Classical Galois Theory aimed to study the solvability of polynomial equations by studying a certain symmetry group associated with the equation. Famously, these ideas allowed Ruffini, Abel, and, of course, Galois to show that there cannot possibly be a closed-form solution to the general quintic equation involving only radicals in the coefficients of the equation; moreover, Galois identified exactly which quintics can be solved explicitly through detailed consideration of the properties of the symmetry group.

Differential Galois Theory is to linear differential equations as Galois Theory is to polynomial equations. The subject was initiated by Picard and Vessiot some 50 years after Galois, and following its ancestor, remained obscure and difficult to understand until later developments, notably 50 some-odd years later by our very own Ritt and Kolchin. As before, the theory is concerned with a certain symmetry group associated with a linear differential equation, and by studying properties of this symmetry group, one can prove, for example, that the equation \(y’ = x - y^2\) cannot be solved in elementary terms; equivalently, the function \(e^{-x^2}\) cannot be integrated in elementary terms!

The approach that we will follow is a slightly more modern approach that crucially uses the Riemann–Hilbert Correspondence. This allows us to realize differential Galois groups very geometrically as coming from monodromy related to differential equations; we have come full circle!

Prerequisites, Format, Expectations, etc.

This seminar should be accessible to one with a good background in linear algebra, some analysis, and basics of modern algebra.

Beyond the mathematical content, these seminars are really about communicating mathematics. In particular, I think that the small seminar environment is a perfect place to develop skills and a sense of comfort in speaking and presenting mathematics. Thus the seminar consists mostly of lectures by the participants. Moreover, this experience is a great place to think carefully about how to explain mathematics to an audience.

References

Our primary reference is the following textbook on the subject:

Another book that will be useful is:

Here is a beautiful survey about the subject:

Finally, the following book is a delightful whimsy:

Schedule

We meet on Tuesdays in Mathematics Room 528 between 7:30PM and 9:30PM. Mostly, there will be a first and second speaker, each speaking for approximately an hour.

09/05
Organization.
09/10
Raymond Cheng,
Introduction.
09/17
Ben Church,
Overview of Tools and the Logarithm,
[Sauloy, §§5,6].
09/24
Shasta Ramachandran,
Introduction to Complex Linear Differential Equations,
[Sauloy, §§7.2–7.3].
09/24
Ben Bellick,
Monodromy of Complex Linear Differential Equations,
[Sauloy, §§7.4–7.5].
10/01
Tasman Fell,
Gauge Transformations and Differential Systems,
[Sauloy, §§7.6–8.1].
10/01
Micah Gay,
Differential Systems, Local Systems, and Fundamental Groups,
[Sauloy, §§8.2–8.4].
10/08
Myeonhu Kim,
Growth Conditions and the Local Riemann–Hilbert Correspondence,
[Sauloy, §9].
10/15
Christian Serio,
Local Riemann–Hilbert Correspondence Categorically,
[Sauloy, §10].
10/15
Raymond Cheng,
Global Riemann–Hilbert Correspondence,
[Sauloy, §12].
10/22
Raymond Cheng,
Differential Galois Groups,
[Sauloy, §13].
10/29
Benjamin Bellick,
Calculating Local Differential Galois Groups,
[Sauloy, §§14.1–14.2].
10/29
Cecilia Wang,
Local Schlesinger Density Theorem,
[Sauloy, §§14.3–14.4].
11/05
No Seminar.
Go Vote!
11/12
Tasman Fell,
The Universal Galois Group,
[Sauloy, §§15, 16.1].
11/19
Myeonhu Kim,
Riemann–Hilbert via Differential Galois Groups, I,
[Sauloy, §§16.2–16.4].
11/19
Christian Serio,
Riemann–Hilbert via Differential Galois Groups, II,
[Sauloy, §§16.2–16.4].
11/26
Shasta Ramachandran,
Picard–Vessiot Theory,
[vdPS, Chapter 1].
11/26
Micah Gay,
\(q\)-Difference Equations.
12/03
TBD.
12/03
TBD.