Abstract:
A tangent category consists of a collection of abstract objects equipped with a structure, called a tangent structure, which makes those abstract objects into locally linear geometric spaces.
In my thesis, I introduced and studied a list of geometric theories from the perspective of tangent category theory. In particular, I showed that each (opposite) category of algebras, such as associative, commutative, or Lie algebras, comes equipped with a tangent structure which allows one to regard those algebraic objects as “operadic” affine schemes; I then extended some constructions of algebraic geometry to this context.
In this talk, I would like to introduce you to the wonderful world of tangent category theory, briefly recall the notion of an operad and its algebras, construct the algebraic and the geometric tangent categories of an operad, and, time permitting, briefly discuss some constructions such as vector fields, the slice tangent category, and differential bundles.
Part of the research for this project was done in collaboration with Sacha Ikonicoff (Université de Strasbourg) and Jean-Simon Pacaud Lemay (Macquarie University).
About the speaker:
Dr. Marcello Lanfranchi is one of the winners of this year’s AARMS Doctoral Thesis Award. He received his PhD from Dalhousie University in October 2024 under the supervision of Dr. Geoffrey Crutwell and Dr. Dorette Pronk for a thesis entitled “A tangent category approach to operadic geometry”.
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