Tensor Networks: a new mathematical tool in quantum many body systems

Tensor Networks have emerged as the natural language to understand and classify exotic properties of quantum many-body systems, such as topological order. In this talk, I will start introducing, in a mathematical language, some relevant questions in the study of quantum many body systems. I will then sketch a roadmap to use Tensor Networks to address such questions and will highlight some of the most important theorems proven so far, some of the mathematical techniques used, and some problems for the future.