In the setting of the wave equation on asymptotically anti-de Sitter spacetimes, it is possible to define an operator which maps Dirichlet data of solutions to corresponding Neumann data at the conformal boundary. The Dirichlet-to-Neumann map (or the boundary-to-boundary propagator) is conjectured to determine the bulk metric up to trivial transformations. In this talk I will report on recent progress on this question, based on the proof that the Dirichlet-to-Neumann map is a fractional power for the boundary metric in a suitable sense. In particular I will discuss the case of Einstein AdS spaces in which case we can prove stronger statements (joint work with Alberto Enciso and Gunther Uhlmann).

This seminar will review recent advances in the development of efficient tree tensor network algorithms and their applications to quantum simulation, benchmarking, and theoretical interpretation [1,2].
In particular, results on two- and three-dimensional systems will be presented, both in and out of equilibrium, including scattering processes and induced false vacuum decay in the two-dimensional quantum Ising model [3]. It will further highlights the use of tree tensor network methods beyond traditional quantum simulation, such as addressing hard classical combinatorial problems through mappings to many-body quantum Hamiltonians, optimizing quantum compilation tasks, integer factorization and enabling quantum equational reasoning [4].
 

[1] Introduction to Tensor Network Methods, Simone Montangero Springer International Publishing
[2] Scattering and induced false vacuum decay in the two-dimensional quantum Ising model, Luka
Pavešić, Marco Di Liberto, Simone Montangero
arXiv:2509.02702
[3] Quantum algorithms for equational reasoning, Davide Rattacaso, Daniel Jaschke, Marco Ballarin, Ilaria Siloi, Simone Montenegro
arXiv:2508.21122
[4] Quantum inspired factorization up to 100-bit RSA number in polynomial time
Marco Tesoro, Ilaria Siloi, Daniel Jaschke, Giuseppe Magnifico, Simone Montangero
arXiv:2410.16355

If you throw a stone into a still pond then the wave set up on the surface of the water
dies away, both at a distance far from the initial disturbance and after a relatively short
interval of time. However, some waves in Nature are not like this. They are, instead,
localised in space and trapped in existence for long periods, their removal costing a great
deal of energy. There are many examples of these long-lived, wave-like topological objects
in physics such as magnetic domain walls, vortices, and skyrmions [1,2]. Recently, many
more have been discovered, including more exotic textures known as hopfions, torons and
blochions. Topology, the study of the shapes of objects, provides an organising principle
to understand the existence and extraordinary properties of these objects. Topology also
allows us to understand and classify large families of unusual quantum-mechanical states
of materials.
In this lecture, we will discuss the existence of topological excitations based on ideas
from symmetry breaking in phase transitions. We will see how topological objects occur
in magnetic materials, where they can be readily measured and visualized. Finally, we will
see how these ideas can be applied to a pressing problem: the possible fate of the
Universe!
[1] Lancaster, T. (2019). Skyrmions in magnetic materials. Contemporary Physics, 60(3),
246–261. https://doi.org/10.1080/00107514.2019.1699352
[2] Lancaster, T. and Blundell, S.J. (2015) Quantum Field Theory for
the Gifted Amateur (OUP, Oxford).