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A non-singular cosmological bounce in the Einstein frame can only take place if the Null Energy Condition (NEC) is violated. I will explore the constraints imposed by demanding tree level unitarity on a cosmological background in single scalar field theories before focusing on the explicit constraints that arise in P(X) theories. In that context, perturbative unitarity makes it impossible for the NEC violation to occur within the region of validity of the effective field theory but I will show explicitly how unitarity may be restored by involving irrelevant operators that arise at a higher scale.

I will explain how cosmological dynamics emerge from the hydrodynamics of isotropic group field theory condensate states in the Gross-Pitaevskii approximation. The correct Friedmann equations are recovered in the classical limit for some choices of the parameters in the action for the group field theory, and quantum gravity corrections arise in the high-curvature regime causing a bounce which generically resolves the big-bang and big-crunch singularities.

There is a huge body of work in Loop Quantum Cosmology comprising of several thousand journal articles. I will provide an overview of conclusions, focusing on the difficult conceptual and mathematical issues that accompany the notion of a bounce and opening the way for phenomenological implications that will be discussed by Ivan Agullo.

Functional renormalization group techniques based on Wetterichs equation provide a powerful tool for studying the properties of gravity in the quantum regime and its connection to the observable low-energy world. Explicit computations in this framework require the evaluation of functional traces over operator-valued functions. In these lectures I will give a pedagogical introduction to the Universal Renormalization Group Machine, a combinatorial algorithm which allows evaluating such traces in a systematic way. Moreover, I will discuss flow equations for “composite operators” which may pave the route towards developing a microscopic description of quantum spacetime within the Asymptotic Safety approach.

Functional renormalization group techniques based on Wetterichs equation provide a powerful tool for studying the properties of gravity in the quantum regime and its connection to the observable low-energy world. Explicit computations in this framework require the evaluation of functional traces over operator-valued functions. In these lectures I will give a pedagogical introduction to the Universal Renormalization Group Machine, a combinatorial algorithm which allows evaluating such traces in a systematic way. Moreover, I will discuss flow equations for “composite operators” which may pave the route towards developing a microscopic description of quantum spacetime within the Asymptotic Safety approach.

In the last decade various cosmological spacetimes have been quantized using the techniques of loop quantum gravity. To understand singularity resolution and decipher reliable Planck scale physics, development of new numerical methods and usage of high performance computing is critical in loop quantum cosmology. In recent years, these developments have robustly demonstrated resolution of singularities in quantum spacetimes. These methods have provided detailed understanding of the emergence of new physics at Planck scale, and of classicality when spacetime curvature becomes very small.  Further, they have validated an effective spacetime description of the underlying quantum geometry -- a key ingredient of phenomenological predictions in loop quantum cosmology. These lectures will introduce these numerical methods.

In the last decade various cosmological spacetimes have been quantized using the techniques of loop quantum gravity. To understand singularity resolution and decipher reliable Planck scale physics, development of new numerical methods and usage of high performance computing is critical in loop quantum cosmology. In recent years, these developments have robustly demonstrated resolution of singularities in quantum spacetimes. These methods have provided detailed understanding of the emergence of new physics at Planck scale, and of classicality when spacetime curvature becomes very small.  Further, they have validated an effective spacetime description of the underlying quantum geometry -- a key ingredient of phenomenological predictions in loop quantum cosmology. These lectures will introduce these numerical methods.

The gauge/gravity duality relates supersymmetric gauge theories to superstring/M-theory. Powerful Monte Carlo methods developed in lattice QCD can be applied to study the former,and from the simulation data we can extract the quantum gravitational effects. In this talk I will give a brief introduction to lattice gauge theory and supersymmetry on a lattice, and show some applications to quantum gravity via the gauge/gravity duality.