Paul Monsky

Tight Closure’s Failure to Localize - a Self-Contained Exposition

Brenner and Monsky have given a negative solution to the localization problem for tight closure. Here I give a treatment of our counterexample that uses only linear algebra, material from an introductory abstract algebra course, and a little local cohomology developed ab initio. But most of this machinery, useful as it is for understanding the counterexample, may be dispensed with; in this paper the author gives a treatment of the example, using only linear algebra, material from an introductory abstract algebra course, and a little local cohomology developed ab initio.

Constructing equidissections for certain classes of trapezoids

We investigate equidissections of a trapezoid T(a), where the ratio of the lengths of two parallel sides is a. (An equidissection is a dissection into triangles of equal areas.) An integer n is in the spectrumS(T(a)) if T(a) admits an equidissection into n triangles. Suppose a is algebraic of degree 2 or 3, with each conjugate over Q having positive real part. We show that if n is large enough, n is in S(T(a)) iff n/(1+a) is an algebraic integer. If, in addition, a is the larger root of a monic quadratic polynomial with integer coefficients, we give a complete description of S(T(a)).

Kepler's First Law¿A Remark

The usual argument giving the motion of a particle under an inverse square central attractive force is a bit involved. Heres a quick derivation from Newtons laws that should appeal to an elementary differential equations class-the key idea is the use of the theory of second order linear equations. Consider the line that passes through the particle at time 0 and whose direction is the initial direction of the particle. The motion stays in a plane containing this line and the central force. So the problem is a planar one, and one may assume the force is located at (0, 0). Newtons equations give

Frobenius’ Result on Simple Groups of Order

Abstract The complete list of pairs of non-isomorphic finite simple groups having the same order is well known. In particular, for p > 3, PSL2(ℤ/p) is the “only” simple group of order Its less well known that Frobenius proved this uniqueness result in 1902. This note presents a version of Frobenius’ argument that might be used in an undergraduate honors algebra course. It also includes a short modern proof, aimed at the same audience, of the much earlier result that PSL2(ℤ/p) is simple for p > 3, a result stated by Galois in 1832.