# BIMSA-AEI 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.

**Upcoming Talks**

**Past Talks**

**Title:** Blowing up extremal Kähler manifolds

**Speaker:** Lars Martin Sektnan (University of Gothenburg)

**Time:** 17:15-18:30, Dec. 7, 2022

**Venue:** Online

**Zoom:** 839 2831 2262 **Passcode:** BIMSA

**Abtract：**

Extremal Kähler metrics were introduced by Calabi as a type of canonical Kähler metric on a Kähler manifold. They generalise constant scalar curvature Kähler metrics. A natural question is when the blowup of a manifold in a point admits an extremal Kähler metric. I will discuss sufficient conditions as well as obstructions to producing extremal metrics on the blowup in the compact setting, coming from works of Arezzo-Pacard, Arezzo-Pacard-Singer, Stoppa, Székelyhidi and joint work with Dervan. I will also discuss the non-compact setting of Poincaré type metrics, where there is an additional obstruction not present in the compact setting.

**Title:** A number theory problem arising in quantum information theory

**Speaker:** Ingemar Bengtsson (University of Stockholm)

**Time:** 17:15-18:30, Nov. 30, 2022

**Venue:** Online

**Zoom:** 839 2831 2262 **Passcode:** BIMSA

**Abtract：**

A maximal regular simplex inscribed in the set of quantum states has some engineering applications --- if it exists. Attempts to prove that it does, in all finite dimensional Hilbert spaces, have revealed an unexpected connection to an open problem in algebraic number theory. The whole story is quite new, and it it may have ramifications that we have not thought of yet.

**Title:** Geometry of quantum state spaces

**Speaker:** Ingemar Bengtsson (University of Stockholm)

**Time:** 17:15-18:30, Nov. 23, 2022

**Venue:** BIMSA 1120

**Zoom:** 839 2831 2262 **Passcode:** BIMSA

**Abtract：**

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.