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Speaker: Donald Richards (Penn State) 

Title: Hyperdeterminantal total positivity

Abstract: This talk will cover some recent joint research with K. W. Johnson. 

Let m be a positive integer, and let K be real-valued function defined on 2m-dimensional Euclidean space.  We define the concept of hyperdeterminantal total positivity for the kernel K, thereby generalizing the classical concept of total positivity. We construct examples of hyperdeterminantal totally positive kernels, extending in particular the fundamental classical example, K(x,y) = exp(xy), x,y ∈ ℝ, of a totally positive kernel.  By applying a hyperdeterminantal Binet-Cauchy argument, a generalization of Karlin's Basic Composition Formula is derived and then applied to construct numerous additional examples of hyperdeterminantal totally positive kernels.  Further generalizations of the concept of hyperdeterminantal total positivity by means of the theory of finite reflection groups are described, and some open problems are posed.