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The fundamental idea of deformation theory is to slightly modify an algebraic or a geometrical object so that the new one is still of the same kind of the original one, but its properties are different.
For example, the deformation of an affine scheme corresponds to slightly changing the relations that define the corresponding coordinate ring.
In this talk, we would like to present some interesting connections between algebraic deformation theory and tangent categories. We will show how to relate infinitesimal deformations of algebras to vector fields of a suitable tangent category. To explore this idea, we will make use of the theory of operads, by showing that the category (and also it's opposite one) of all algebraic operads over a fixed commutative and unital ring R, form a tangent category, naturally included in the tangent category of tangent monads over a tangent category.
As part of my PhD research, this work is in collaboration with my supervisors Dorette Pronk and Geoffrey Cruttwell.
I would like also to thank Geoff Vooys for the useful discussions I had with him on deformation theory.
Updated Jan 3, 2023 by Frank Fu