Tag Archives: Math

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

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Gauge Theory and Low-Dimensional Topology (Part I: Historical Context)

Hi! This month, I thought I would start a brief series of articles describing the uses of gauge theory in mathematics. Rather than discuss current research directions in gauge theory (of which there are many), I hope to give an … Continue reading

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The “Idea” of a Scheme

The mathematical concept of a “scheme” seems to pop up everywhere, but it’s hard to get a good grasp on what a scheme actually is. Any time you might ask someone what a scheme is in passing, there never seems … Continue reading

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Optimal Control Theory to Settle Reinhardt’s Conjecture

The 2010’s are a Golden Age for packing problems. In 2014, Hales announced the long-awaited completion of a high-profile machine proof project called FlySpeck, which verified his proof of Kepler’s conjecture. Johannes Kepler, in 1600, conjectured that the densest way to pack … Continue reading

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What is a Manifold? (6/6)

In posts 1-3 we were able to reduce all of the geometry of a curve in 3-space to an interval along with two or three real-valued functions. We also discussed when two sets of such data give equivalent (overlapping) curves. This … Continue reading

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