Recent years have seen a surge of interest in developing measures of complexity for the unitary evolution in quantum mechanics. One approach, known as Nielsen complexity, visualizes the evolution as a continuous "program" executed by the system and applies measures developed in computational complexity theory to judge whether the unitary evolution operators are simple or complex. Another approach, known as Krylov complexity, tracks how rapidly quantum states and operators spread over different independent directions as they evolve. One key question is whether these quantities are capable of distinguishing the evolution operators of solvable (simple) systems from generic (chaotic, complicated) ones. I will describe the challenges and achievements within this line of pursuit, as well as interrelations between the two apparently different approaches.

Seminar funded by La Caixa: fellowship from the ”la Caixa” Foundation (ID 100010434) with code LCF/BQ/PI24/120

I will describe recent results on applications of spread complexity to local quantum quenches in 2D CFTs. In particular, for holographic CFTs, I will discuss the relation between the rate of growth spread complexity and proper radial momentum in AdS.

The age of gravitational-wave astronomy is now in full swing: For the first time, we gain observational access to the dynamical strong-field regime of the gravitational interaction. Constraining potential deviations from General Relativity (GR) requires reliable waveform predictions, not just in GR, but also when higher-curvature corrections contribute to the dynamics. I will present an overview of recent progress on

(i) classical time evolution in the presence of ghosts,

(ii) mathematical well-posedness of the initial value problem,

(iii) numerical relativity simulations to obtain full nonlinear waveforms.

In combination, this opens up a feasible pathway to use current and future gravitational-wave observations to constrain effective field theories of gravity.

Semi-classical gravity serves as a valuable framework for exploring quantum effects in gravity. However, finding solutions to the semi-classical Einstein equations that consistently incorporate backreaction remains a formidable challenge, limiting our understanding of quantum corrections to black hole physics. In this talk, I present the construction of three-dimensional quantum black holes —exact solutions that emerge within holographic end-of-the-world branes. These spacetimes arise from an induced higher-derivative theory of gravity, consistently coupled to a large-c conformal field theory with an ultraviolet cutoff, capturing all-order effects in semi-classical backreaction. Remarkably, these quantum-corrected black holes remain well above the Planck scale, making them robust against full-fledged quantum gravitational effects. I discuss their geometric and thermodynamic properties across various exact solutions and highlight key applications of these constructions.