Harry Altman's webpage

I can be reached via email at . (If you can't see my email address, that's because it's been Javascript-obfuscated to fool spammers; take a look at the page source and you can find it ROT13ed in there.) My PGP public key can be found here.

Recent updates

• Beren came up with a cool tree!
• Probabilities in groups page has been updated to note the recent work of Thomas Browning!
• Snake page has been updated with new minimal winnables!
• Equational probabilities in various algebraic structures
• Integer complexity, stability, and self-similarity is now up on arXiv!

Math papers and drafts

Published papers

• Numbers with Integer Complexity Close to the Lower Bound (with Joshua Zelinsky)
  • "Online" version
  • arXiv version
• Integer Complexity and Well-Ordering
  • arXiv version
• Integer Complexity: Representing Numbers of Bounded Defect
  • arXiv version
• Intermediate Arithmetic Operations on Ordinal Numbers
  • arXiv version
• Internal Structure of Addition Chains: Well-Ordering
  • arXiv version
• Integer Complexity: Algorithms and Computational Results
  • My implementation of the described algorithms
    • Written in Haskell. See the readme for usage instructions. A warning: I really haven't made the code very readable, or put the functions in a sensible order. I have at least made a point of describing what each of them do, though.
  • arXiv version
• Integer Complexity: The Integer Defect
  • arXiv version

Preprints

• Maximum linearizations of lower sets in N^m with application to monomial ideals (with Andreas Weiermann)
• Integer complexity: Stability and self-similarity (with Juan Arias de Reyna)

Dissertation

• Integer Complexity, Addition Chains, and Well-Ordering -- My dissertation.
  • A note: This dissertation takes the form of four papers "stapled together", together with an introduction and a conclusion. Chapter 2 is "Numbers with Integer Complexity Close to the Lower Bound", above. Chapter 3 is "Integer Complexity and Well-Ordering", above. Chapter 4 is "Internal Structure of Addition Chains: Well-Ordering." Chapter 5 has been split into "Integer Complexity: Representing Numbers of Bounded Defect" and "Integer Complexity: Algorithms and Computational Results".
  • Another note: Some of the problems in Chapter 6 have actually already been solved, either by myself or by Juan Arias de Reyna... more on that some other time...
  • My implementation of the described algorithms
    • Written in Haskell. See the readme for usage instructions. A warning: I really haven't made the code very readable, or put the functions in a sensible order. I have at least made a point of describing what each of them do, though.

Drafts

Polished drafts, should be readable

Old drafts, may be incomprehensible and out of date

• Numbers with integer complexity close to the lower bound I (with Joshua Zelinsky): In which we devise a method for classifying numbers whose integer complexity is close to the lower bound, and derive various consequences. This is out of date and not very readable.
  • Classification of n with d(n)≤22d(2): The table of results from the computation described in the above draft. Currently not included in the draft proper, as even in this compressed form it's 7 pages long. We have since come up with a way of doing these computations algorithmically, so this is no longer fully necessary. Also I may not have corrected the few mistakes in it.
  • Numbers with integer complexity close to the lower bound: A rather longer version of the above paper in which we discuss some things we didn't have room for in the short version. Currently in the process of being turned into multiple papers which will actually be readable. Some of the conjectures in here have since been proved.
• Highest few sums and products of ones: A write-up of an idea I don't intend to publish -- the method doesn't generalize; my newer results with Joshua Zelinsky extend and obsolete it. Still I think the method is interesting and was worth writing up. If you want a better presentation of the results themselves, proved via a different method, see "Integer Complexity: The Integer Defect" above.
  • Here is the Haskell code used for the computations.

"Problem dump"

• Which graphs are winnable in Snake?: You know the game of Snake? What if we tried playing it on an arbitrary finite graph?
• Are probabilities in groups well-ordered?: Wherein I speculate irresponsibly about a paper by Sean Eberhard about probabilities in finite groups.
  • More on equational probabilities in algebraic structures
• The PDPP model of computation: What is the complexity of search in this strange unphysical model of computation?
• What if we generalized the definition of homotopy?: What if we allowed any connected space instead of just [0,1]? I never got any good answer to this.
• A very silly construction: In which I reconsider the construction I answered a question about earlier here, and ask just what it can make.
• Notes on a silly problem: OK, the original problem here has since been solved, but I'm keeping this up because some variants may still be interesting.
• Beren's Fibonacci tree: In this case, I have figured out what's going on, but I thought it was worth writing about here!

Things I wrote for a former job

• Data representation in Solidity -- just how does the Solidity language (as of version 0.8.20) represent data, under the hood? Written for work but possibly of more general interest.

Other stuff

• For the paranoid, my PGP/GPG public key NEW PUBLIC KEY AS OF 23 JUL 2010 (HD crashed, old private key unrecoverable)
• Internet What Database?
• Spelling Elements with Elements
• Everything I owe the Internet Oracle
• The Co-Files: A list of mathematical terms where "co-" has its ordinary English meaning
• Minimum and maximum possible scores in Space Alert

Here are some drawings that people have given me which I have bothered to scan:

• by Linda Cai
• by Linda Cai
• by Guillermo whose last name I no longer recall

And here are some assorted other images.