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Given two embedded projective varieties X and Y, their join X*Y is the Zariski closure of union of all lines spanned by a point on X and a point on Y. The r-th secant variety of X is the iterated join of a variety with itself: X^{r} = X*X*...*X. While secant varieties are well-known objects in algebraic geometry, they also arise as special cases of (the Zariski closures of) statistical models. I will give an overview of the different ways that these secant varieties arise in statistical contexts and report on some of the new algebraic techniques that are developing to study these varieties. Some of the statistical models that I will mention include mixture models, phylogenetic models, and the factor analysis model.

Job Talk, Refreshments provided.