[Binghamton Geometry Topology Seminar] Geometry and Topology seminar: Michael Ching

[Cary Malkiewich]

Hi everyone,

This week we are very pleased to have Michael Ching (Amherst College)
speaking in the geometry and topology seminar. The title and abstract are
below. As usual, the talk is at 2:50pm in WH-100E. Please let me know if
you would like to come to the dinner after!

Cheers,
Cary

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Title: Tangent ∞-categories and Goodwillie calculus
Abstract: The theory of Goodwillie calculus uses an analogy between
homotopy theory and differential geometry to make systematic decompositions
of homotopy-theoretic functors into “polynomial” pieces. For example, a
suitable functor (say from the category of topological spaces to itself)
has a “Taylor tower”, a sequence of polynomial approximations that, in good
cases, can be used to recover information about the original functor.

Cockett and Cruttwell (following Rosický) have developed an abstract
framework which axiomatizes the categorical properties of the tangent
bundle functor on the category of smooth manifolds, and includes other
“tangent bundle” constructions in areas such as algebraic geometry and
synthetic differential geometry, among others.

In this talk I will describe joint work with Kristine Bauer and Matthew
Burke that puts Goodwillie calculus into this same “tangent category”
framework (or, rather, its ∞-categorical counterpart) and thus makes
precise the hitherto informal analogy developed by Goodwillie. I will
argue, in particular, that the Taylor tower construction can be recovered
in a formal way from this underlying tangent structure. This work sets the
scene for importing other concepts from differential geometry, such as
connections and curvature, into homotopy theory.
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