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An elliptic operator is a type of differential operator that generalizes the notion of a Laplacian and is characterized by its symbol being invertible everywhere except possibly at infinity. These operators are crucial in the study of partial differential equations as they often yield well-posed problems, leading to smooth solutions under appropriate boundary conditions.
A Batalin-Vilkovisky algebra is a graded commutative algebra equipped with a second-order differential operator called the BV operator, which satisfies a specific compatibility condition with the algebra's product. It plays a crucial role in the mathematical formulation of gauge theories and quantization, particularly in the context of the Batalin-Vilkovisky formalism for handling systems with symmetries.
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