First published 2022
Leonard Gross works in his laboratory, staring at equations that describe how gauge fields evolve through time and space. He must prove that solutions exist and remain unique when the Yang-Mills heat equation operates over three-dimensional regions. The initial conditions live in a Sobolev space with index one half, placing him at the critical threshold where standard methods fail. Gross develops a two-step approach: he first solves an augmented parabolic equation, then transforms this solution using gauge functions to recover the original Yang-Mills equation. These gauge transformations require him to work within a complete topological group of Sobolev regularity three halves, though this group lacks the structure of a Hilbert Lie group. He establishes energy inequalities and Neumann domination principles to control the behavior of his augmented solutions. The mathematical landscape becomes treacherous as Gross navigates between existence proofs and uniqueness guarantees, knowing that strong solutions emerge only after accounting for gauge equivalences.
Genres: science-fiction, mathematics, physics, academic
Vibes: thought-provoking
Tropes: academic, mathematical
111 pages · Paperback · Amer Mathematical Society