## Classical Invariant Theory

### Course description:

We shall study the book `Classical Invariant Theory' by Peter Olver (Cambridge University Press, 1999). Classical invariant theory is the study of the intrinsic properties of polynomials. Intrinsic properties are those which are unaffected by change of variables and hence are purely geometric - not tied to an explicit coordinate system. For example, multiplicity of roots is intrinsic. Also factorizability is intrinsic. On the other hand, explicit values of roots and particular coefficients of polynomials are not intrinsic. The study of invariants is closely tied to the problem of equivalence - when can one polynomial be transformed into another by a suitable change of coordinates. There is also an associated canonical form problem - to find a system of coordinates in which the polynomial takes on a particularly simple form. The solutions to these intimately related problems, and much more, are governed by the invariants, and so the first goal of classical invariant theory is to determine the fundamental invariants. With a sufficient number of invariants in hand, one can effectively solve the equivalence and canonical form problems, and, a least in principle, completely charterize the underlying geometry of a given polynomial.

Olver writes (p. x): `I wrote this introductory textbook in the hope of furthering the recent revival of classical invariant theory in both pure and applied mathematics. The presentation is not from an abstract, algebraic standpoint, but rather as a subject of interest for applications in both mathematics and other scientific fields.... The purpose of the book is to provide the student with a firm grounding in the basics of classical invariant theory.'

Concerning prerequisites Olver writes (p. xv): `I have tried to keep the prerequisites to a minimum, so that the text can be profitably read by anyone trained in just the most standard undergraduate material. Certainly one should be familiar with basic linear algebra.... No knowledge of the general theory of polynomial equations is assumed. An introductory course in group theory could prove helpful to the novice but is by no means essential since I develop the theory of groups and their representations from scratch. All constructions take place over the real or complex numbers, and so no knowledge of more general abstract field theory is ever required.'

### Prerequisites:

Math 504 Abstract Algebra, Math 510 Linear Algebra or consent of the instructor

For more information about the course contact Dr. Kenneth R. Driessel (office: Carver 410, e-mail: driessel@iastate.edu, internet: http://www.public.iastate.edu/~driessel.)