Statistical models are used in phylogenetics to recover the evolution of species. These models are generally highly complicated and the inference involves using some fragile numerical algorithms. The proposed project aims at an in-depth study of phylogenetic models from a mathematical point of view. We find it convenient to take a perspective of algebraic statistics, which is an emerging field focused on solving statistical inference problems using concepts from algebraic geometry. It allows us to identify discrete statistical models with geometric objects. A new approach we propose is based on the analysis of these models in spaces expressed by moments. Using this approach we managed to obtain a good understanding of these models in the multivariate binary case. In this project want to generalise this for arbitrary discrete data. Similarly as in the binary case we want to obtain the full set of defining polynomial equations and inequalities. In addition, we want to understand identifiability of these models and various group acting on it. This concepts are very important from practical point of view as they will enable us to enhance the existing inference procedures.This objective has two important extensions, one motivated by biology and the other by algebraic geometry. First, we want to propose an alternative, simple and robust method to model evolution in biology. We can do it using our understanding of the geometry of this models and links to the underlying evolutionary process. Second, we want to obtain a better understanding of certain classical projective varieties.Algebraic statistics and especially its applications in computational biology have been a very important part of the research activity at UC Berkeley. The outgoing part of the program will allow the applicant to obtain a relevant training needed to succeed with the proposed project.