What kind of project? I want you to write a survey article on the topic.
How long? 15-20 pages long (without the bibliography and the abstract)
Which software should I use? LaTex
Copyright. The text should be original, which means copy-paste is not
allowed, beacause this would be an important copyright issue, which
would imply failing the exam. There should be citations for all
references you use in the text.
Bibliography. At least 3 (non-greek) books and 2 articles.
TOPIC: DIFFERENTIAL GALOIS THEORY
Differential Galois theory studies the algebraic properties of
solutions to differential equations. It extends classical Galois theory
to differential equations.A central notion is that of a differential
field, which is a field equipped with a derivation operator satisfying
certain properties.
Just as in classical Galois theory, for
each differential equation, there is a corresponding differential Galois
group, which captures the symmetries of its solutions.The differential
Galois group is typically a matrix group over a differential field
associated with the equation.
The Picard-Vessiot theory
provides a fundamental framework for understanding the relationship
between differential fields and their differential Galois groups.It
establishes a correspondence between the solutions of a linear
differential equation and the structure of its differential Galois
group.
Differential Galois theory provides tools for
determining whether a given differential equation is integrable in terms
of elementary functions.This involves studying the differential Galois
group and analyzing its properties, such as solvability and simplicity.
Differential Galois theory has applications in various areas of
mathematics and physics, including mathematical physics, algebraic
geometry, and computer algebra systems.It plays a crucial role in the
study of integrable systems, quantum mechanics, and differential
equations arising in theoretical physics.
Some References
`Galois Theory of Linear Differential Equations` by Marius van der Put
and Michael F. Singer provides a comprehensive introduction to the
subject.
`Lectures on Differential Galois Theory` by A. Magid is a classic text that covers the basics of the theory.
`Galois Theory of Linear Differential Equations` by Marius van der Put
and Michael F. Singer focuses on the linear case and its connections to
algebraic geometry.
`Differential Algebra and Algebraic Groups`
by E. Kolchin discusses the relationship between differential algebra
and algebraic groups, which is foundational for differential Galois
theory.
