The preferred model for many years for studying the evolutionary past of species has been a phylogenetic tree. Such structures are essentially graphs such as the one above whose tips represent species. The closeness of two species in such a tree is then used as a measure for the evolutionary relatedness. Different approaches exist to build such trees from biological data. Noise in the data can affect the correctness of such a tree and to get a better understanding of how small changes in a tree can affect the conclusions we draw from them, we also studied neighborhoods of trees  in a more recent project. This also included the development of a parsimony-based measure for trees .
Given the many data sets that have been analyzed over the years, one approach to build the 'tree of life' is to somehow combine the evolutionary insights that have resulted from them with the new findings from NGS generated data sets. This is a formidable task in which many challenges have to be overcome one of which is the problem of partial information. In a series of papers [3-7] we recently study this problem for the case that the information on which the overall tree is built is given in terms of distances between species.
Branching processes underlay population dynamics and are an important aspect of modelling evolution. In a recent project, we analysed age-dependent branching processes, where we developed techniques to identify complete population-size-age-structure probability densities .
- Bastkowski, S., Moulton, V., Spillner, A., Wu, T., Neighborhoods of trees in circular orderings, Bulletin of Mathematical Biology, in press.
- Moulton, V., Wu, T., A parsimony-based metric for phylogenetic trees, Advances in Applied Mathematics, In press.
- Dress, A.W.M., Huber, K.T., Steel, A., Lassoing' a phylogentic tree I: Basic properties, shellings and covers, Journal of Mathematical Biology, 65(1), 2012, 77-105.
- Popescu, A.-A., Huber, K.T., Lassoing and corralling rooted phylogenetic trees, Bulletin of Mathematical Biology, 75(3), 2013, 444-465.
- Dress, A.W.M., Huber, K.T., Steel, A matroid associated with a phylogenetic tree, Discrete Mathematics & Theoretical Computer Science, 16(2), 2014, 41-56.
- Huber, K.T., Steel, M., Reconstructing fully-resolved trees from triplet cover distances. The Electronic Journal of Combinatorics. 21(2), 2014, #P2.15.
- Huber, K.T., Kettleborough, G., Distinguished minimal topological lassos, Siam Journal on Discrete Mathematics, 29(2), 2015, 940-961.
- Greenman, C., Chou, T., A kinetic theory for age-structured stochastic birth-death processes (www.arxiv.org/abs/1506.02111)