Generalized Geometry has become a powerful tool to study solutions of supergravity and their transformation under duality symmetries of string and M-theory. Its success originates from unifying all bosonic symmetries, consisting of diffeomorphisms and form-field gauge transformations. This idea can be pushed further by including supersymmetry to obtain supergeneralized geometry. After reviewing its salient features, I show how it captures fermionic T-dualities and allows to describe integrable superstrings (called eta- and lambda-deformation) without the ambiguities one faces in generalized geometry.

We present a bicrossproduct basis for the $\varrho$-Poincaré quantum group, comparing the structure with the previously known one obtained through a Drinfel'd twist. We consider, then, a new $\star$-product associated with the time-to-the-right ordering of noncommutative exponentials in the $\varrho$-Minkowski spacetime, showing how it is related to the bicrossproduct basis and some of its relevant properties.

Generalized Geometry has become a powerful tool to study solutions of supergravity and their transformation under duality symmetries of string and M-theory. Its success originates from unifying all bosonic symmetries, consisting of diffeomorphisms and form-field gauge transformations. This idea can be pushed further by including supersymmetry to obtain supergeneralized geometry. After reviewing its salient features, I show how it captures fermionic T-dualities and allows to describe integrable superstrings (called eta- and lambda-deformation) without the ambiguities one faces in generalized geometry.

A major application of extended geometries in supergravity is the construction of consistent truncations. Remarkably, all this construction's ingredients also appear in the context of generalised T- and U-dualities. After a quick review of both concepts, I will discuss a conjectured correspondence between consistent truncations and generalised dualities in string and M-theory. We will prove that all known generalised dualities give rise to consistent truncations. Moreover, we will see evidence that after incorporating higher derivative correction, all consistent truncations might be based on generalised dualities.

Poisson-Lie groups emerge naturally in the classical limit of quantum groups. Besides their important role in mathematics, they are also central to the phenomena of T-duality in physics. Originally, T-duality arises in the context of string theory but over that last decade it has also become an essential tool to study integrable two-dimensional $\sigma$-models. While this approach works very well in the classical regime, we only started to understand its implications for quantum corrections last year. After giving an introduction to Poisson-Lie T-duality and integrable $\sigma$-models, I will discuss these recent developments and their implications.

Poisson-Lie T-duality was originally introduced to identify the dynamics of closed strings probing different target spaces. But nowadays it also has become a crucial ingredient in the construction of integrable, two-dimensional $\sigma$-models. After reviewing this intriguing connection from a worldsheet perspective, I will switch to the target space. There we are going to see that Poisson-Lie T-duality naturally appears in the context of gauged SUGRAs and consistent truncations which help to construct new AdS vacua.

Both constituents of my title are well established areas of research with a wide range of applications. Unfortunately, in the current standard formulation of DFT only the tiny subset of PL symmetry which gives rise to abelian T-duality is manifest. In my talk, I present an altered DFT version, DFT on group manifolds, which make the full PL symmetry manifest. We discuss both the NS/NS and R/R sector of the theory. Later allows us to derive the transformation rules for R/R field strengths under full PL T-duality for the first time. If time permits, I will also comment on applications in integral deformations and the extension of the framework to also capture dressing cosets.

A formulation of Double Field Theory is presented which makes Poisson-Lie T-duality manifest. It allows to identify the doubled space with a Drinfeld double and provides a powerful tool to extract the transformation of the metric, B-field, dilaton and R/R potentials under Poisson-Lie T-duality.

I review how integrability allows us to explore the planar limit of the AdS/CFT correspondence for arbitrary values of the t'Hooft coupling. In string theory integrability of the 2D $\sigma$-model is closely related to Poisson-Lie Symmetry. Double Field Theory can be used to make this symmetry manifest and therewith provides a new tool to study the implications for the gravity side of the correspondence.

A formulation of Double Field Theory is presented which makes Poisson-Lie T-duality manifest. It allows to identify the doubled space with a Drinfeld double and provides a powerful tool to extract the transformation of the metric, $B$-field, dilaton and R/R potentials under Poisson-Lie T-duality.

Theories of class S are 4D N=2 SCFTs which result from the compactification of 6D N=(2,0) SCFTs on punctured Riemann surfaces. They provide a geometric perspective on S-duality and are essential in the AGT correspondence. I will present the first step in extending this construction to N=1. To this end, we discuss the punctures relevant in the compactification of the world-volume theory of M5-branes probing an ADE-singularity. They are closely related to the time evolution of a dynamical system and exhibit a surprisingly rice and complex structure compared to N=2.

Theories of class S are 4D N=2 SCFTs which result from the compactification of 6D N=(2,0) SCFTs on punctured Riemann surfaces. They provide a geometric perspective on S-duality and are essential in the AGT correspondence. I will present the first step in extending this construction to N=1. To this end, we discuss the punctures relevant in the compactification of the world-volume theory of M5-branes probing an ADE-singularity. They are closely related to the time evolution of a dynamical system and exhibit a surprisingly rich and complex structure compared to N=2.

Besides propagating in target space like a point particle, a closed string is also able to wind around non-contractible circles. A direct consequence thereof is T-duality. In the textbook example, it identifies the closed string dynamics on a large and a small circle by interchanging its winding and momentum modes. Patching a background by such dualities clearly goes beyond the notion of conventional geometry. However, there are extensive efforts to embed them into a framework called string geometry. It provides access to a vast number of new backgrounds with intriguing phenomenology, like e.g. the possibility to obtain de Sitter vacua. Double Field Theory (DFT) is the most promising approach to describe these backgrounds and their properties. But still, it is closely related to the torus. I will present a theory based on Closed String Field theory starting from a Wess-Zumino-Witten model which goes beyond the torus. It plays an important role in clarifying the recent confusion about different constraints in DFT. Furthermore it allows to uplift a large class of new backgrounds to string theory. These backgrounds are not T-dual to any geometric ones.

One of the biggest unsolved mystery in physics is understanding all fundamental forces in nature in one unified framework. It kept generations of physicists busy, including famous ones like Albert Einstein, but we still don't know a definite answer. I will explain why this problem is so hard, but at the same time extremely important to understand the origins of our universe and its future. On the way, we will meet the who is who of physics' hall of fame and bust the myth of the lonely professor working in complete isolation from the world.

Over the last years, we could witness stunning experimental evidence for the existence of black holes. At the singularity in their centre, gravitational forces are so strong that they even rip spacetime apart. Our conventional approach to gravity, general relativity, fails here and must be replaced by a new framework. Substituting point particles by extended strings is a potential cure. However, it is currently not feasible to obtain direct conclusions from this idea and explain what happens at the singularity. To rectify this situation, I show how four different arenas of string theory (dualities, geometry, supergravity, and integrability) are secretly governed by one central concept, Generalised Homogeneous Spaces (GHS). They are the string's version of homogeneous spaces with a remarkable variety of new properties. I summarise my current efforts to incorporate their quantum corrections due to the extended nature of the string and how they eventually should resolve singularities in black holes or cosmology.

Symmetries are one of the major tools to understand the structure of physical theories. But even the most powerful symmetry is useless if it is hidden and therefore not accessible in calculations. Prominent examples are S-, T- and U-dualities of superstrings and branes. We know that they unify the five perturbative superstring theories and M-theory into a single framework, but their imprints on the low-energy effective supergravity theories are subtle and easy to miss. A framework which addresses this issue is (exceptional) generalized geometry. Although thought as a natural extension of geometry to an extended tangent bundle, it still lacks fundamental objects of differential geometry like a Riemann tensor. After a short review of the most important differences between generalized and standard differential geometry, I will present the underlying cause for this trouble and present a proposal for a solution which gives a new, geometric perspective on duality symmetries in supergravity. This approach will not just resolve some old puzzles but it also has direct applications, leading to a much broader notion of dualities in supergarvity that can be used to generate new solutions using a tool known as consistent truncations.

A major application of extended geometries in supergravity is the construction of consistent truncations. Remarkably, all this construction's ingredients also appear in the context of generalised T- and U-dualities. After a quick review of both concepts, I will discuss a conjectured correspondence between consistent truncations and generalised dualities in string and M-theory. We will prove that all known generalised dualities give rise to consistent truncations. Moreover, we will see evidence that after incorporating higher derivative correction, all consistent truncations might be based on generalised dualities.

Over the last years, we could witness stunning experimental evidence for the existence of black holes. At the singularity in their centre, gravitational forces are so strong that they even rip spacetime apart. Our conventional approach to gravity, general relativity, fails here and must be replaced by a new framework. Substituting point particles by extended strings is a potential cure. However, it is currently not feasible to obtain direct conclusions from this idea and explain what happens at the singularity. To rectify this situation, I show how four different arenas of string theory (dualities, geometry, supergravity, and integrability) are secretly governed by one central concept, Generalised Homogeneous Spaces (GHS). They are the string's version of homogeneous spaces with a remarkable variety of new properties. I summarise my current efforts to incorporate their quantum corrections due to the extended nature of the string and how they eventually should resolve singularities in black holes or cosmology.

I will discuss a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of 2-dimensional $\sigma$-models giving rise to integrability-preserving transformations. Assuming a generalised Scherk-Schwarz ansatz, in fact, the two problems admit essentially the same algebraic formulation, emerging from an underlying double Lie algebra. After presenting the derivation of the classification, we discuss in detail the relation to modified supergravity and the additional conditions to recover the standard (unmodified) supergravity. Starting from a master equation - that encodes all the possible continuous deformations allowed in the family of solution-generating techniques - I show that these are classified by the Lie algebra cohomologies $H^2(\mathfrak(h),\mathbb{R})$ and $H^3(\mathfrak{h},\mathbb{R})$ of the maximally isotropic subalgebra $\mathfrak{h}$ of the double Lie algebra. Finally, I introduce a non-trivial example, the integrable bi-Yang-Baxter-Wess-Zumino model.

Higher derivative corrections in gravity and gauge theories are key to approach some of the big questions in theoretical physics, like the evolution of the universe or the resolution of black hole singularities. I explain why they are so important but also why they are very hard to obtain. Dualities, which for example arise naturally in string theory, have recently provided a powerful tool to get a better handle on these corrections. Unfortunately the most well understood duality in this context, abelian T-duality, is very restrictive. There is the more versatile framework of generalised T-dualities with a wide range of applications, but its relevance for higher derivative corrections just started to emerge last year. We discuss these new developments and point out some of their applications.

Symmetries play a central role in theoretical physics. But, we have to understand how they are realized in order to benefit from them in calculations. An interesting example along this line is a large class of two-dimensional quantum field theories, called $\sigma$-models. I will show that they exhibit a hidden, continues symmetry which is governed by the Lie group O($D$,$D$). It tightly constrains both, the classical and the quantum regime of these models. As an example, I will demonstrate how it affects the one- and two-loop $\beta$-functions. The resulting insights are indispensable to compute the RG flows of integrable $\sigma$-models.

Abstract: There is an intriguing connection between integrable $\sigma$-models and Poisson-Lie (PL) symmetry. As I will review, the latter is manifest in the $\mathcal{E}$-model, rendering it a powerful tool to construct a variety of integrable models which recently have been identified with surface defects in 4d Chern-Simons theory. Manifest PL symmetry facilitates computations which would be forbiddingly complex without it. Important examples, which I will discuss in detail, are one and two-loop beta-functions. We will see that they underpin a deep connection between classical integrability and the corresponding quantum regime.

S- and (abelian) T-duality play a central role in string theory, but their scope is limited to highly constrained spacetimes. Generalised T-dualities, which include non-abelian and Poisson-Lie T-duality, apply to a significantly larger class of target spaces with a wide range of applications. Classically they are on an equal footing with abelian T-duality, but their quantum corrections are much more mysterious and mostly unexplored. I will review the main problems which have to be solved to make progress in this direction. Afterwards, I demonstrate how recently explored connections between generalised T-dualities, double field theory, and consistent truncations allow to prove that two-loop RG flows are preserved under these dualities. We will discuss the implications of this result and emphasise how the intriguing mathematical structures that govern all the current
applications influence quantum corrections in a highly non-trivial way.

Abelian T-duality is well appreciated among string theorists. But it forms only a very small part of a much larger family of generalised T-dualities which however attracted much less attention until recently. The main reason for their marginalization is that in contrast to abelian T-duality they are not a symmetry of full string theory, i.e. they do not hold for the full $\alpha’$ and $g_s$ expansion. Still they turned out to be essential in the recent quest for integrable $\sigma$-modes and also in the construction of explicit solutions to supergravity. Thus, I want to present you the idea behind generalised T-dualities on the worldsheet and show how they are connected to integrability and supergravity.

Generalised Scherk-Schwarz reductions are a powerful tool to construct consistent truncations in Double and Exceptional Field Theories. Recently, it turned out that they are also closely related to Poisson-Lie T-duality. However, the most general form of Poisson-Lie T-duality, the dressing coset construction, can not be implemented in terms of a generalised Scherk-Schwarz ansatz. I will show that implementing it in generalised geometry leads to a natural extension of the generalised Scherk-Schwarz ansatz which comes with many new features: 1) Partial or full breaking of SUSY which allows to find many new examples of generalised Kähler or Calabi-Yau Manifolds. 2) Singular backgrounds with localised sources. 3) Localised vector multiplets while still resulting in consistent truncations.

A formulation of Double Field Theory is presented which makes Poisson-Lie T-duality manifest. It allows to identify the doubled space with a Drinfeld double and provides a powerful tool to extract the transformation of the metric, B-field, dilaton and R/R potentials under Poisson-Lie T-duality.

I review how integrability allows us to explore the planar limit of the AdS/CFT correspondence for arbitrary values of the t'Hooft coupling. In string theory integrability of the 2D $\sigma$-model is closely related to Poisson-Lie Symmetry. Double Field Theory can be used to make this symmetry manifest and therewith provides a new tool to study the implications for the gravity side of the correspondence.

Poisson-Lie T-duality is a generalization of traditional non-abelian T-duality and enjoys, at least at the classical level, all features of abelian T-duality. Recently, it got a lot of attention in the context of integrable $\eta$- and $\lambda$-deformations. I review this intriguing framework, outline its applications and show that it naturally admits a double field theory description. Latter has many applications in the realm of abelian T-duality but did not really touch Poisson-Lie T-duality until now. I present an extension of the current double field theory formulation which makes Poisson-Lie T-duality manifest. It gives rise to various new applications and also introduces powerful new mathematical structures, like Drinfeld doubles and quantum groups, in the theory.

While only three spheres, $S^1$, $S^3$ and $S^7$, are parallelizable, it was recently shown that all spheres are generalized parallelizable. In addition to its beautiful mathematical structure, this extended notion of parallelizability allows to identify certain maximal gauged supergravities as consistent truncations of 10/11D supergravity. In this talk, I present the first systematic construction of generalized parallelize spaces and demonstrate how this string theory inspired concept captures T-duality in a natural way.

Dualities are at the heart of string theory. They identify closed string theories in different target spaces (T-duality) and at inverse values of the string coupling (S-duality). To exploit their full potential, it would be great to make them manifest already at the level of the low energy effective action. Double Field Theory (DFT) exactly follows this idea for T-duality on a torus. Incorporating also S-duality in this framework, Exceptional Field Theory (EFT) is born. I review these two approaches and show how they are expect to give new insights in subject like flux compactifications, moduli stabilization and cosmology. I emphasize their weak spots and present my humble contribution to deal with some of these problems.

Only a fraction of the vast landscape of vacua in string theory is accessible from supergravity. Stringy geometries, whose properties are governed by the extended nature of the string, are beyond its scope. Double Field Theory (DFT), which makes T-duality on tori manifest at the level of an effective field theory, provides a convenient tool to explore vacua beyond the supergravity regime. Despite the substantial progresses made in this direction, there are still open questions and technical ambiguities. Some of them can be solved by extending the derivation of DFT from a torus to more general non-abelian group manifolds. After a short review of the existing formalism, I derive such a theory, called DFT$_{\mathrm{WZW}}$, using Closed String Field Theory applied to Wess-Zumino-Witten models. Further, I discuss its connection to half maximal gauged supergravities in lower dimensions. Even for those of them who can not be generated by a supergravity compactification, DFT$_{\mathrm{WZW}}$ provides a higher dimensional origin.

Besides propagating in target space like a point particle, a closed string is also able to wind around non-contractible circles. A direct consequence thereof is T-duality. In the textbook example, it identifies the closed string dynamics on a large and a small circle by interchanging its winding and momentum modes. Patching a background by such dualities clearly goes beyond the notion of conventional geometry. However, there are extensive efforts to embed them into a framework called string geometry. It provides access to a vast number of new backgrounds with intriguing phenomenology, like e.g. the possibility to obtain de Sitter vacua. Double Field Theory (DFT) is the most promising approach to describe these backgrounds and their properties. But still, it is closely related to the torus. I will present a theory based on Closed String Field theory starting from a Wess-Zumino-Witten model which goes beyond the torus. It plays an important role in clarifying the recent confusion about different constraints in DFT. Furthermore it allows to uplift a large class of new backgrounds to string theory. These backgrounds are not T-dual to any geometric ones.