Methods of Mathematical Physics, Tallinn University of Technology, Fall 2011

Synopsis of lectures

Students' presentations:

• December 6. Piret Avila, Introduction to GAP and Sage.
• December 13. Alari Varmann, Simplicity of the alternating groups.

Prerequisites. Working knowledge of basic mathematical concepts: vector spaces, matrices, groups, fields, differential equations.

The book by Robert Gilmore Lie Groups, Physics, and Geometry. An Introduction for Physicists, Engineers, and Chemists, Cambridge Univ. Press, 2008, is available from the author's website (some links there are broken, the full list of files is available here). Note that the layout and pagination of those files are different from the published version. (Thanks to Riivo Must for the pointer).

Some basic literature and additional material:

1. I.R. Shafarevich, Basic Notions of Algebra, Springer, 1990 (this is translation from the 1st Russian edition, VINITI, 1986; there is also a 2nd, corrected and augmented Russian edition, Regular & Chaotic Dynamics, 2001). A very good conceptual introduction, without much technical details, painting a broad picture and interconnections between different mathematical concepts, and their relation to science.
2. M.I. Kargapolov and Yu.I. Merzlyakov, Fundamentals of the Theory of Groups, Springer, 1979 (translation from the 2nd Russian edition; there is also a 3rd Russian edition, Nauka, 1982). A good textbook on group theory.
3. M. Hall, The Theory of Groups, Macmillan, 1959. Ditto. (Russian translation is available in Tallinn libraries).
4. J.D. Dixon, Problems in Group Theory, Blaisdell Publ. Co., 1967 (republished by Dover in 1973). Each chapter contains a short list of facts from group theory, and then a lot of (easy) problems. Good for practicing. (I have this book and can lend it).
5. Wikipedia. List of group theory topics.
6. The Group Properties Wiki.
7. J. Baez, This Week's Finds in Mathematical Physics. A big collection of online notes (blog, as we would say these days) of various degree of sophistication and often of highly entertaining character. Groups, Lie algebras, symmetries, and their relations to physics appear prominently.
8. Computer algebra programs: GAP and Sage. Great for experimentation with groups and Lie algebras.

Suggestions for students for a lecture topic:

• Galois theory of databases (after Plotkin).
• More examples from symmetries of differential equations. Suggested sources, in addition to Gilmore, are: B.J. Cantwell, Introduction to Symmetry Analysis, Cambridge Univ. Press, 2002, and H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge Univ. Press, 1989.
• Chapter 14 (Hydrogenic atoms) and Chapter 15 (Maxwell's equations) of Gilmore.
• (As)symmetries of genetic code.
• Lie groups and error analysis (after Schiff and Shnider).
• Examples of computer calculations with Lie groups from W.-H. Steeb, Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, 2nd ed., World Scientific, 2007.
• Groups and symmetries in music theory (see, e.g., arXiv:1111.4451 and references therein).
• Algorithm for isomorphism of Lie algebras utilizing solutions of polynomial systems (after de Graaf).

N.B. All the books mentioned above are available in electronic form in multiple places.