openings:**
- One post-doc position starting from September 2010 on ``Motion
Planning in Quantum Control''

- One post-doc position starting from September 2011 on
``sub-Riemannian geometry and hypoelliptic heat kernels 1''

- One post-doc position starting from September 2012 on
``sub-Riemannian geometry and hypoelliptic heat kernels 2''

- One post-doc position starting from September 2013 on
``sub-Riemannian geometry and hypoelliptic heat kernels 3''

- One (3 years) PhD position starting from September 2010 on
``measure theory in sub-Riemannian geometry''

- One (3 years) PhD position starting from September 2011 on
``Control of the rotation of molecule''

**Activities:**
- Workshop on
Quantum Control IHP, Paris Dec. 8-10, 2010.
- Workshop on Sub-Riemannian Geometry (spring 2012)
- School on Geometric Control Methods in PDEs (spring 2014)

**Main collaborator**
Mario Sigalotti

**Other members of the team:**
Andrei Agrachev
Riccardo Adami
Thomas Chambrion
Gregoire Charlot
Yacine
Chitour
Jean-Paul Gauthier
Frederic Jean

**Abstract ** The aim of this project of 5 years is to create a research group on geometric
control methods in PDEs with the arrival of the PI at the CNRS Laboratoire CMAP (Centre
de Mathematiques Appliquees) of the Ecole Polytechnique in Paris (in January 09). With the
ERC-Starting Grant, the PI plans to hire 4 post-doc fellows, 2 PhD students and also to organize
advanced research schools and workshops. One of the main purpose of this project is to facilitate
the collaboration with my research group which is quite spread across France and Italy.

The PI plans to develop a research group studying certain PDEs for which geometric control
techniques open new horizons. More precisely the PI plans to exploit the relation between the
sub-Riemannian distance and the properties of the kernel of the corresponding hypoelliptic heat
equation and to study controllability properties of the Schroedinger equation.

In the last years the PI has developed a net of international collaborations and,
together with his collaborators and PhD students, has obtained a certain number of results via a
mixed combination of geometric methods in control (Hamiltonian methods, Lie group techniques,
conjugate point theory,
singularity theory etc.) and noncommutative
Fourier analysis.

This has allowed to solve open problems in the field, e.g., the definition of an intrinsic
hypoelliptic Laplacian, the explicit construction of the hypoelliptic heat kernel for the most
important 3D Lie groups, and the proof of the controllability of the bilinear
Schroedinger equation with discrete spectrum, under some ``generic'' assumptions. Many more
related questions are still open and the scope of this project is to tackle them.

All subjects studied in this project have real applications: the problem of controllability of the
Schroedinger equation has direct applications in Laser spectroscopy and in Nuclear Magnetic
Resonance; the problem of nonisotropic diffusion has applications in cognitive neuroscience (in
particular for models of human vision).

**Keywords** Geometric control theory, Sub-Riemannian geometry, Carnot Caratheodory distance,
Conjugate points, Cut locus, Controllability of the bilinear Schroedinger equation, Hypoelliptic
heat equation, Quantum Control.