The −1 Dimensional Reduced Homology Group of the Empty Set

Hatcher’s Algebraic Topology unfortunately hadn’t been written when I studied the subject in grad school, but a few years ago I participated in a meetup that went through about half of it. I wrote up notes on things I puzzled over, or just observations I found helpful. Here they are. Besides the homology group mentioned in the title, some other tidbits: a figure to elucidate the calculation of the fundamental group of the complement of the Alexander Horned Sphere, and more details for the intuitive proof Hatcher sketches of Poincaré duality.

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The Peloponnesian War

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Socrates, Bad Guy

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The Death of Socrates, Jacques Louis David.
(From Wikimedia Commons)

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Gracefully Insane

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Topics in Nonstandard Arithmetic 9: Tricks with Quantifiers

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Every specialty has its tricks of the trade. They become second nature to practitioners, so they often don’t make it into the textbooks. Quantifiers rule in logic; here are some of the games we can play with them. I’ll start with tricks that apply in logic generally, then turn to those specific to Peano arithmetic.

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The Second French Revolution

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Delacroix’s Liberty Leading the People

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Topics in Nonstandard Arithmetic 8: Extensions and Substructures

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Substructures and extensions loom large in math: subgroups, subrings, extension fields, submanifolds, subspaces of topological spaces… So too in the model theory of PA.

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Nonstandard Models of Arithmetic 24

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MW: Indicators: we don’t need to discuss these, to prove the Paris-Harrington theorem. But I think they offer valuable insight.

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The Decision to Drop the Bomb

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Another post from the History Book Club.

(Why ‘atomic bomb’ rather than ‘nuclear bomb’? See this post.)

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The Making of the Atomic Bomb

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For a few years, I belonged to a history book club. Unlike many book clubs, we didn’t all read the same book. Instead, we’d pick a topic for the next meeting, at which the participants would each give short presentations on books of their choosing.

Recently I ran across my write-ups. As the internet has yet to run out of space, I thought I’d post them. I begin with two on the atomic bomb.

(Why ‘atomic bomb’, rather than ‘nuclear bomb’? See this post.)

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