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Author Archives: apgoucher
Cyclotomic fields
The nth cyclotomic field is the field generated by a primitive nth root of unity, ζ. It is an example of a number field, consisting of algebraic numbers, and its dimension is φ(n) when regarded as a vector space over … Continue reading
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Urbit for mathematicians
The three months since the last cp4space article have largely been spent learning about an interesting project called Urbit. My attention was drawn to its existence during a talk by RivaMelissa Tez (formerly of Intel) on the importance of continued … Continue reading
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A curious construction of the Mathieu group M11
Previously, we discussed which regular polytopes have vertexsets that occur as proper subsets of the vertexset of another regular polytope in the same dimension. In particular, when there is a Hadamard matrix of order 4k, then then the (4k−1)dimensional simplex … Continue reading
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Assorted topics
This is a digest of things that have happened this month, but which are individually too small to each warrant a separate cp4space post. Firstly, there have been a couple of exciting results in the field of combinatorics: The ErdősFaberLovász … Continue reading
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Meagre sets and null sets
There are two competing notions for describing a subset of the real numbers as being ‘small’: a null set is a subset of the reals with Lebesgue measure zero; a meagre set is a countable union of nowheredense sets. Both … Continue reading
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Keep your public keys private
Yes, the title sounds very counterintuitive. After all, don’t digital signature schemes require the general public to know your public key so that they can verify your signatures? That is correct, but importantly they don’t need to know your public … Continue reading
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The neural network of the Stockfish chess engine
Last time, we briefly mentioned the highlevel differences between Stockfish and Leela Chess. To recap, Stockfish evaluates about 100 million positions per second using rudimentary heuristics, whereas Leela Chess evaluates 40 000 positions per second using a deep neural network … Continue reading
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Rigid heptagon linkage
In January 2000, Erich Friedman considered the problem of finding a rigid unitdistance graph G containing a regular heptagon as a subgraph. That is to say, the graph is immersed in the plane such that: every edge of G must … Continue reading
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BarretoNaehrig curves and cryptographic pairings
There’s a very elegant cryptographic construction discovered by Barreto and Naehrig in a 2005 paper. It is beautiful from a pure mathematical perspective, but also has an impressive application: it was* part of the ingenious mechanism by which Zcash supports … Continue reading
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Shallow trees with heavy leaves
There are two very different stateoftheart chess engines: Stockfish and Leela Chess Zero. Stockfish searches many more positions (100 000 000 per second) and evaluates them using computationally cheap heuristics. The tree search methodology is a refinement of alphabeta pruning. … Continue reading
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