Quantum cosmology has gradually emerged as the focus of devoted research, mostly within the second half of last century. As we entered the 21st century, the subject is still very much alive. The outcome of results and templates for investigation have been enlarged, some very recent and fascinating. Hence this book, where the authors bequeath some of their views, as they believe this current century is the one where quantum cosmology will be fully accomplished.Though some aspects are not discussed (namely, supersymmetry or loop structures), there are perhaps a set of challenges that in the authors' opinion remain, some since the dawn of quantum mechanics and applications to cosmology. Others could have been selected, at the readers' discretion and opinion. The authors put herewith a chart and directions to explore, some of which they have worked on or aimed to work more, in the twilight of their current efforts. Their confidence is that someone will follow in their trails, venturing in discovering the proper answer, by being able to formulate the right questions beforehand. The authors' shared foresight is that such discoveries, from those formulations, will be attained upon endorsing the routes within the challenges herewith indicated.

The aim of this book is to convey a broad perspective of what quantum cosmology is and how it can be a relevant field of research in the 21st century. A series of challenges (or research directions to follow) are provided, in the form of chapters, as the means of forwarding some of the main elements of quantum cosmology. This is therefore a more 'practical' oriented book, where a reasonable amount of technical features are presented in a compact appendix.

Following the fundamental insights from quantum mechanics and general relativity, geometry itself should have a quantum description; the search for a complete understanding of this description is what drives the field of quantum gravity. Group field theory is an ambitious framework in which theories of quantum geometry are formulated, incorporating successful ideas from the fields of matrix models, ten-sor models, spin foam models and loop quantum gravity, as well as from the broader areas of quantum field theory and mathematical physics. This special issue collects recent work in group field theory and these related approaches, as well as other neighbouring fields (e.g., cosmology, quantum information and quantum foundations, statistical physics) to the extent that these are directly relevant to quantum gravity research.

We read in order to know we are not alone, I once heard, and perhaps it could also be suggested that we write in order not to be alone, to endorse, to promote continuity. The idea for this book took about ten years to materialize, and it is the author’s hope that its content will constitute the beginning of further explorations beyond current horizons. More speci cally, this book appeals to the reader to engage upon and persevere with a journey, moving through the less well explored territories in the evolution of the very early universe, and pushing towards new landscapes. P- haps, during or after consulting this book, this attitude and this willingness will be embraced by someone, somewhere, and this person will go on to enrich our quantum cosmological description of the early universe, by means of a clearer supersymm- ric perspective. It is to these creative and inquisitive ‘young minds’ that the book is addressed. The reader will not therefore nd in this book all the answers to all the problems regarding a supersymmetric and quantum description of the early universe, and this remark is substantiated in the book by a list of unresolved and challenging problems, itself incomplete.

"It would be hard to imagine a better guide to this difficult subject." -- Scientific American In Three Roads to Quantum Gravity, Lee Smolin provides an accessible overview of the attempts to build a final "theory of everything." He explains in simple terms what scientists are talking about when they say the world is made from exotic entities such as loops, strings, and black holes and tells the fascinating stories behind these discoveries: the rivalries, epiphanies, and intrigues he witnessed firsthand. "Provocative, original, and unsettling." -- The New York Review of Books "An excellent writer, a creative thinker." -- Nature

We read in order to know we are not alone, I once heard, and perhaps it could also be suggested that we write in order not to be alone, to endorse, to promote continuity. The idea for this book took about 10 years to materialize, and it is the author’s hope that its content will constitute the beginning of further explorations beyond current horizons. More speci cally, this book appeals to the reader to engage upon and persevere with a journey, moving through the less well explored territories in the evolution of the very early universe, and pushing towards new landscapes. P- haps, during or after consulting this book, this attitude and this willingness will be embraced by someone, somewhere, and this person will go on to enrich our quantum cosmological description of the early universe, by means of a clearer supersymm- ric perspective. It is to these creative and inquisitive ‘young minds’ that the book is addressed. The reader will not therefore nd in this book all the answers to all the problems regarding a supersymmetric and quantum description of the early universe, and this remark is substantiated in the book by a list of unresolved and challenging problems, itself incomplete.

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