Speaker: Pavel Etingof (MIT)

Title: Which linear differential equations can be explicitly solved?

Abstract:

In kindergarten we learn to solve first order linear differential equations 
$$
y'+p(x)y=q(x)
$$ 
(by Euler's formula), and even some higher order ones, such $y''=y$ or $x^2y''=y$. But what about $(x^2-1)y''=y$? Can it be solved explicitly, perhaps in elementary school? Middle school? High school? MIT?
 
Well, it depends on what you mean by explicit. If you want to write solutions in elementary functions, then the answer is no (one can only write them in terms of so-called hypergeometric functions). But how to prove such a claim, or even precisely state it?
 
This should remind us the story with algebraic equations, 
$$
x^n+a_1x^{n-1}+...+a_n=0.
$$ 
In nursery school we learn how to solve them for $n=1,2$, and later some learn how to do it for $n=3$ (Cardano's formula) and $n=4$ (Ferrari's formula). But for $n\ge 5$ this cannot be done for general coefficients using only arithmetic operations and radicals. This is a great theorem of N. H. Abel which is one of the cornerstones of modern algebra and mathematics in general. 
 
The idea of proving Abel's theorem is to use symmetries that permute the $n$ roots of the equation preserving relations between them. These symmetries form a finite group $G$, a subgroup of the symmetric group $S_n$, which for general coefficients is the full $S_n$; it is called the Galois group. If the equation is solved in radicals, then $G$ should be ``glued" from cyclic groups, because the symmetries of the roots of the equation $x^n=a$ satisfied by $\sqrt[n]{a}$ form a cyclic group $\Bbb Z/n$, rotating the regular $n$-gon spanned by these roots in the complex plane. Such groups are called solvable; they are characterized by the property that their iterated commutator subgroups eventually peter out (become the trivial group $\lbrace 1\rbrace$). The reason we can solve equations of degree $\le 4$ but not $5$ and higher is that the group $S_n$ is solvable if and only if $n\le 4$. This is a starting point of Galois theory. 
 
It is remarkable that Galois theory can be extended to differential equations, and this extension can be used to state and prove non-solvability of certain linear differential equations, such as $(x^2-1)y''+y=0$, in elementary functions. This is also based on considering the group $G$ of symmetries that permute solutions of the equation preserving relations between them. However, in this case, if the equation has order $n$, solutions form an $n$-dimensional complex vector space $\mathbb{C}^n$, so the group $G$ is a subgroup of $GL_n(\mathbb{C})$ (the group of invertible matrices of size $n$), which is the group of symmetries of this space. And it is no longer a finite group but rather an (algebraic) Lie group. For example, for the equation $(x^2-1)y''+y=0$ this group is $G=SL(2,\Bbb C)$, the subgroup of matrices with determinant $1$. This group is not solvable, which prevents us from having an explicit formula for solutions. 
 
In the talk I will explain the basics of Galois theory for differential equations and its applications. I will try to make it as elementary as possible, but some basic knowledge about groups and analytic functions will probably be needed.

Reception following talk in room 2-290.