Abstract: How many tangent lines does a plane curve have at a given point? And how can we effectively solve polynomial equations with integral coefficients? These seemingly unrelated problems from algebraic geometry and number theory can both be solved using tools from abstract algebra, through the notion of Rees algebras.
In this talk, I will first discuss the fundamental role of Rees algebras in the study of commutative rings and singularities of algebraic varieties. I will then focus on the problem of finding the implicit equations of Rees algebras, discussing some algebraic and combinatorial methods.