Fri, 16 Feb 2024

15:00 - 16:00

L5

Morse Theory for Tubular Neighborhoods

Antoine Commaret

(INRIA Sophia-Antipolis)

Abstract

Given a set $X$ inside a Riemaniann manifold $M$ and a smooth function $f : X -> \mathbb{R}$, Morse Theory studies the evolution of the topology of the closed sublevel sets filtration $X_c = X \cap f^{-1}(-\infty, c]$ when $c \in \mathbb{R}$ varies using properties on $f$ and $X$ when the function is sufficiently generic. Such functions are called Morse Functions . In that case, the sets $X_c$ have the homotopy type of a CW-complex with cells added at every critical point. In particular, the persistent homology diagram associated to the sublevel sets filtration of a Morse Function is easily understood. 
 

In this talk, we will give a broad overview of the classical Morse Theory, i.e when $X$ is itself a manifold, before discussing how this regularity assumption can be relaxed. When $M$ is a Euclidean space, we will describe how to define a notion of Morse Functions, first on sets with positive reach (a result from Joseph Fu, 1988), and then for any tubular neighborhood of a set at a regular value of its distance function, i.e when $X = \{ x \in M, d_Y(x) \leq \varepsilon \}$ where $Y \subset M$ is a compact set and $\varepsilon > 0$ is a regular value of $d_Y$ the distance to $Y$ function.
 

 

If needed, here are three references :
 

Morse Theory , John Milnor, 1963
 

Curvature Measures and Generalized Morse Theory, Joseph Fu, 1988

Morse Theory for Tubular Neighborhoods, Antoine Commaret, 2024, Arxiv preprint https://arxiv.org/abs/2401.04034