Mathematics Colloquia and Seminars

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The integer points within a convex cone form a conical semigroup. When the (pointed) cone is polyhedral, there is a unique inclusion minimal integer generating set known as the Hilbert Basis. This means that every integer point in the polyhedral cone can be represented uniquely as a positive integer combination of the elements of the Hilbert basis. Thus, it is natural to ask: do we preserve any notion of finite generation for conical semigroups when we relax the polyhedral condition? Do we have any analogue to the Hilbert Basis? In this talk, we will introduce the notion of $(R,G)$-finite generation and present recent results showing that the conical semigroup given by the second order cone, $SOC(n)$, is $(R,G)$-finitely generated in dimensions three through ten. Lastly, we will present an analogous result for the conical semigroup given by the cone of positive semidefinite matrices, showing that it is also $(R,G)$-finitely generated. 

 

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