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Category Archives: Math
Real Numbers Base…Factorials! And A Byproduct
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
Gauge Theory and LowDimensional Topology (Part II: Smooth FourManifolds)
In the last post, I attempted to give an overview of the state of affairs in fourmanifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) fourmanifolds and their intersection forms afforded by … Continue reading
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
Donaldson Turns 60
“Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4dimensions have been discovered. Moreover, the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory … Continue reading
Posted in Algebraic Geometry, Math, Topology
Tagged Differential Geometry, Math, Topology
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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
Posted in Math, Math in Pop Culture, Mathematics in Society
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