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12 answers

Is pi a good random number generator?

Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, which makes me wonder whether I should just use…
James Propp
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56
votes
12 answers

Homological Algebra texts

I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe). As usual, the rule is one reference per post. Please include…
alekzander
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56
votes
1 answer

History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here: http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf Allyn Jackson writes about how Mumford was profoundly impressed: Mumford found the leaps into abstraction to be breathtaking.…
Dikran Karagueuzian
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56
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8 answers

Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity. Apparently, the reason why we usually…
Wojowu
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56
votes
4 answers

Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML. There is a term "Partially Observed Groups" in machine learning theory which has been popularized by recent work to understand deep learning. The idea…
drosophyllum
  • 577
56
votes
10 answers

What are dessins d'enfants?

There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led Grothendieck to define a special class of these mappings, called the Children's Drawings, or, in French, Dessins d'Enfants…
Ilya Nikokoshev
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56
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14 answers

Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the only group satisfying the given constraints is…
Matthieu Romagny
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56
votes
6 answers

Can the symmetric groups on sets of different cardinalities be isomorphic?

For any set X, let SX be the symmetric group on X, the group of permutations of X. My question is: Can there be two nonempty sets X and Y with different cardinalities, but for which SX is isomorphic to SY? Certainly there are no finite examples,…
Joel David Hamkins
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56
votes
4 answers

Are there refuted analogues of the Riemann hypothesis?

The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important analogues that are now known to be false?
Colin McLarty
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56
votes
28 answers

Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course. Here is a (not so good) example of the sort of thing I am looking for. a) Prove that every…
Donald
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56
votes
17 answers

Atiyah-Singer index theorem

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of time discussing the topology and algebra, but very…
Andy Putman
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55
votes
10 answers

How often do people read the work that they cite?

I have the following question: How likely it is that an author carefully read through a paper cited by him? Not everyone reads through everything that they have cited. Sometimes, if one wants to use a theorem that is not in a standard textbook,…
Vidit Nanda
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55
votes
6 answers

Poincaré Conjecture and the Shape of the Universe

Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?
Shake Baby
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55
votes
14 answers

'Important' applications of p-adic numbers outside of algebra (and number theory).

Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily explain to a student)? Here I do not demand the…
Mikhail Bondarko
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55
votes
5 answers

Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its expected topology where "expected" makes sense because…
Jonny Evans
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