A¹ homotopy is currently still a research area, but if you read the linked Wikipedia article:
"The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A¹, which is."
In other words: it's a novel approach towards homotopy theory, which does have applications in the physical world.
Even better (Wikipedia article):
"[A¹ homotopy theory] has also recently revolutionized the theory of enumerative geometry problems."
Enumerative geometry does have applications in the physical world.
https://www.ecfr.gov/current/title-7/subtitle-B/chapter-XVII...
A¹ homotopy is currently still a research area, but if you read the linked Wikipedia article:
"The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A¹, which is."
In other words: it's a novel approach towards homotopy theory, which does have applications in the physical world.
Even better (Wikipedia article):
"[A¹ homotopy theory] has also recently revolutionized the theory of enumerative geometry problems."
Enumerative geometry does have applications in the physical world.