What I learned from Zhiwei Yun about Hilbert schemes

One knows, of course, that Hilbert schemes of smooth curves and smooth surfaces are nice, and Hilbert schemes of varieties of dimension greater than two are terrifying.

Zhiwei Yun was here giving a talk about his work with Davesh Maulik on Hilbert schemes of curves with planar singularities, and he made a point I’d never appreciated; it’s not the dimension of the variety, but the dimension of its tangent space that really measures the terrifyingness of the  Hilbert space.  Singular curves C with planar singularities are not so bad — you still have a nice Hilbert scheme with an Abel-Jacobi map to the compactified Jacobian.  But let C be the union of the coordinate axes in A^3 and all bets are off.  Hideous extra high-dimensional components aplenty.  If I had time to write a longer blog post today I would think about what the punctual Hilbert scheme at the origin looks like.  But maybe one of you guys will just tell me.

Update:  Jesse Kass explains that I am wrong about C; its Hilbert scheme has a non-smoothable component, but it doesn’t have any components whose dimension is too large.

Is Arrow’s Theorem interesting?

Suppose a group of people has to make a choice from a set S of options.  Each member of the group ranks the options in S from best to worst.  A “voting system” is a mechanism for aggregating these rankings into a single ranking, meant to represent the preferences of the group as a whole.

There are certain natural features you’d like a voting system to have.  For instance, you might want it to be “monotone” — if a voter who likes option A better than B switches those two in her ranking, that shouldn’t improve A’s overall position or worsen B’s.

Kenneth Arrow wrote down a modest list of axioms, including monotonicity, that seem like pretty non-negotiable features you’d want a voting system to have.  Then he proved that no voting system satisfies all the axioms when S consists of more than two options.

Why wouldn’t that be interesting?

Well, here are some axioms that are not on Arrow’s list:

• Anonymity (the overall outcome is invariant under permutation of voters)
• Neutrality (the overall outcome is invariant under permutation of options)

Surely you don’t really want to consider a voting system that doesn’t meet these requirements.  But if you add these two requirements, the resulting special case of Arrow’s theorem was proved more than 150 years earlier, by Condorcet!  Namely:  it is not hard to check that when |S| = 2, the only anonymous, neutral, Arrovian voting system is majority rule.  Add to that Arrow’s axiom of  “independence of irrelevant alternatives” and you get

(*) if a majority of the population ranks A above B, then A must finish above B in the final ranking.

But what Condorcet observed is the following discomfiting phenomenon:  suppose there are three options, and suppose that the rankings in the population are equally divided between A>B>C, C>A>B, and B>C>A.  Then a majority ranks A over B, a majority ranks B over C, and a majority ranks C over A.  This contradicts (*).

Given this, my question is:  why is Arrow’s theorem considered such a big deal in the theory of social choice?  Suppose it were false, and there were a non-anonymous or non-neutral voting mechanism that satisfied Arrow’s other axioms; would there be any serious argument that such a voting system should be adopted?

Thanks to Greg Kuperberg for some helpful explanation about this stuff on Google+.  Relevant reading:  Ben Webster says Arrow’s Theorem is a scam, but not for the reasons discussed in this post.

Put the second law of thermodynamics down and slowly step away, New York Times

Yet given her professional background, Dr. Oakley couldn’t help doubting altruism’s exalted reputation. “I’m not looking at altruism as a sacred thing from on high,” she said. “I’m looking at it as an engineer.”

And by the first rule of engineering, she said, “there is no such thing as a free lunch; there are always trade-offs.” If you increase order in one place, you must decrease it somewhere else.

Moreover, the laws of thermodynamics dictate that the transfer of energy will itself exact a tax, which means that the overall disorder churned up by the transaction will be slightly greater than the new orderliness created. None of which is to argue against good deeds, Dr. Oakley said, but rather to adopt a bit of an engineer’s mind-set, and be prepared for energy losses and your own limitations.

Stop hurting physics!

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It’s like a state, a state of Kong

For the few people who will get nostalgic pleasure out of this, a 1992 article from the Harvard Crimson in which I am extensively quoted about my love for the Hong Kong restaurant in Harvard Square.  I still go there just about every time I’m in town, most recently with Steve Burt.  The once-great “Top of the Kong” comedy club appears no longer to exist, sadly.  Update: No, apparently there’s still a comedy club there, it just changed its name to The Comedy Studio!

Glossary for non-Boston people: “Peking ravioli” is New England nomenclature for fried dumplings, developed by Joyce Chen in order to get Italian people to come to her restaurant.  If an entrepreneur of her caliber had ever lived in Milwaukee, I could probably get Shanghai spaeztle around here.

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Subjective probabilities: point/counterpoint

• Adam Elga:  “Subjective Probabilities Should Be Sharp” — at least for rational agents, who are vulnerable to a kind of Dutch Book attack if they insist that there are observable hypotheses whose probability can not be specified as a real number.
• Cosma Shalizi:  “On the certainty of the Bayesian Fortune-Teller” — People shouldn’t call themselves Bayesians unless they’re committed to the view that all observable hypotheses have sharp probabilities — even if they present their views in some hierarchical way “the probability that the probability is p is f(p)” you can obtain whatever expected value you want by integrating over the distribution.  On the other hand, if you reject this view, you are not really a Bayesian and you are probably vulnerable to Dutch Book as in Elga, but Shalizi is at ease with both of these outcomes.