Harry Altman at the University of Michigan
My name is Harry Altman. I recently completed my PhD in mathematics at the
University of Michigan, studying under Jeffrey Lagarias. Recently
I've been studying some combinatorial number theory involving the complexity of
computing natural numbers in some simple computational models.
Some recent updates (I appear to have missed some):
2017 October 30 -- Intermediate
Arithmetic Operations on Ordinal Numbers is now published!
2016 June 24 -- Integer Complexity: Algorithms and Computational Results is now on arXiv.
2016 April 10 -- Added more images.
2016 March 22 -- "Integer
Complexity: Representing Numbers of Bounded Defect" is on arXiv.
2015 September 9 -- Published version of "Integer Complexity and Well-Ordering"
2015 July 1 -- More questions about Snake.
2015 January 26 -- The super-Jacobsthal exponentiation paper is now on arXiv!
2014 September 9 -- Added more images.
2014 September 8 -- The addition chain
defects paper is on arXiv!
2014 May 27 -- My dissertation can now be found
below, along with the corresponding code!
2014 January 8 -- Minimum and maximum possible scores in Space Alert
2013 December 29 -- Two more words for the Co-files.
I can be reached via email at . (If you can't see my email address, that's because it's been
URL.) My PGP public key can be found here.
Papers and drafts
- 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 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
- 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.
Polished drafts, should be readable
Nothing to go here at the moment.
Old drafts, may be incomprehensible and out of date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 somewhat out of date and not very readable; I am
currently working on fixing that.
- 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 in the above draft extend and obsolete it
(hence all the references to the improvements made in our forthcoming paper).
Still I think the method is interesting and was worth writing up.
- Here is the Haskell code used for the computations.
Informal bloggy writing on problems I don't necessarily know much of anything about.
- Notes on a silly problem: A silly problem in
combinatorial number theory; coloring the natural numbers such that n+m gets a
different color from n*m.
- Some questions about Snake: You know the game of
Snake? What if we tried playing it on an arbitrary finite graph? There's a
lot you could ask about this, but I have two particular questions.
Here are miscellaneous files I wish to put in public view:
Here are some drawings that people have given me which I have bothered to scan:
And here are some assorted other images.