Lars Martin Sektnan
(University of Gothenburg)
Titel: Blowing up extremal Kähler manifolds
Abstract: 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.
(University of Stockholm)
Titel: A number theory problem arising in quantum information theory
Abstract: 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.
(University of Stockholm)
Titel: Geometry of quantum state spaces
Abstract: 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.
Titel: On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr-Newman spacetime
Abstract: In relativistic quantum mechanics, the discrete spectrum of the Dirac Hamiltonian with a Coulomb potential famously agrees with Sommerfeld’s fine structure formula for the hydrogen atom. In the Coulomb approximation, the proton is assumed to only have a positive electric charge. However, the physical proton also has a magnetic moment which yields a hyperfine structure of the hydrogen atom that’s normally computed perturbatively. Aiming towards a non-perturbative approach, Pekeris in 1987 proposed taking the Kerr-Newman spacetime with its ring singularity as a source for the proton’s electric charge and magnetic moment. Given the proton’s mass and electric charge, the resulting Kerr-Newman spacetime lies well within the naked singularity sector which possess closed timelike loops. In 2014 Tahvildar-Zadeh showed that the zero-gravity limit of the Kerr-Newman spacetime (zGKN) produces a flat but topologically nontrivial spacetime that’s no longer plagued by closed timelike loops. In 2015 Tahvildar-Zadeh and Kiessling studied the hydrogen problem with Dirac’s equation on the zGKN spacetime and found that the Hamiltonian is essentially self-adjoint and found a single bound state for the discrete spectrum. In this talk, we show how their ideas can be extended to classify the discrete spectrum completely and relate it back to the known hydrogenic Dirac spectrum but yielding hyperfine-like and Lamb shift-like effects.
Titel: Graviton amplitudes from twistor sigma models
Abstract: I will discuss a new tool that helps us obtain Andrew Hodges' extremely compact formulae for tree-level graviton MHV amplitudes in GR. Unlike previous on-shell methods like recursion or worldsheet models, our approach starts directly from space-time perturbation theory. It systematically constructs a generating functional for these MHV amplitudes by means of hyperkähler geometry and twistor theory. This generating functional takes the form of the on-shell action of a 2d defect CFT that we call a twistor sigma model. We show how computing this on-shell action results in a tree-diagram expansion of the MHV amplitudes anticipated by Bern, et al in 1998. The diagrams, when resummed via the matrix tree theorem, magically condense into Hodges' formulae. Time permitting, I will also comment on a flat holographic interpretation of the twistor sigma model.
(University of Potsdam)
Titel: On the index theorem and the Rokhlin theorem
Titel: Kähler geometry, black holes, and twistor integrals
Abstract: Using complex methods in relativity, we show that black hole geometries can be encoded in a single scalar function, called Kähler potential. We show that this function encodes the Newman-Janis shift from Schwarzschild to Kerr, as well as different dualities within the Plebanski-Demianski family of solutions. At the linear level, we show that these structures can be nicely understood in terms of twistor theory. This is joint work with Steffen Aksteiner.
Titel: On black holes and Buchdahl stars
Abstract: Buchdahl star is the most compact non black hole object, and the two are respectively defined by gravitational potential, $\Phi(R) = 4/9, 1/2$. We would first argue that the Buchdahl star is a Virial distribution where its equilibrium is due to the Virial theorem, and then attempt to see what kind of black hole properties like extremalization and over extremalization could be carried over to the Buchdahl star. Could Buchdahl star serve as a precursor to black hole, and in particular its role in formation of extremal black holes.
(Universidad de la República, Montevideo, Uruguay)
Titel: Numerical Studies of Axisymmetric Periodic Analogues of Kerr Black Holes
Abstract: In this talk we present some results concerning the numerical study of solutions to the axisymmetric stationary Einstein Equations in a coaxial periodic set up. Since the discovery of the periodic static solutions by Myers, and independently by Korotkin and Nicolai, the existence of solutions in the stationary case has been elusive, for one particular reason: the kind of asymptotic behaviour of the solutions implies the divergence in the usual barrier methods used to prove existence. We adopt a numerical approach, working with two possible configurations: one with equidistant horizons with the same angular momentum, and one with equidistant horizons with alternating opposite angular momentum. The parameters are the period and the length, area and angular momentum of the horizons. We will show how the numerical implementation is done via a harmonic map heat flow, and the introduction of new boundary conditions that are suitable for the problem will be discussed. Finally, we will show the results we obtained.
Titel: Electromagnetic and gravitational Hopfions
Abstract: Hopfions are a family of field solutions which have non-trivial topological structure. Their connections with Hopf fibration will be presented. I will focus on two physical applications of Hopfions: electromagnetism and linear gravitation. The issue of topological charges will be briefly discussed.
(Jagiellonian University, Krakow)
Titel: Bound states of nonlinear Schrödinger equations with trapping potentials in higher dimensions
Abstract: In this talk I would like to discuss some results regarding a particular nonlinear Schrödinger equation, dubbed by us the Schrödinger-Newton-Hooke equation. It is usually encountered as a description of various quantum-mechanical systems but it can also be obtained as a nonrelativistic limit of small scalar perturbations of the anti-de Sitter spacetime. This observation gives us a motivation to investigate this equation in higher dimensions. However, it turns out that for dimensions d>6 not very much is known about it, since the usually employed approach based on the variational methods ceases to work. I would like to show how one can overcome this problem by focusing on spherically symmetric solutions and using the classical methods coming from the field of ordinary differential equations. In particular, I will prove the existence of the whole ladder of excited solutions and show the uniqueness of the ground states. I will also discuss how the frequencies and masses of the solutions vary for different dimensions. Most of the presented methods and results hold also for other similar systems with trapping potentials.
(University of Vienna)
Titel: Spin Hall effects and gravitational lensing
Abstract: Spin Hall effects represent a diverse class of physical phenomena related to the propagation of wave packets carrying intrinsic angular momentum. These effects have been experimentally observed in optics and condensed matter physics, but they are also expected to occur for wave packets propagating in gravitational fields. In this talk, I will introduce the equations of motion describing the gravitational spin Hall effect, and I will discuss their properties, physical interpretation, as well as the relation to the Mathisson-Papapetrou equations. Based on this, I will present recent results regarding the strong lensing of gravitational waves. Given the typical wavelengths of observed gravitational waves, the gravitational spin Hall effect is expected to have a significant effect that could lead to experimental observation.
Titel:On $S^1$-symmetric gravitatonal instantons
Abstract: Gravitational instantons are complete 4-dimensional Ricci-flat manifolds with Riemannian signature and curvature decaying sufficiently fast. I shall present ongoing work on the classification of $S^1$-symmetric gravitational instantons. The approach taken makes use the $G$-signature theorem and an identity of Israel-Robinson type.
Titel: G-to-zero limit of Kerr-Newman solution
Abstract: The limit of vanishing G of the maximal analytically extended Kerr–Newman solution in Boyer–Lindquist coordinates leads to the so-called 'magic electromagnetic field' (or the square root of Kerr) on a topologically nontrivial flat spacetime. The background, in this case, is a two-sheeted spacetime with special leaves represented in oblate spheroidal coordinates. The magic electromagnetic field can also be generated by a complex shift in the Coulomb field, parallel to the Newman and Janis transformation. I will review some of the interesting questions and results along these lines.
Titel: Belinski-Zakharov method revisited
Abstract: The Belinski-Zakharov (BZ) method is among the various solution generating techniques for 4-dimensional, Ricci flat metrics with two Killing vectors. It was used by Chen and Teo to find a new asymptotically flat gravitational instanton solution. In this talk we review the BZ method and conjecture a simple combinatorical form for the generated metrics.
Title: An introduction to spinor and twistor methods for gravitation
Abstract: We give an elementary introduction to spinors in 4 complex dimensions, focusing on the different reality structures associated with Lorentzian and Riemannian signatures. We discuss connections with almost-complex structures and with integrability aspects relevant to gravitation, especially regarding algebraically special geometries. The emphasis is on a pedagogical presentation; no previous knowledge of spinors or complex geometry is assumed.
Title: Price's law for Teukolsky master equation in Kerr spacetimes
Abstract: Teukolsky master equation governs the dynamics of the spin $s$ components in Kerr spacetimes. I will show their precise late time asymptotic profiles, i.e., the conjectured Price's law in the physics literature.