Abstract: The derivative of a vector field on flat Euclidean n-space may be viewed as a function with values in the nxn matrices. This matrix may be broken into pieces (symmetric part, skew part, trace, and so on) and we can set some parts to be zero to impose various natural PDE on the original vector field. Prolongation is a technique that allows us to investigate and sometimes solve these PDE, often with a geometric interpretation in terms of symmetries. This is just the start of a general theory, to be explored in these lectures.
Lecture 1, Lecture 2, Lecture 3, Lecture 4
Srni 2004 notes (Srni04)
Abstract: This mini-course will be an introduction to rigid differential-geometric structures and their automorphisms. It will cover some of the basic theory of G-structures of finite type in the sense of Cartan and a variety of examples. A few major theorems on automorphisms of such structures, such as Zimmer’s Embedding Theorem and Gromov’s Open-Dense Theorem, will be presented, along with some applications. The last part will cover a recent generalization of Zimmer’s Embedding to the tractor setting and an application in conformal pseudo-Riemannian geometry (joint work with K. Neusser).
Abstract: Non-linear wave equations are partial differential equations that arise naturally as equations of motion in mathematical physics, for example when considering charged particles interacting with their self-induced electromagnetic field. In these lectures I will try to give an overview of some techniques and ideas often used in the study of non-linear wave equations on the Minkowski spacetime. Classically, local existence is handled using energy estimates and Sobolev inequalities. To obtain global existence results, it is necessary to exploit also dispersive properties of the solutions. One way to do this is by using the Klainerman-Sobolev inequality, where the Sobolev norms are defined in terms of vector fields that are the generators of the Poincaré group. The higher the space dimension, the stronger is the dispersion, and we will see that in high space dimensions, say 4 or greater, it is fairly easy to prove global existence for small initial data. In lower dimensions, an additional algebraic condition on the non-linear part of the equation, called the null condition, may be needed.
Lecture 1, Lecture 2, Lecture 3
Abstract: In this course we will present some methods and ideas at the crossroads of differential geometry, topology, analysis and dynamical systems, motivated by open problems in fluid mechanics and magnetohydrodynamics. Some topics to be covered:
For 3D flows: Local and global invariants of the action of the group of volume preserving diffeomorphisms: asymptotic linking number, KAM invariants, contact-type invariants. Magnetic relaxation, equilibrium states, and dynamics of the 3D Euler equations. Open problems concerning foliation cycles of 3D div-free fields.
For 2D flows: Variational problems on the orbits of the group of area preserving transformations. Long-time behavior of the 2D Euler equation.
Lectures 1-3, Lecture 4
Ana Cristina Ferreira: Dynamics and completeness of the geodesic flow on pseudo and holomorphic Riemannian manifolds
Seung-Yeon Ryoo: Quantitative nonembeddability of nilpotent groups into uniformly convex spaces
Wojciech Kamiński: Conformal boundary of solutions to Einstein's equations
Jan Slovák: Geometry of diffusion tensor imaging
Jie Xu: A codimension 2 approach to the S^1-Stability Conjecture