Mini Course: Convexity and semiconvexity in the Heisenberg group and Carnot groups with applications

This is Session 2

Please REGISTER here if you plan on coming...

10:00-12:00, Monday & Friday

July 13 and July 17

Room No. Lab 4, F01.

• 4 hours, 2 days in total

Description

This four-lecture mini-course provides an introduction to convexity and semiconvexity in the setting of the Heisenberg group and, more generally, Carnot groups, with an emphasis on recent developments and applications to Hamilton–Jacobi equations. Beginning with the classical Euclidean theory of semiconcavity and viscosity solutions, the course then introduces the geometry of Carnot groups, including homogeneous and Carnot–Carathéodory (CC) distances. It proceeds to the notions of horizontal convexity and semiconcavity before presenting recent results on the horizontal semiconcavity of the squared CC distance, including a sketch of the proof in the case of ideal groups (joint work with Qing Liu and Ye Zhang). The course concludes by discussing applications of these results to Hamilton–Jacobi equations.

Lecture Outline

• Lecture 1: The Euclidean case: Euclidean distance semiconcavity and viscosity solutions.
• Lecture 2: Carnot groups: an introduction, homogeneous distance, and Carnot–Carathéodory (CC) distance.
• Lecture 3: Horizontal convexity and semiconcavity.
• Lecture 4: Horizontal semiconcavity of the squared CC distance; sketch of the proof for the case of ideal groups (joint work with Qing Liu and Ye Zhang). Applications to Hamilton–Jacobi equations.

Intended Audience

This mini-course is aimed at both PhD students and researchers, but you should have a solid grasp of partial differential equations to benefit!

Bio

Professor Federica Dragoni is a TSVP scholar at OIST and a professor (personal chair) in the School of Mathematics at Cardiff University. Her research focuses on nonlinear partial differential equations, geometric analysis, and sub-Riemannian geometry, with particular expertise in the Heisenberg group, Carnot groups, and Hamilton–Jacobi equations. She is a member of Cardiff's Analysis Group and has made significant contributions to the theory of degenerate PDEs, convexity, and geometric properties of non-Euclidean spaces. She also leads the Welsh node of the EPSRC Network on Generalised and Low-Regularity Solutions of Nonlinear PDEs.