## A Pretty Lemma About Prime Ideals and Products of Ideals

I was trying to prove a theorem in algebraic geometry which basically held if and only if this lemma held. Here’s the lemma:

Lemma: Given any ring $A$, a prime ideal $\mathfrak{p} \subset A$, and a finite collection of ideals $I_j,$ where $j \in \{1, 2, … , n\}$, then if $I$ is the intersection of the ideals, then $I \subset \mathfrak{p}$ implies that $I_j \subset \mathfrak{p}$ for some $j \in \{1, 2, … , n\}$. Continue reading “A Pretty Lemma About Prime Ideals and Products of Ideals” »

Posted in Algebra, Algebraic Geometry, Math | Tagged , , | 2 Comments

## Donaldson Turns 60

Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover, the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry. It is remarkable and encouraging that such a young mathematician can understand and harness such a wide range of ideas and techniques in so short a time and put them to such brilliant use. It is an indication that mathematics has not lost its unity, or its vitality.” – Sir Michael Atiyah

## What to Do When a Group Gets Stuck Working on a Task

In my previous post, I discussed how to adapt a problem that you have found in order to make the problem groupworthy. One of the important things to consider when adapting real-world problems is to avoid giving step-by-step instructions and formulas to students.  Instead, a teacher should maximize the opportunities for groups to make their own decisions about problems. In other words, in order to have challenging and productive group discussions, there must be an element of uncertainty so that students engage with the problem and with each other.