We shall study the book `Equivalence, Invariants, and Symmetry' by Peter Olver (Cambridge University Press, 1995). The fundamental equivalence problem is to determine whether two geometric objects can be transformed into each other by a suitable change of variables. For example, one might be interested in whether two given differential equations are the same equation rewritten in terms of different independent and dependent variables. The determination of the symmetry group of a geometric object can be regarded as a special case of the general equivalence problem - a symmetry is merely a self-equivalence. Thus the solution of the equivalence problem for differential equations will include a determination of all symmetries of a given differential equation. Besides addressing the general equivalence problem, we shall also develop methods to directly calculate the symmetry group of a given object.
When a student first encounters ordinary differential equations, they are presented with a bewildering variety of special techniques designed to solve certain particular types of equations. Around the middle of the 19th century, Sophus Lie discovered that these special methods were all special cases of a general procedure based on invariance of the differential equation under a continuous group of symmetries. These groups, now called Lie groups, have had a profound inpact on all areas of mathematics.
We shall also consider the problem of determining canonnical forms for the objects of interest. By definition, a `canonical form' is provided by a particularly simple equivalent object. The Jordan canonical form for matrices is an example.
By definition, a coframe on a manifold is a `complete' collection of one-forms in the sense that, at each point, it provides a basis for the cotangent space. We say that two coframes are equivalent if thay are mapped into each other by a diffeomorphism. The equivalence problem for coframes is the nost important of equivalence problems because it ultimately includes all the others as special cases. Olver writes (p.5): `One important goal of this book is to present an understandable introduction to the Cartan method, illustrated by many applicatons, with the hope of disseminating this powerful and useful theory yet wider in the mathematical community and beyond.'
Math 504 Abstract Algebra, Math 510 Linear Algebra, Math 534 Topology, Math 557 Ordinary Differential Equations, Math 562 Manifolds, Tensors and Differential Geometry
For more information about the course contact Dr. Kenneth R. Driessel (office: Carver 410, e-mail: firstname.lastname@example.org, internet: http://www.public.iastate.edu/~driessel.)