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Most practical studies in Shape Dynamics involve an N-body Newtonian interaction which is described by a homogeneous potential. This property allows one to proof several interesting features like the emergence of an arrow of time. However, more generic interactions are not described by these kind of potentials and introduce additional dimensionful coupling constants. Thus, it is an open question whether more generic interactions can be written in a fully relational manner. By studying the concrete example of the gravitational Weber interaction which is, in a sense, a more realistic theory of gravity, we show that it is possible translate non-Newtonian interactions, which have inhomogeneous potentials and additional coupling constants, into a relational language. This opens the door to study other interactions and may shed light into the relationalization of gravity as described by general relativity.

Inflation is the leading paradigm of the early Universe, according to which the tiny temperature fluctuations observed in the cosmic microwave background (CMB) originate from quantum vacuum fluctuations at very early times. Recent observations show that the Starobinsky potential is favored among the single field inflationary models. However, the calculations that match the data exclude the Planck era. I will explain why this era is important and how using techniques from loop quantum gravity, the effects of this period can be studied. In particular, we find that for a large part of the initial data surface predictions for the power spectrum of the CMB are indistinguishable from predictions neglecting the pre-inflationary era. However, initial conditions exist for which the quantum gravitational corrections to the power spectrum are potentially observable.

Relationalism is the strict disentanglement of physical law from the definition of physical object. This can be formalized in the shape dynamcis postulate that the objective evolution of the universe is described by an "equation of state of a curve in relational configuration space." The application of this postulate to General Relativity implies that gravity is described by an equation of state of a curve on conformal superspace. It turns out that the naive quantization of these equations of state introduces an undesired preferred time parametrization. However, it turns out that one can still describe the quantum evolution of the system as an equation of state of the Bohmian trajectory which remains manifestly parametrization independent. These quantum systems generically develop quasi-isolated bound states (atoms) that can be used as reference systems. It turns out that the system as a whole expands if described in units defined by these atoms. This produces phenomenological effects that are usually ascribed to the presence of a cosmological constant. This "effective cosmological constant" is however unaffected by vacuum energy. I pesent the formal argument for this statement and show this explicitly by remormalizing a scalar field coupled to shape dynamics.

Shape Dynamics(SD) can be derived from principles that differ in significant respects from Einstein's derivation of GR. It requires a spatially closed universe and allows a smaller set of solutions than GR does for this case. There are indications that its solution space can be fully characterized and endowed with a measure. These architectonic features suggest that SD can explain the arrows of time as direct consequences of the law of the universe. They do not require special initial conditions. I will discuss these and other major issues on which SD may cast light. I will also discuss the problems that face SD.

I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multi-scale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for AdS/CFT. Novel holographic features of this tensor network class will be examined, such as an equivalence between the causal cone C[R] and the entanglement wedge E[R] of connected boundary regions R.

We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the Bruhat-Tits tree as a tensor network, we recover (partially) features of the p-adic AdS/CFT dictionary discussed recently in the literature.

According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the Einstein-Hilbert action of a certain region in the dual space-time. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries. By applying Tensor Network Renormalization to the discretized Euclidean path integral that prepares the CFT state, I will justify the recent suggestion by Caputa et al. that the complexity of a path integral is quantified by the Liouville action. The Liouville field specifies the conformal frame in which the path integral is evaluated; in the most efficient / least complexity frame, the Liouville field is closely related to entanglement entropies of CFT2 intervals. Assuming the Ryu-Takayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual space-time. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesic-wise rewriting of the non-linear vacuum Einstein's equations in 3d with a negative cosmological constant. I emphasize that this is very much work in progress; I hope the audience will help me to sharpen the arguments.

I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.

In this talk, I would like to discuss how we can realize the correspondence between AdS/CFT and tensor network in quantum field theories (i.e. the continous limit). As the first approach I will discuss a possible connection between continuous MERA and AdS/CFT. Next I will introduce the second approach based on the optimization of Euclidean path-integral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.

We will review the topic of tensor network renormalization, relate it to real space Hamiltonian flows, and discuss the emergence of matrix product operator algebras as symmetries of the renormalization fixed points. joint work with Matthias Bal, Michael Marien and Jutho Haegeman