#### PDE in Lie groups

**Giovanna Citti (Università di Bologna, Italia)**

Viernes, 20 de octubre, Dpto. de Matemáticas, Aula 520, 12:00h. |

The motivation of the problem comes from a problem of visual perception. Indeed the visual cortex has been modelled as a Lie group, with a totally degenerate metric, also called subriemannian. The associated PDE can be expressed as a sum of squares of vector fields, and the study of these type of equations under suitable geometric condition, has been started by Hormander and Rothschild – Stein. However regularity at the boundary, well known in the Euclidean setting, was still open even for the subriemannian Laplacian. We will face the problem with a suitable extension of a result of Caffarelli and Silvestre to this degenerate setting.

La profesora Giovanna Citti ha sido invitada por Davide Barbieri, profesor de nuestro departamento, a él le debemos las siguientes lineas sobre el próximo coloquio:

Giovanna Citti is full professor of mathematical analysis at the University of Bologna, where she coordinates the PhD program in mathematics. She has given important contributions to the study of linear and nonlinear PDE in different settings, from complex analysis, with crucial results on the Levi equation, to Hormander-type PDE. A PDE on a manifold that is defined by the sum of squares of vector fields, or even in more general forms, can have many of the properties of an elliptic equation (or of a parabolic one, if the associated diffusion is considered) if the involved vector fields together with their commutators span the entire tangent space at any point of the manifold. In this case it is called a Hormander-type PDE. Such vector fields can be recast, up to an approximating limiting procedure, into elements of a nilpotent Lie algebra with a notion of dilations, whose corresponding Lie groups are called Carnot groups, and their left invariant metrics are called subriemannian. Giovanna is one of the main world expert in the analysis of linear and nonlinear PDE on such structures. Her results have appeared in most of the more relevant journals of the area, such as Acta Math., J. Math. Pures Appl., Adv. Math., Calc. Var. Partial Differential Equations, J. Funct. Anal., Crelle's J., Trans. Amer. Math. Soc.

She is also, thanks to her long term collaboration with A. Sarti (CNRS, Paris), one of the founders and leading exponents of the so-called neurogeometry. Neurogeometry is a mathematical modelling approach to visual perception which relies on both the phenomenology of vision as a cognitive experience and the quantitative analysis of the physiological properties of families of cells in the brain's cortical area dedicated to the processing of visual stimuli. One of their most important works consists of a neural model for the generation of visual illusions showing the appearance of illusory contours by means of an algorithm of Hormander-type diffusion and concentration to a regular surface. More recently, they have introduced neural models based on spectral analysis and gauge theory in subriemannian structures that reproduce perceptual image segmentations.

Giovanna is also very active in the organization of initiatives for the stimulus of mathematical research. In the recent years she has coordinated an FP6 Adventure STREP project, an FP7 ITN project, a EU-US Atlantis project and recently a H2020 RISE project, the last two with the participation of the department of mathematics at UAM.

In this colloquium she will address the solution of one of the oldest open problems concerning PDE in Carnot groups. The problem of Hoelder regularity of solutions at the boundary of smooth domains is indeed a central issue in the theory of classical solutions to elliptic PDE. However, it was still open for Carnot groups. Only when the vector fields are the generators of the Lie algebra of the Heisenberg group this could be solved, more than 35 years ago, by D. Jerison, but with techniques that can not be extended to other settings because they rely on the special relationship between the Heisenberg group and the Fourier transform. G. Citti, together with A. Baldi and G. Cupini at the University of Bologna, have recently developed a new set of techniques that allow to control the restrictions of heat kernels to smooth manifolds for all Carnot groups. This turned out to be a powerful instrument to attack several problems, and in particular to obtain boundary regularity.

Giovanna Citti is full professor of mathematical analysis at the University of Bologna, where she coordinates the PhD program in mathematics. She has given important contributions to the study of linear and nonlinear PDE in different settings, from complex analysis, with crucial results on the Levi equation, to Hormander-type PDE. A PDE on a manifold that is defined by the sum of squares of vector fields, or even in more general forms, can have many of the properties of an elliptic equation (or of a parabolic one, if the associated diffusion is considered) if the involved vector fields together with their commutators span the entire tangent space at any point of the manifold. In this case it is called a Hormander-type PDE. Such vector fields can be recast, up to an approximating limiting procedure, into elements of a nilpotent Lie algebra with a notion of dilations, whose corresponding Lie groups are called Carnot groups, and their left invariant metrics are called subriemannian. Giovanna is one of the main world expert in the analysis of linear and nonlinear PDE on such structures. Her results have appeared in most of the more relevant journals of the area, such as Acta Math., J. Math. Pures Appl., Adv. Math., Calc. Var. Partial Differential Equations, J. Funct. Anal., Crelle's J., Trans. Amer. Math. Soc.

She is also, thanks to her long term collaboration with A. Sarti (CNRS, Paris), one of the founders and leading exponents of the so-called neurogeometry. Neurogeometry is a mathematical modelling approach to visual perception which relies on both the phenomenology of vision as a cognitive experience and the quantitative analysis of the physiological properties of families of cells in the brain's cortical area dedicated to the processing of visual stimuli. One of their most important works consists of a neural model for the generation of visual illusions showing the appearance of illusory contours by means of an algorithm of Hormander-type diffusion and concentration to a regular surface. More recently, they have introduced neural models based on spectral analysis and gauge theory in subriemannian structures that reproduce perceptual image segmentations.

Giovanna is also very active in the organization of initiatives for the stimulus of mathematical research. In the recent years she has coordinated an FP6 Adventure STREP project, an FP7 ITN project, a EU-US Atlantis project and recently a H2020 RISE project, the last two with the participation of the department of mathematics at UAM.

In this colloquium she will address the solution of one of the oldest open problems concerning PDE in Carnot groups. The problem of Hoelder regularity of solutions at the boundary of smooth domains is indeed a central issue in the theory of classical solutions to elliptic PDE. However, it was still open for Carnot groups. Only when the vector fields are the generators of the Lie algebra of the Heisenberg group this could be solved, more than 35 years ago, by D. Jerison, but with techniques that can not be extended to other settings because they rely on the special relationship between the Heisenberg group and the Fourier transform. G. Citti, together with A. Baldi and G. Cupini at the University of Bologna, have recently developed a new set of techniques that allow to control the restrictions of heat kernels to smooth manifolds for all Carnot groups. This turned out to be a powerful instrument to attack several problems, and in particular to obtain boundary regularity.