I will present a supertranslation-invariant and Lorentz-covariant definition of angular momentum in asymptotically flat 4D spacetime. This definition uses only asymptotic metric data and reproduces the flux necessary to obtain known radiation reaction effects. The formula has an appealing physical interpretation, it extends to Lorentz boost charges and integrated fluxes, and agrees with other existing definitions in appropriate reference frames.

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