Geometry of classical mechanics usually means symplectic geometry. In quantum mechanics this geometry is inseparably joined to the geometry of probability theory, creating a very rich structure. I will introduce this structure from scratch, assuming that the audience has not spent any thought on the geometry of finite dimensional quantum mechanics. But I will be gently leading up to a point of view from where you can see some unsolved problems that I am working on.
"My name is Ingemar Bengtsson, and I have been a lecturer at Fysikum in Stockholm since '93 (and a professor since '00). My previous 'career' was at Chalmers, CERN and Imperial College.
The research areas that I like the best usually have something to do with geometry. General relativity is a favourite. Most of my work there is on black holes. My strongest prejudice is that the world has four dimensions; this is the direction in which I look for clues about quantum gravity. Then I work on quantum information theory, since the geometry of the space of quantum states is wonderful and rather mysterious. What I find fascinating about relativity and quantum mechanics---as it happens, the two deepest theories we have---is that their basic equations have been around for a century, and yet they keep springing conceptual surprises on us. I am looking for the next surprise there, but I do keep a weather eye open on other subjects as well."
Geometry and Gravitation Seminar
Einstein's 1915 theory of general relativity, together with much of modern physics, is built on a geometrical foundation. Lorentz geometry, spin geometry, symplectic geometry and calculus of variations, and the theory of partial differential equations, are examples of the tools which are used in the research group Geometry and Gravitation, to study fundamental questions in the theory of gravity, and related physical theories.
The gravitational force, described by general relativity, plays an essential role in the current models of the universe. It manifests itself on a large range of scales, from the big bang model of cosmology to the theory of black holes and gravitational waves, and general relativity impinges on our daily lives through its role in for example the GPS system.
Following a lengthy period during which the properties of exact solutions was investigated, leading to an analysis of global nature of black hole spacetimes, including the Schwarzschild and Kerr black hole solutions, as well as of families of cosmological models, the theory of general relativity now has a well-developed formal and conceptual basis. The observation of general relativistic effects in binary pulsar systems, gravitational lensing, as well as the recent observation of gravitational waves, has led to the development of a large community studying the problems of general relativity, from a physical and mathematical perspective.
Several important and well-motivated conjectures in general relativity, including cosmic censorship, black hole uniqueness and stability, as well as the Penrose inequality, are among the most important open problems in modern mathematics, and attract the attention of a growing community of researchers. These main conjectures remain open, in spite of major progress, including the proof of non-linear stability of Minkowski space in the early 1990’s, the proof of the Riemannian version of the Penrose Inequality and special cases of the Cosmic Censorship Conjecture around 2000, as well as recent progress on the black hole stability and uniqueness problems, since 2010, and remain an important source of inspiration for further work.
The research group Geometry and Gravitation carries out research on the fundamental problems of general relativity and related physical theories, including the black hole stability problem, and the dynamics of self-gravitating matter systems, both from a theoretical point of view, as well as with a view towards experiment.