REU 2000 Topics
This summer's topic were invariant theory of homogeneous polynomials
(forms) and algebraic invariants of knots and links.
Invariant Theory
Invariant Theory Classical and Modern
Invariant theory is a subject that relates to many area of
mathematics - Algebraic Geometry, Algebraic Topology, Combinatorics,
Differential Equations and Representation Theory, to name five. In
the nineteenth century several methods were developed to generate
invariants, and general properties, such as Gordon's and Hilbert's
finiteness results, were discovered and understood. However, the
subject more or less died out, in part because of the complexity of
the computations involved in this subject. In recent years, there
has been a revival of interest in Invariant Theory. This stems from
two reasons: (1) Modern computers have made feasiible many
computation that could only be dreamed of at the turn of this
century. (2) New applications have been found for Invariant Theory,
ranging form Mumford's construction of moduli spaces, symmetry
groups of differential equations to computer vision.
We will begin with an exposition of the main algorithms for
generating invariants. From here, there are many avenues to
explore. One is simply to try to extend some of the calculations to
other cases. For instance, the complete invariant system for binary
septic was worked out by Dixmier and Lazard only about a decade ago;
the binary octic was done a little earlier by Shioda. But the
complete invariant system for binary nonics is not worked out. One
can also consider ternary or quaternary forms. For ternary forms,
Clebsch and Gordon determined the covariants for the ternary cubic.
This is important for the theory of elliptic curves, since each of
these is represented as the zero - locus of such a form. Only
recently was the ternary quartic understood, by Dixmier (the
corresponding curves now have genus 3, so this is, in effect, a
description of the moduli space of genus 3 curves.) For quaternary
forms, the geometric objects are projective algebraic surfaces.
The other directions to pursue are in the applications of
invariant theory. This is very open - ended, and could depend on the
interests of the students. A student interested in this ought to
take a look at the just - published book by Peter Olver, ``Classical
Invariant Theory'', London Math. Soc. Student Texts, number 44. In
the extensive bibliography of this book are references
ranging over a very wide domain, which could provide material for N
REU programs, where N is a very large number.
Algebraic Invariants of Knots and Links
The area of classical knot and link theory is attractive to many
undergraduates, but it requires limitation in scope as well as substantial
simplification in the introductory development in order to be accessible.
Our REU has chosen research directions t hat take off from linear algebra,
elementary group theory and combinatorics. For example, the algebraic
invariants for knots, links and braids, as traditionally developed,
require a substantial amount of point-set and algebraic topology
(including homolo gygroups). While this background is important for a
thorough development, a equivalent approach to knots developed by
Reidemeister considers equivalence classes of graphs under the
Reidemeister moves and (isotopy classes) of planar embeddings.
As an example, the projects for the Summer of 1999 was to investigate
certain matrix representations of the braid group and develop geometric
interpretations applicable to knots and links. Professor Stoltzfus
developed an elementary approach to the Gassn er representation (as well
as others related to the lower central series) using two additional,
little studied, ingredients: twisting of the matrix multiplication and
invariance of the ``representation" under a sesqui-linear form. The
problems for the REU are algebraic, with heuristic allusions to their
geometric interpretations. The interplay of algebra and geometry in this
setting, as always, is a powerful tool. An important open question
(despite recent purported attempts at proof) is whether the Gassner
representation is injective or not.
An approach was developed that is accessible to undergraduates having
taken only a first course in abstract algebra.
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Last Update: 23 January, 2004