@article{bernstein_long_2017, title={L-INFINITY OPTIMIZATION TO LINEAR SPACES AND PHYLOGENETIC TREES}, volume={31}, ISSN={["1095-7146"]}, DOI={10.1137/16m1101027}, abstractNote={Given a distance matrix consisting of pairwise distances between species, a distance-based phylogenetic reconstruction method returns a tree metric or equidistant tree metric (ultrametric) that best fits the data. We investigate distance-based phylogenetic reconstruction using the $l^\infty$-metric. In particular, we analyze the set of ultrametrics and tree metrics $l^\infty$-closest to an arbitrary dissimilarity map to determine its dimension and the tree topologies it represents. In the case of ultrametrics, we decompose the space of dissimilarity maps on three elements and on four elements relative to the tree topologies represented. Our approach is to first address uniqueness issues arising in $l^\infty$-optimization to linear spaces. We show that the $l^\infty$-closest point in a linear space is unique if and only if the underlying matroid of the linear space is uniform. We also give a polyhedral decomposition of $\mathbb{R}^m$ based on the dimension of the set of $l^\infty$-closest points in a linea...}, number={2}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Bernstein, Daniel Irving and Long, Colby}, year={2017}, pages={875–889} }
 @article{bernstein_ho_long_steel_st john_sullivant_2015, title={BOUNDS ON THE EXPECTED SIZE OF THE MAXIMUM AGREEMENT SUBTREE}, volume={29}, ISSN={["1095-7146"]}, DOI={10.1137/140997750}, abstractNote={We prove polynomial upper and lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and Yule-Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.}, number={4}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Bernstein, Daniel Irving and Ho, Lam Si Tung and Long, Colby and Steel, Mike and St John, Katherine and Sullivant, Seth}, year={2015}, pages={2065–2074} }
 @article{long_sullivant_2015, title={Identifiability of 3-class Jukes-Cantor mixtures}, volume={64}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2014.12.003}, abstractNote={We prove identifiability of the tree parameters of the 3-class Jukes–Cantor mixture model. The proof uses ideas from algebraic statistics, in particular: finding phylogenetic invariants that separate the varieties associated to different triples of trees; computing dimensions of the resulting phylogenetic varieties; and using the disentangling number to reduce to trees with a small number of leaves. Symbolic computation also plays a key role in handling the many different cases and finding relevant phylogenetic invariants.}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Long, Colby and Sullivant, Seth}, year={2015}, month={Mar}, pages={89–110} }
 @article{long_sullivant_2015, title={Tying up loose strands: Defining equations of the strand symmetric model}, volume={6}, DOI={10.18409/jas.v6i1.34}, abstractNote={The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show that the set of known phylogenetic invariants for the general strand symmetric model of the three leaf claw tree entirely defines the ideal. This knowledge allows one to determine the vanishing ideal of the general strand symmetric model of any trivalent tree. Our proof of the main result is computational. We use the fact that the Zariski closure of the strand symmetric model is the secant variety of a toric variety to compute the dimension of the variety. We then show that the known equations generate a prime ideal of the correct dimension using elimination theory. }, number={1}, journal={Journal of Algebraic Statistics}, author={Long, C. and Sullivant, S.}, year={2015}, pages={17–23} }