Professor Emeritus and Adjunct Professor

Contact information

Department of Pure MathematicsPeter in the pot
University of Waterloo
Waterloo, Ontario, Canada
N2L 3G1

Office room number: Mathematics & Computer (MC) 5407
Office telephone number: 519-888-4567 ext. 37741
Fax number: 519-725-0160

Research interests

  • Logic and applications to CS
  • Group representation theory
  • K-theory
  • Algebraic topology

Papers and books to view and/or download

Below is a list of various pieces of work from the past 40 years, some published, at least one rather light, and a couple of fairly polished (indexes and all) undergraduate level books. The list is somewhat more in a "braggadocio" vein than I'd maybe like it to be. But a major purpose of these webpages sponsored by our institution (the braggadocio culture certainly being a long tradition at this institution!) is presumably to help produce more `business', particularly students and postdocs, and to help attract the best applicants when we have jobs advertised. So this is an advance apology for any impression that I might be overly impressed with my own accomplishments. (Whether I actually am or not is obviously not for me to judge!)

The first three are quite old research papers for which I still have some fondness :

1. (1969---5 pages) The unsolved problem mentioned in the last paragraph is still unsolved and continues to have quite a few papers published attacking it from various viewpoints in algebraic topology.

On the realizability of Steenrod modules (PDF).

2. (1975---with J.F. Adams---9 pages)

Operations on K-theory of torsion-free spaces.

For a list of nearly all my early publications in Algebraic Topology, K-Theory and Hopf Algebras, see the reference list in my Springer Lecture Notes in Mathematics, #746, $\tau $-rings and Wreath Product Representations, Springer (1979).

3. (1989---9 pages) A record for me---Rota's acceptance letter arrived exactly 6 days after submission.

A Bernstein-type Formula for Projective Representations of $S_n$ and $A_n$.

The next two are research papers related to representation theory and Hopf algebras.

4. (1996---38 pages)

On the Schur index of graded representations (PDF).

For a complete list of my publications in this area, see the reference lists to these, plus that in my joint Oxford University Press book with John Humphreys: Projective Representations of the Symmetric Groups Oxford Mathematical Monographs (1992).

5. (2003---with Katherine E. Stange and Chris Wooff---23 pages)

General Zelevinski Algebras (PDF)..

The next one is much lighter, not unrelated to a major recreational interest.
It describes the only mathematical `object' named after me---unfortunately, since it's the Hoffman handicap, and named by a non-mathematician, my good friend Al White from Orangeville, who uses it in his software for race timing! The handicapping gets used fairly often, particularly in each recent year to decide the male and female age-group X-C ski champions for Ontario.

6. (2006 update---7 pages)

Experiments and formulas concerning age handicapping in ski races (PDF).

Papers 7 to 12 are mainly expository, written over the past 6 or 7 years (except for 11), as I have become more interested in logic and physics---sins of old age! In particular, number 12 is fairly often used as the textbook for PMATH 330, the basic logic course for mathematics students not specializing in pure mathematics.

7. (2003---95 pages)

Floyd-Hoare Logic (PDF).

8. (2002---118 pages)

The $\lambda $-calculus (PDF).

9. (2004---412 pages)

Computability for the Mathematical (PDF).

10. (Jan 2007---78 pages)

Newton $\Leftrightarrow $ Kepler `Logic', and an enlightening loiter with Emmy Noether (PDF).

People outside U of W who download either 11 or 12 might wish to consider donating to the Pure Math scholarship fund.
Make checques payable to U of Waterloo and send to: Chair, Pure Math. Dept., University of Waterloo, Waterloo, ON, Canada, N2L 3G1

11. (1973 to 2006---282 pages) This is written in a very terse style, one which I don't necessarily go for any more, but it does have some advantages. Anyway it's an antidote for the exact opposite in number 12 below.
There is maybe 5 times the amount of content in this, in 70% as many words, compared to the logic text.

Bare Bones Algebra, Eh!? (PDF).

12. (latest update 2005---420 pages)


Logic for the Mathematical (PDF).

The next six are fairly recent work. The three of them have considerable overlap: 16 is just (14 plus 15 minus many details).

13. (June, 2007---26 pages) There is little really new here beyond a lengthy syntactic proof (pp. 7 to 11) that a well-known system needs a change to one of its proof rules in order to be complete. (At least one referee for a prestigious CS journal could improve his/her erudition by knowing about this.)

Systems for 'while' - Total Correctness with Quantifiers and Connectives (PDF).

14. (Nov. 2007---92 pages) The length here is caused by giving much more detail than is customary. The paper does the case of a single procedure variable (which seems to be the standard `procedure', accompanied by a vague reference to generalizability, in the CS literature).

Deterministic Dynamic Logic Imperative Recursive Programming Proof Systems (PDF).

15. (Dec. 2007---30 pages) This gives all the extra details necessary for the general situation of arbitrarily many procedure variables, removing the need for a ``vague reference" as just above.

Imperative Mutual Recursion Proof Systems (PDF).

16. (Jan. 2008---41 pages) This is the more reasonable-length version of 14 and 15. The theoretical CS literature is full of erroneous results on this subject, so the extra details in 14 and 15 seemed important to record publicly somewhere.

A Proof System for Recursive Programming with Local Declarations (PDF).

17. (April, 2008---68 pages) This is entirely expository. It seems impossible to find, in either research papers or texts, thoroughly detailed readable treatments of the semantics of either of the two main non-trivial basic command constructors used by deterministic imperative computer languages, namely ITERATION, e.g. "while G, do C", and RECURSION, e.g. "begin DECLARE X TO BE C; DO X end". The latter is somewhat subtle when it comes later to the so-called "axiomatic semantics", i.e. Floyd-Hoare logic, which 14, 15 and 16 above deal with.

Detailed Semantics of Toy Command Languages for Iteration and Recursion (PDF).

18. (Sept. 2008---69 pages) The main new material here is defining the semantics of the most general form of recursion by use of a depth measure, where "depth" refers to `how may times the program cycles' when doing the recursion, so to speak. This leads to an apparently new way to `unfold' a recursive command, and to the only way I know to prove the soundness of the rule I use for partial correctness of recursive commands, as done fora less general form of recursion in 14, 15 and 16 above. (I am hoping that I'll be able to generalize that latter work by removing the simplicity requirement, perhaps based partly on the present paper.) We also include a lot of comparison of that recursion rule with those of other authors from earlier work, the basic idea for all of these being due to Dana Scott back in the '60's.

Depth Measure Semantics and Partial Correctness Rules for Imperative Recursion. (PDF).

19. (Jan. 2009---23 pages) This paper consists mainly of a few examples of verifications, using a modification of the system, in #16 above, which we conjecture to finally do the job of producing a Cook-complete system where we no longer need the recursions to satisfy the technical condition of being simple. It also includes a short proof of soundness for the special case used here of a new rule for partial correctness of recursive commands.

Challenge Examples and a Conjecture re: Correctness of Imperative Recursive Programs (both Partial and Total) (PDF).

20. (Jan. 2009---40 pages) This sketches the main details in a proof that the system from #19 above is actually Cook-complete. A more painstaking version will be forthcoming. Surprisingly, the proof seems to be somewhat shorter and less tortuous than the one given for the simple case in #16 above.

Variable Depth Recursion Semantics and Correctness (PDF).

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