Abstract: How many tangent lines does a plane curve have at a given point? And how does this relate to swapping vertices of a polyhedron in $R^n$? These seemingly independent problems can be both approached using tools from abstract algebra, through the notion of Rees algebras.
In this talk, I will discuss the fundamental role of Rees algebras in the study of commutative rings and of singularities of algebraic varieties. I will then give an overview of how one can use methods from algebraic combinatorics and polyhedral geometry to understand the algebraic properties of Rees algebras.
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