I study commutative algebra. Much of my work has been joint with
Prof. William Heinzer of Purdue University. Together we have
studied many kinds of rings (all commutative with unity) that are
not quite Noetherian: Laskerian rings, Artinian rings, rings with
the ascending chain condition on ideals generated by a fixed number
of elements, rings with the ascending chain condition on colon
ideals, etc. More recently we have focused on regular local domains
(which of course are very Noetherian) and a class of ideals primary
for the maximal ideal. We called this class "first coefficient ideals",
because they are maximal among those having the same first two
coefficients in the Hilbert polynomial. We frequently used the
computer algebra system MACAULAY in our study of this class of ideals.
I have taught many of the mathematics courses offered at Colgate;
my favorites are both semesters of abstract algebra (especially the
second, on Galois theory), geometry and combinatorics. I have used a
computer algebra system in several
courses, especially in applied mathematics for the social sciences.
I should also mention two more elementary projects in which I have
participated: I assisted in the translation of an approximate
solution by Casanova (yes, that Casanova) on the Delian
Problem ("Construct with straightedge and compass a cube with
exactly twice the volume of a given cube"). And an exposition that
I wrote for distribution to one of my classes, on why it is impossible
to cut a cube into a finite number of polyhedral pieces and
reassemble them into a regular tetrahedron, was later published in
the South African recreational journal Wistukkie.
My wife Clara
and I are very interested in drama, from both sides of the footlights.
Here is a link to the
.pdf file of the transparencies I used for
the Gehman lecture at the Mathematical Association of America
Seaway Section meeting at Queen's University in Kingston, Ontario,
on April 1-2, 2005.
Here are links to the web pages of some of the courses I've taught:
From the Math Dept's "junior" seminar (now replaced by Math 399):
Math 102 /Core 143 : Introduction to Statistics --
based on Freedman, Pisani and Purves, Statistics, Fourth
Edition. I am one of several instructors, past and
present, who have contributed to the substantial web
component of this course. We are proud of the course.
Math 111 : Calculus I and
Math 112 : Calculus II -- based on James Stewart's Single
Variable Calculus, Early Transcendentals, Fourth Edition.
(For Calc II, Fifth Edition.)
Math 213: Multivariate Calculus -- based on William
G. McCallum et al., Multivariable Calculus.
Math 320 : Abstract Algebra I --
based on Saracino, Abstract Algebra: A First Course.
I supplied a substantial number of graphics; they
are linked to this page, and my assignments appear here.
Math 323 : Real Analysis I --
based on Abbott, Understanding Analysis. Not a "webby"
course, at least as I taught it, but you are welcome to look
at my assignments and the few graphics linked to this page.
Math 421 : Abstract Algebra II (Galois theory) -- this
is the 37-page "text", in .pdf format, that I wrote
for a course on Galois theory
using the "Moore method" of teaching mathematics, i.e., a list
of definitions, theorems, and discussions, with most of the
proofs omitted so that students can be asked to supply them.
Michael Weiner, now at Penn State University, Altoona campus,
taught the course, using this
"text". My sense is that it went well, but I don't have
any pedagogical suggestions to make to anyone else who
might want to try it. I should note that my colleague
Dan Saracino, on whose book this "text" is based, has
expanded his book in a second edition to include Galois theory;
it should appear shortly.
I much enjoyed the play The Discovery of
the Calculus: The Battle Between Wilhelm Leibniz and Isaac Newton,
written by H. W. Straley, Charlene B. Straley, and F. A. "Chip" Straley,
of Woodberry Forest School, Woodberry Forest, VA. To make the play
more available, I have digitized the script and images provided by
the Straleys (with their kind permission) and arranged them in several
file formats. Here is the index to these files:
The Discovery of the Calculus
(with William Heinzer) Factorization of monic
polynomials, Proceedings of the American Mathematical Society,
131(4) (2003), 1049-1052.
Example of an interpolation domain, Journal of Pure
and Applied Algebra, 174 (2002), 149-152.
(with William Heinzer and Roger Wiegand) Residue fields of a
zero-dimensional ring, Journal of Pure and Applied Algebra,
129 (1998), 67-85.
(with William Heinzer) Ideal theory in two-dimensional regular
local domains and birational extensions, Communications in Algebra
23(8) (1995), 2863-2880.
(with William Heinzer and Sylvia Wiegand) Prime ideals in
birational extensions of polynomial rings, Contemporary Mathematics
159 (1994), 73-93.
(with William Heinzer) ACCP in polynomial rings: a
counterexample, Proceedings of the American Mathematical Society
121 (1994), 975-977.
(with Robert Gilmer and William Heinzer) The Noetherian property
in rings of integer-valued polynomials, Transactions of the American
Mathematical Society 338 (1993), 187-199.
(with William Heinzer) Exceptional prime divisors of
two-dimensional local domains, in Commutative Algebra:
Proceedings of a Microprogram Held June 15 - July 2, 1987
M. Hochster, C. Huneke and J. D. Sally, eds., Springer-Verlag, New York,
Finite Krull dimension, complete integral closure and
GCD-domains, Communications in Algebra 3(10) (1975), 951-958.
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