![[picture]](http://maths.dur.ac.uk/pics/va.jpg)
Project IV topics 2023-24
Area: Algebra and Analysis
Topic: Galois theory and Differential Equations.
Numerous unsuccessful attempts to solve a series of algebraic and differential
equations ‘in explicit form’ led mathematicians to the belief that explicit solutions
for these equations simply do not exist.
The first proofs of unsolvability of algebraic equations by radicals were found
by Abel and Galois. While considering the problem of explicitly finding an indefinite
integral of an algebraic differential form, Abel founded the theory of algebraic
curves. Liouville continued Abel’s research and proved that indefinite integrals of
many algebraic and elementary differential forms are non-elementary. The unsolvability by quadratures of some linear differential equations was also first proved by
Liouville. Arnol’d discovered that many classical problems in mathematics are unsolvable
for topological reasons. In particular, he showed that it is for topological reasons
that the general algebraic equation of degree 5 cannot be solved by radicals.
When working on this Project we are going to consider three versions of Galois theory.
Namely, the
usual, the differential, and the topological versions. These versions are unified
by a general approach to problems concerning the solvability and unsolvability of
equations, based mainly on group theory.
Prerequisites:
Algebra and Number Theory II, Galois theory III
Basic resources: