# Category Archives: Math

## AMS Notices Spotlight October 2017

Hello and welcome to the October AMS Notices Spotlight. As we are now into the swing of the busyness of the semester it is sometimes nice to take a break and think about math not related to our classes. With … Continue reading

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## Real Numbers Base…Factorials! And A By-product

PROPOSITION 1:  For a real number  x  there exists a sequence $x_1, x_2, x_3,…$ of integers such that $\hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots, \hspace{2cm} (*)$ where $x_1$ can be any integer, but for … Continue reading

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## Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds)

In the last post, I attempted to give an overview of the state of affairs in four-manifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) four-manifolds and their intersection forms afforded by … Continue reading

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## 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 … Continue reading

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## An Infinite Understanding

“Have you ever thought about how strange it is that we think about infinity every day, but most people think about it only on the rarest of occasions, if ever?” This is the text message I recently sent two of … Continue reading