Consider the category of algebraic integrable connections on a smooth connected algebraic variety $X$ with a base point $x$ (over a field of characteristic zero). This is a Tannakian category with a fibre functor given by taking fibres over $x$ and is hence equivalent to the representations of some pro-algebraic group. This group is the wild fundamental group. One may consider the subcategory of integrable connections with regular singularities (which is a condition at infinity wrt $X$). This subcategory (when the base field is the complex numbers) is equivalent to the category of finite dimensional complex representations of the fundamental group and is hence described by the "Tannakian hull" of the fundamental group, in general a very large pro-algebraic group. In any case, the fact that the regular connections form a Tannakian subcategory makes this Tannakian hull a quotient of the wild fundamental group. The kernel somehow reflects the possible irregularities of general connnections on $X$. These can also be described exactly in terms of Stokes phenomena.

The reason why one would be interested in the wild fundamental group and not just in irregular connections per se (and their associated Stokes phenomena) is on the one hand that it essentially gives a description of the differential Galois group of integrable connections, on the other hand that there is a very visible but still imperfectly understood connection (sic!) between irregular singularities and wild ramification in positive characteristic. This incidentally is the reason for the terminology "wild".

Addendum: As for references there are several books by N. Katz (in the Annals of Mathematics Study series) that take up both the differential Galois group side and characteristic $p$ side of (what should be) the same systems. They give, I feel, a good feeling for how these ideas can be used in concrete situations. There are of course also several relevant books on differential Galois theory. However, many of them tend to be less geometric (in particular the (sometimes only implicitly chosen) base point is usually the generic point of $X$ which gives a less geometric flavour to the whole business. The book by van der Put and Singer; "Galois theory of linear differential equations" tries to find some middle road.

Consider the category of algebraic integrable connections on a smooth connected algebraic variety $X$ with a base point $x$ (over a field of characteristic zero). This is a Tannakian category with a fibre functor given by taking fibres over $x$ and is hence equivalent to the representations of some pro-algebraic group. This group is the wild fundamental group. One may consider the subcategory of integrable connections with regular singularities (which is a condition at infinity wrt $X$). This subcategory (when the base field is the complex numbers) is equivalent to the category of finite dimensional complex representations of the fundamental group and is hence described by the "Tannakian hull" of the fundamental group, in general a very large pro-algebraic group. In any case, the fact that the regular connections form a Tannakian subcategory makes this Tannakian hull a quotient of the wild fundamental group. The kernel somehow reflects the possible irregularities of general connnections on $X$. These can also be described exactly in terms of Stokes phenomena.

The reason why one would be interested in the wild fundamental group and not just in irregular connections per se (and their associated Stokes phenomena) is on the one hand that it essentially gives a description of the differential Galois group of integrable connections, on the other hand that there is a very visible but still imperfectly understood connection (sic!) between irregular singularities and wild ramification in positive characteristic. This incidentally is the reason for the terminology "wild".

Addendum: As for references there are several books by N. Katz (in the Annals of Mathematics Study series) that take up both the differential Galois group side and characteristic $p$ side of (what should be) the same systems. They give, I feel, a good feeling for how these ideas can be used in concrete situations. There are of course also several relevant books on differential Galois theory. However, many of them tend to be less geometric (in particular the (sometimes only implicitly chosen) base point is usually the generic point of $X$ which gives a less geometric flavour to the whole business). The book by van der Put and Singer; "Galois theory of linear differential equations" tries to find some middle road.

Consider the category of algebraic integrable connections on a smooth connected algebraic variety $X$ with a base point $x$ (over a field of characteristic zero). This is a Tannakian category with a fibre functor given by taking fibres over $x$ and is hence equivalent to the representations of some pro-algebraic group. This group is the wild fundamental group. One may consider the subcategory of integrable connections with regular singularities (which is a condition at infinity wrt $X$). This subcategory (when the base field is the complex numbers) is equivalent to the category of finite dimensional complex representations of the fundamental group and is hence described by the "Tannakian hull" of the fundamental group, in general a very large pro-algebraic group. In any case, the fact that the regular connections form a Tannakian subcategory makes this Tannakian hull a quotient of the wild fundamental group. The kernel somehow reflects the possible irregularities of general connnections on $X$. These can also be described exactly in terms of Stokes phenomena.

The reason why one would be interested in the wild fundamental group and not just in irregular connections per se (and their associated Stokes phenomena) is on the one hand that it essentially gives a description of the differential Galois group of integrable connections, on the other hand that there is a very visible but still imperfectly understood connection (sic!) between irregular singularities and wild ramification in positive characteristic. This incidentally is the reason for the terminology "wild".

Consider the category of algebraic integrable connections on a smooth connected algebraic variety $X$ with a base point $x$ (over a field of characteristic zero). This is a Tannakian category with a fibre functor given by taking fibres over $x$ and is hence equivalent to the representations of some pro-algebraic group. This group is the wild fundamental group. One may consider the subcategory of integrable connections with regular singularities (which is a condition at infinity wrt $X$). This subcategory (when the base field is the complex numbers) is equivalent to the category of finite dimensional complex representations of the fundamental group and is hence described by the "Tannakian hull" of the fundamental group, in general a very large pro-algebraic group. In any case, the fact that the regular connections form a Tannakian subcategory makes this Tannakian hull a quotient of the wild fundamental group. The kernel somehow reflects the possible irregularities of general connnections on $X$. These can also be described exactly in terms of Stokes phenomena.

The reason why one would be interested in the wild fundamental group and not just in irregular connections per se (and their associated Stokes phenomena) is on the one hand that it essentially gives a description of the differential Galois group of integrable connections, on the other hand that there is a very visible but still imperfectly understood connection (sic!) between irregular singularities and wild ramification in positive characteristic. This incidentally is the reason for the terminology "wild".

Addendum: As for references there are several books by N. Katz (in the Annals of Mathematics Study series) that take up both the differential Galois group side and characteristic $p$ side of (what should be) the same systems. They give, I feel, a good feeling for how these ideas can be used in concrete situations. There are of course also several relevant books on differential Galois theory. However, many of them tend to be less geometric (in particular the (sometimes only implicitly chosen) base point is usually the generic point of $X$ which gives a less geometric flavour to the whole business. The book by van der Put and Singer; "Galois theory of linear differential equations" tries to find some middle road.