Why Isn't There an Infinite Dimensional Determinant? - Vaughn Gerrits скачать в хорошем качестве

Why Isn't There an Infinite Dimensional Determinant? - Vaughn Gerrits

Abstract: The determinant is a very powerful tool used in finite dimensional linear algebra. It is essential to much of the theory of finite dimensional differential forms. It is used to define many physically relevant Lie Groups. It is used to define like the characteristic polynomial, which enjoys many very useful properties, gives us strong algebraic results like the Cayley-Hamilton theorem, and is the reason that the still unsolved invariant subspace problem is trivial over finite dimensional spaces. However there is no generalization to operators on infinite dimensional vector spaces, or even to Hilbert spaces. In this talk we will first construct the determinant, and try to understand the object philosophically, after which we will attempt to generalize the idea and see where it goes wrong. It often requires a much sturdier rope to cross the ravine between finite and infinite dimensional geometry than one might hope, and by investigating this particular gap, we hope to better understand the chasm itself. The "We promise this applies to Physics" seminar series YT playlist:    • The "We promise this applies to Physics" S...