# Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with me, after he sent me a few papers about the topic of lie group analysis in epidemic models.

Since it has been approved I got cold feet. In all honesty, my experience in analysis is not very good. I have only taken a 200-level Analysis course with Steven Lay's book and barely got through.

The textbook we are using is Applications of Lie Groups to Differential Equations by Peter J Olver. The textbook says they only assume an elementary understanding of analysis. I have taken a lot of algebra (rings, fields, groups, galois theory) and have done much better in those classes. I'm not entirely worried about the groups/algebra part. I did go over some of the theorems Olver stated we should know in the opening and I understood the statements and proofs quite easily.

Anyway, my question is how should I prepare for this course? Is there a sort of textbook which will go over preliminaries to this textbook? Is there other more applied/easier textbooks to go along with this textbook?

• Related: this question on getting started with Lie Groups. – Damien Feb 4 '16 at 1:49

I've been working my way through Sophus Lie's Differential Invariant paper of 1884, and it's pretty tough going. His style was more that of the 17th century, and many of the assumptions of his day are lost on modern readers, but there are some ideas in his work that have barely been touched since his death. I studied under Lawrence Dresner in the early 90's, and most of his work deals with applications of Lie groups to nonlinear PDE's, but he did some work with ODE's as well. I highly recommend Dresner's Similarity Solutions of Nonlinear Partial Differential Equations from Pitman Advanced Publishing Program, ISBN 0-273-08621-9.

I've been using Lie's THM 4.4.1 to solve a variety of linear and nonlinear ODE's on math.stackexchange. Feel free to go to my homepage and look at some of the solutions. The basic idea of this thm. is that if you can find an infinite continuous group that preserves the structure of the smooth manifold (a.k.a., a Lie group) which leaves a DEQ invariant, the DEQ itself is in the kernel of the map with the group stabilizers. Lie calls these group stabilizers "differential invariants." If you can find such an invariant group, you can rewrite the DEQ in terms of the group stabilizers and then use the Lie algebra between the stabilizers to solve the DEQ. At singularities, saddle points and along separatrices the Lie algebra is particularly simple and can be used to find special solutions with very little effort. Olver's book gives a way to start with the DEQ and find an invariant Lie group,and Emmy Noether's work suggests that this should always be possible, but it's over my head as of now and I'm still working to understand it. If you figure out how it works, please let me know!

That be a tough book, friend. I'd recommend going straight at it, but if you really start getting slayed, start off with Symmetry Methods for Differential Equations.

The book of Peter Olver is too much sophisticated for beginners in Lie theory, The book by Peter Hydon is little bit elementary, but between these two books there is a book written by George Bluman which I must say is elegant piece of work by author. Moreover, author Bluman is very responsive to every single query on his book.

• btw, what you link to is Bluman & Anco (2002), the newer version of Blumen & Kumei (1989), which is more cited / well-known. The sequel to Bluman & Anco is Bluman & Cheviakov (2010). – Geremia Feb 28 '17 at 22:46
• @Geremia : The sequel of Bluman & Anco you have mentioned is hard one, it can only be understood if you have good knowledge of Symmetry methods which can only be learnt from earlier books of Bluman and Hydon. – IgotiT Mar 10 '17 at 1:18
• I'm quite impressed with Bluman & Anco (2002). Its first chapter on measurement and dimensional analysis motivates symmetry methods quite well. – Geremia Mar 10 '17 at 1:30
• @Geremia : 7 year ago I started working on Symmetry methods, and somebody advised me the book Bluman & Anco (2002), he also suggested to avoid first chapter. But when I read the book two three times I realized that I have missed the most important first chapter of book. I agree with you that the first chapter motivates symmetry methods. – IgotiT Mar 10 '17 at 1:47
• Was he a pure mathematician? Bluman & Anco (2002) is excellent for applied. – Geremia Mar 10 '17 at 2:33