2023/2024
Research Seminar "An Introduction to Goodwillie Calculus"
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Alexei Gorinov
Language:
English
ECTS credits:
6
Contact hours:
72
Course Syllabus
Abstract
"Given a smooth function $f:\mathbb{R}^n\to \mathbb{R}$, one can construct its Taylor series at the origin. The homogeneous components of the series are homogeneous polynomials with coefficients obtained by taking the derivatives of $f$ at $0$. Moreover, under some assumptions on $f, x\in\mathbb{R}^n$ and $k$ the value of the degree $k$ Taylor polynomial at $x$ approximates $f(x)$ reasonably well. Remarkably, under some hypotheses which are not too restrictive, all these statements have analogues for functors from (certain subcategories of) topological spaces to topological spaces. For example, the linear approximation of the functor of embeddings in a given manifold $M$ turns out to be the functor of immersions in $M$.
These phenomena are described by the Goodwillie calculus. Our goal is to understand several examples in some detail with a focus on the embedding functor, and then to discuss the general formalism behind these constructions."