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I come from an applied math background and my interest lately have been in the application of PDE theory to machine learning problems. This has led me down a more "pure" route concerning solutions of differential equations when the coefficients come from more general algebraic structures (such as rings).

A simple example comes in the one dimensional case.

$$ y' - \lambda y = 0 $$

Which has solutions of the form

$$ y = y_0 e^{\lambda x} $$

Now if we look at the similar equation

$$ D_x \mathbf{Y} - \Lambda\mathbf{Y} = \mathbf{0} $$

As long as $\Lambda$ is a constant matrix, we have a solution similar to the one dimensional problem.

$$ \mathbf{Y} = \exp(\Lambda x) \mathbf{Y}_0 $$

This similarity fails when $\Lambda$ is not constant. I'm interested in exploring the question of when (and how) we can solve differential equations when the coefficients are not necessarily real numbers but can be more general such as elements of a ring (such as in the example above)

I have a copy of Differential Algebra by Joseph Ritt, a collection of notes on the application of algebraic methods to differential equations, but was wondering if there were other resources out there.

I may be out of my depth on tackling these kind of problems (as Ritt's notes have been a little tough to get through), but for reference purposes my mathematical background includes:

  • Graduate-level
    • PDE Theory
    • Measure Theory
    • Functional Analysis
  • Undergraduate level
    • Analysis (Point set topology on metric spaces, etc)
    • Abstract Algebra (Groups, Rings, and Fields)

I'm currently in my 3rd year of an applied mathematics undergraduate program.

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  • $\begingroup$ A student and I wrote a paper on solving ODEs over an algebra. This is not quite what you're asking about, but, I do use a synthesis of abstract algebra and advanced calculus. You might be interested: arxiv.org/abs/1708.04137 (this is part of my larger goal of understanding calculus where real numbers are replaced with an algebra $\mathcal{A}$, I call this the $\mathcal{A}$-calculus. I'm especially proud that we can solve ALL the usual cases by a universal algebraic method. $\endgroup$ – James S. Cook Jan 18 '18 at 0:30
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Maybe a good route for you would be to start studying Lie Groups and Lie algebras to get a feeling of the exponential map and what it does. Good refferences abound, I would suggest Lee's Introduction to Smooth Manifolds (in which after you cover the first chapters can skip right to the theory that interests you) and to pair it of with Arnold's Partial differential equations.

It may seem a bit of a hassle to study manifolds in order to get what you want , but in the end, it is something you will have to do if you want to get deeper int the theory of partial differential equations.

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