Tag Archives: Math

Intersection of a Chain of Subsets

Assume $\{F_x\}_{x \in \Gamma}$ is a collection of subsets (of a not-so important set!) such that every two are comparable, i.e for any $x$ and $y$, either $F_x \subset F_y \ \ $ or $\ \ F_x \supset F_y \ … Continue reading

Posted in Analysis, Math | Tagged , , , | 1 Comment

Machine-Checked Proof

“In my view, the choice between the conventional process by a human referee and computer verification is as evident as the choice between a sundial and an atomic clock in science.” – Tom Hales (from [4]) “The rapid advance of … Continue reading

Posted in Math, Technology & Math | Tagged , , , | Leave a comment

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

Posted in Math, Topology | Tagged , | 2 Comments

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

Posted in Algebra, Algebraic Geometry, Math | Tagged , , | 3 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 … Continue reading

Posted in Algebraic Geometry, Math, Topology | Tagged , , | Leave a comment