Tropical geometry and dissimilarity vectors of trees

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I would like to talk about tropical geometry and dissimilarity vectors of trees. At first sight, these two mathematical topics seem to have no relation. However, nice connections are given by tropical Grassmannians. Indeed, for example, the space of n-trees is equal to the tropical Grassmannian G_{2,n}. In recent articles (partially joint work with C. Bocci of U. Siena in Italy), we investigate the relationship between tropical Grassmannians G_{m,n} and m-dissimilarity vectors of n-trees for the case m>2. The motivation comes from phylogenetics. From an alignment of DNA sequences from the genomes of n species, one wants to construct the suitable phylogenetic tree. The so-called distance based approach for doing this, consists of two steps. Firstly, one collapses the data into a dissimilarity matrix and secondly, one searches for the weighted tree which represents this matrix, provided such a tree exists. Therefore, the study of the space of n-trees is of crucial importance.