REU 1995: Curvature of algebraic singularities (A. Durfee)
The REU project for 1995 directed by Alan Durfee investigated the following topics:
 Curvature near singular points of algebraic curves and surfaces, both real and complex.
(This project continued the 1988 and 1990 REU projects of O'Shea, and 1994 REU project of Durfee and O'Shea.
It was followed by the 1996 REU project of O'Shea.)
Let f(x_{1}, x_{2}, ..., x_{n}) be
a real or complex polynomial which vanishes at the origin. If the
partial derivatives vanish at the origin then this point is a singular
point of the hypersurface defined by the zero locus of the polynomial
in nspace. One can investigate the curvature of this hypersurface
near the origin, and also the curvature of nearby smooth
hypersurfaces. In particular one can look at the limit of the
Gaussian and principal curvatures along paths in nspace ending at the
origin.
This group investigated this problem in the cases n=2 and
n=3. In particular, they investigated whether these limiting
curvatures depended just on the tangent vector of the path at the
origin and not the path itself.
(See the papers by Hemmer, Starr and Tsai below.)
 Polynomial realizations of knot types (See the paper by Auerbach below.)
 Total curvature of algebraic knots (See the paper by Brutt below.)
Student participants
 Ruth Auerbach, Haverford College '96
 LeeAnne Brutt, Allegheny College '96
 David Hemmer, Dartmouth College '96
 Jason Starr, University of California, Berkeley '96
 Harrison Tsai, Harvard College '96
Ruth Auerbach is currently a graduate student in mathematics at UCLA, email rauerbac@capaccess.org;
David Hemmer is currently a graduate student in mathematics at the University of Chicago, email hemmer@math.uchicago.edu;
Jason Starr is currently a graduate student in mathematics at Harvard; and
Harrison Tsai is currently a graduate student in mathematics at UC Berkeley.
(The picture is of both 1995 REU groups)
Reports
The group produced the following papers, which are available in gzipped postscript. (To unzip, execute the command "gunzip".)
 David Hemmer, Limiting curvature near
singular points of algebraic curves
 Abstract: This paper examines the
limiting curvature of algebraic plane curves near a singular
point p.
The limiting curvature is shown to be determined completely by the
limiting tangent directions of the curve at p
except for at finite number of ``bad''
directions, which are completely classified. The results easily
extend to limiting Gaussian curvatures of complex algebraic curves.
Using these results, a similar analysis is performed on limiting
Gaussian curvature of real surfaces of the form f(x,y) + z^{n }= 0
near (0,0,0). This analysis permits several results
to be obtained about limiting curvatures of
Brieskorn singularities x^{a}+y^{b}+z^{c}=0.
Finally, for these examples, a conjecture is given relating these
bad directions to the exceptional lines of the surface.
 Jason Starr and Harrison Tsai,
Rational functions on algebraic varieties
 Abstract:
Let f(x,y,z) be a real or complex polynomial with f(0,0,0)= 0, and
let T be a rational function on three space which is possibly
undefined at the origin (for example, the curvature of the level
surfaces of the above polynomial). This paper investigates the
limiting behavior of T along paths lying in the surface f(x,y,z)=0
which end at the origin.
A tangent direction at the origin is called ``good'' if the limiting behavior
of T is independent of the path chosen, and ``bad'' otherwise.
An algorithm is presented for finding the bad directions.
 Ruth Auerbach, Looseended knots

Abstract: A (polynomial) looseended knot is a (polynomial) embedding
of an interval into the threeball whose endpoints, and only these
points, lie on the boundary of the ball. It is shown that looseended
knots are the same as usual knots.
The paper contains two
(unpublished) results due to Lee Rudolph relating the degree of a
polynomial looseended knot to standard knot invariants. Polynomial
representatives of the (5,2) and (7,2) torus knot are given.
 LeeAnne Brutt, Total curvature of curves
parameterized by polynomials
 Abstract:
The total curvature of a polynomial embedding of the line into
threespace is related to the degree of the polynomial using a result
of Milnor, and a conjecture is given.
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