Galois Theory, Differential Equations, and Lie Groups?

In summary, the conversation discusses the connection between Galois Theory over Lie Groups and solutions of differential equations, known as Differential Galois Theory. The article also mentions Picard-Vessiot theory as a way to deal with solutions of differential equations.
  • #1
"pi"mp
129
1
I apologize for the informal and un-rigourous question. I have heard, in passing, that doing Galois Theory over Lie Groups instead of discrete groups is connected to solutions of differential equations instead of algebraic equations.

First of all, is this correct? If so, what is this correspondence called and where can I learn about it?
Much thanks!
 
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  • #2
I think the term you are looking for is Differential Galois Theory.
Don't ask me how it works, because I only know a bit of the "normal" Galois Theory (over finite field extensions), and I don't know how it's being used here.

http://en.wikipedia.org/wiki/Differential_Galois_theory

The article also mentions Picard-Vessiot theory, which seems to deal with solutions of differential equations in terms of integrals.
 

What is Galois Theory and why is it important in mathematics?

Galois Theory is a branch of abstract algebra that studies the properties of field extensions, which are mathematical structures that extend the properties of the familiar real and complex numbers. It is important in mathematics because it provides a powerful tool for understanding the solvability of polynomial equations and has applications in various areas such as number theory, algebraic geometry, and cryptography.

How does Galois Theory relate to differential equations?

Galois Theory has applications in differential equations as it can be used to determine whether a given differential equation can be solved using elementary functions, such as polynomials, exponential, and trigonometric functions. This is known as the Galois theory of differential equations.

What is the significance of Lie groups in mathematics?

Lie groups are mathematical structures that have applications in various areas of mathematics, including differential equations, geometry, and physics. They are important because they provide a framework for understanding symmetries and transformations in mathematics, which have important implications in the study of physical systems and mathematical models.

How do Lie groups relate to differential equations?

Lie groups are intimately related to differential equations as they can be used to study the symmetries and solutions of differential equations. In particular, the theory of Lie groups provides a powerful tool for finding exact solutions to differential equations and understanding the properties of their solutions.

What are some real-world applications of Galois Theory, Differential Equations, and Lie Groups?

Some real-world applications of these mathematical concepts include cryptography, coding theory, control systems, and mathematical modeling in physics, engineering, and economics. They also have applications in computer graphics and animation, as well as in understanding the behavior of complex systems such as weather patterns and biological systems.

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