Gröbner degeneration in algebra and geometry

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Abstract: Gröbner bases are a special kind of generating set of an ideal in a polynomial ring. Introduced by Buchberger in 1965, they have played an essential role in the development of computational commutative algebra and algebraic geometry. Moreover, they serve as a valuable tool in theoretical inquiry by allowing us to replace questions about arbitrary affine varieties with questions about closely-related affine varieties defined by monomial ideals, which admit study by topological and combinatorial techniques. We will discuss the recent history of Gröbner degenerations in generalized determinantal settings and then conclude with an application in Schubert calculus.