Mathematics Colloquia and Seminars

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The Hilbert scheme of points on a surface S is a variety that parametrizes finite closed subschemes of S. Even for simple surfaces S, these Hilbert schemes have surprising connections to symmetric functions, representation theory, knot theory, and more. In this talk, I will discuss an approach to study these spaces using polyhedral combinatorics via Newton-Okounkov bodies. I will share polyhedral models for the coordinate rings of several such Hilbert schemes obtained in this way, as well as applications to well-studied questions about effective divisors and Verlinde series associated to these moduli spaces. Some of these results are based on joint work with Eugene Gorsky, Alexei Oblomkov, and Joshua P. Turner.