This is an introduction to loop quantum cosmology (LQC) reviewing mini- and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self-consistent algorithm, working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra. It requires inclusion of new fields in the dynamics and the system is free of gauge redundancies. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. The dynamical realization of the non-semi-simple infinite dimensional symmetry algebra at the level of charge algebra is exact and free from central terms.

Carrollian field theories at the classical level possess an infinite number of space-time symmetries, namely the supertranslations. In this article, we inquire whether these symmetries for interacting Carrollian scalar field theory survive in the presence of quantum effects. For interactions polynomial in the field, the answer is in the affirmative. We also study a renormalization group flow particularly tailored to respect the manifest Carroll invariance and analyze the consequences of introducing Carroll-breaking deformations. The renormalization group flow, with perturbative loop-level effects taken into account, indicates a new fixed point apart from the Gaussian ones.

We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probabilty $P\left(l\right)\mathrm{}\sim l^{-\delta }$ in addition to nearest neighbor connections. This system has been shown to exhibit small-world behavior for $\delta <2,$ above which its behavior is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at $\delta =1.\mathrm{}$ The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicates that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behavior of a single quantity, the average bond length. The phase transitions in all dimensions are nontrivial in nature.