We explore the emergence of integrable structures in three-dimensional gravity with asymptotically AdS_3 boundary conditions. In this setting, the asymptotic symmetries are governed by the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, which is an infinite family of nonlinear, integrable equations that are characterized by an infinite set of abelian, conserved charges. 

We discuss the explicit construction of the asymptotic symmetries and their canonical charges, which describe an abelian charge algebra. Due to the integrable scheme, we demonstrate that their temperature remains constant, even for non-axisymmetric configurations. This behavior is linked to the presence of hyperelliptic curves related to the integrable models. 

To illustrate the relationship between integrable models and gravity, we present a periodic black hole solution associated with the Korteweg-de Vries (KdV) equation, a reduction of the AKNS hierarchy. The resulting geometry describes a cnoidal black hole, and its thermodynamic properties are encoded in two independent hyperelliptic curves. The talk is based on joint works with M. Cárdenas, K. Lara and M. Pino: Integrable black hole dynamics in the asymptotic structure of AdS3, Integrable Systems and Spacetime Dynamics.

After describing the tiny history of string theory, I will talk about the ads/cmt as a theory to overcome the plank scale mass gap and non-locality induced by the Quantum entanglement of strongly interacting systems. We suggest that condensation of various tensor type can be a rich contents of new experiments. If time allows, I will talk about how string theory gives a drastic contrast with that of usual perturbative field theory in prediction of momentum space topology.

Black holes want to absorb things, often called no-hair theorem. In special situations, however, they do admit stationary hairs outside the event horizon. Both the no-hair theorem and its violation have interesting physical implications. I will explain novel hairy black holes in AdS, their quantum characterization from dual CFT and the implications to the information paradox.

Recent studies show that three-dimensional Einstein gravity with a negative cosmological constant, when written as an SL(2,R)×SL(2,R) Chern–Simons theory, establishes a bridge between general relativity and integrable systems by rewriting the Einstein field equations as an integrable hierarchy known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. Within this framework, different reductions of the hierarchy give rise to familiar 1+1-dimensional integrable equations, among them the sine-Gordon equation.I will discuss the physical motivation for searching for new gravitational configurations guided by this hierarchy, focusing on the sine-Gordon equation due to the structure of its solutions and the possibility of using specific types of periodic solutions to construct asymptotically AdS or near-horizon-type geometries. This talk presents current progress from an ongoing thesis principally based on the articles Integrable Systems and Spacetime Dynamics (Cárdenas et al., 2021) and Integrable Black Hole Dynamics in the Asymptotic Structure of AdS3 (Cárdenas, Correa & Pino, 2025).

Double Field Theory (DFT) has emerged as a comprehensive framework for gravity, presenting a testable and robust alternative to General Relativity (GR), rooted in the O(D, D) symmetry principle of string theory. This talk aims to provide an accessible introduction to DFT, structured in a manner similar to traditional GR courses. Key topics include doubled-yet-gauged coordinates, Riemannian versus non-Riemannian parametrisations of fundamental fields, covariant derivatives, curvatures, and the O(D, D)-symmetric augmentation of the Einstein field equation, identified as the unified field equation for the closed string massless sector. By offering a novel perspective, DFT addresses unresolved questions in GR and enables the exploration of diverse physical phenomena, paving the way for significant future research. Based on a review paper: Gravitational Core of Double Field Theory: Lecture Notes.

In this talk, we explore the emergence of generalized Korteweg-de Vries (KdV) symmetries in the near-horizon region of extremal black holes. We show specific diffeomorphisms resolving the throat region lead to KdV-type asymptotic conditions. These symmetries influence the variational principle and define an infinite hierarchy of KdV Hamiltonians with implications for the path integral formulation. By relating the near-horizon dynamics to the Jackiw-Teitelboim (JT) dilaton equations, we reveal the integrable structure underlying the field equations and interpret the KdV modes as dynamical degrees of freedom associated with axisymmetric gravitational perturbations. Finally, we discuss the quantum perturbation theory of the generalized KdV action and compute the one-loop correction to the partition function.

The Katz vector in Einstein-Hilbert gravity allows to construct a boundary term that solves the Dirichlet problem and yields the correct (finite) mass for static, spherically symmetric black hole solutions. Here we show that this boundary term can be obtained as a dimensional continuation of a transgression form, which depends on two suitably chosen Lorentz connections. After discussing and comparing this result with other regularization methods known in the literature, we generalize the Katz vector to a generic Lovelock theory. We also comment on ongoing research regarding the applicability of this method to more general solutions beyond the static and spherically symmetric case, particularly within the family of asymptotically locally Anti-de Sitter (AlAdS) spacetimes.

The talk illustrates how a generative AI model can be trained to learn the relationship between geometry and quantum field theory, producing Type IIB brane configurations in string theory that realize these field theories and tracking variations of these brane configurations that distinguish gauge theory phases related by duality. We focus on a particular family of 4‑dimensional supersymmetric gauge theories associated with Calabi-Yau geometries, which are realized by brane configurations that depend on the shape of the corresponding mirror curve of the Calabi-Yau. The generative AI model takes the complex‑structure moduli of the Calabi-Yau mirror curve as input and generates the shape of the mirror curve from which we read off the corresponding gauge theory Lagrangian and phase. We illustrate how we can extend this method to gauge theories in different spacetime dimensions, leading to the discovery of a more general family of gauge theory dualities.

We show that de Sitter space can decay into smooth geometries containing self-gravitating BPS vortices. In this setting the matter sector reduces to a set of first-order vortex equations, while the gravitational field is governed by a Liouville equation with a smooth source. The resulting spacetime combines a factor with a deformed two-sphere whose solid-angle deficit is fixed by the vortex number. The Euclidean continuation defines a regular instanton mediating the thermal decay of de Sitter into these defect geometries, and its on-shell action yields the leading decay rate. This provides a fully analytic example of a gravitational instanton involving topological defects in a cosmological background.

We begin with the motivation for the needs and possibilities of going beyond Einstein's Gravity both theoretical and obervational point of view. Then we introduce the Dilaton-Einstein-Gauss-Bonnet (dEGB) gravity as one of the simplest extensions of Einstein’s theory of gravity that incorporates higher-curvature terms. We mention its black holes, pointing out the existence of a minimum mass threshold, below which black holes cannot form—unlike in Einstein’s gravity. We then overview the implications of this framework for cosmological evolution. A key new aspect is the emergence of novel phases at high temperatures. We mention in particular gravitational waves, which allow us to place constraints on the parameters of the dEGB gravity theory.

Krylov complexity has recently emerged as a new paradigm for characterizing quantum chaos in many-body systems. Recent studies have shown that, in quantum chaotic systems, the Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work, we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaos and suggest that its height can serve as an effective order parameter. We investigate the Krylov complexity of thermofield double states in systems with mixed phase space, revealing a clear correlation with the Brody distribution, which interpolates between Poisson and Wigner level statistics. Our analysis spans two-dimensional random matrix models featuring (i) GOE-Poisson and (ii) GUE-Poisson transitions, and extends to higher- dimensional systems, including a stringy matrix model (GOE-Poisson) and the mass-deformed SYK model (GUE-Poisson). These results establish Krylov complexity as a powerful diagnostic of quantum chaos, highlight its interplay with level statistics in mixed phase systems, and offer deeper insights into the general properties of quantum chaotic systems.

We investigate the bulk reconstruction of AdS black hole spacetime emergent from quantum entanglement within a machine learning framework. Utilizing neural ordinary differential equations alongside Monte-Carlo integration, we develop a method tailored for continuous training functions to extract the general isotropic bulk metric from entanglement entropy data. To validate our approach, we first apply our machine learning algorithm to holographic entanglement entropy data derived from the Gubser-Rocha and superconductor models, which serve as representative models of strongly coupled matters in holography. Our algorithm successfully extracts the corresponding bulk metrics from these data. Additionally, we extend our methodology to many-body systems by employing entanglement entropy data from a fermionic tight-binding chain at half filling, exemplifying critical one-dimensional systems, and derive the associated bulk metric. We find that the metrics for a tight-binding chain and the Gubser-Rocha model are similar. We speculate this similarity is due to the metallic property of these models.