David Pierce | Matematik | M.S.G.S.Ü.

Mathematics page

This page is obsolete; I have no links to it, as far as I know; and if you are looking for something, you should probably go through my homepage.

Contents

  1. Advice to mathematicians (written by others)
  2. Remarks on model-theory (what I do)
  3. My own work (research, expositions, talks)
  4. TeX and other typesetting resources
  5. Links

Advice

☛ Writers of mathematics might do well to read “Writing Mathematics” and “Preparation of Papers” (both pdf files) from the London Mathematical Society. (However, although ¶ 2 of the latter article offers sound advice—

If your native language is not English, try to get someone to check your manuscript: at least compare it with papers in the same area written by English-speaking authors

—I do not think that people who grew up speaking English for all sorts of purposes should have exclusive right to determine what is good mathematical English. I frequently see turns of phrase that do not strike me as idiomatic English, but I cannot say that there is anything wrong with them. The flexibility of English is an advantage.)

☛ Speakers of mathematics should read John E. McCarthy, “How to give a good colloquium”, CMS Notes, vol. 31, no 5, September 1999. (I formatted and saved the article in HTML.) The key points can be extracted:

Here are some suggestions on giving a colloquium. They are guidelines, not absolute rules.

  1. Don't be intimidated by the audience.
  2. Don't try to impress the audience with your brilliance.
  3. The first 20 minutes should be completely understandable to graduate students.
  4. Carry everyone along.
  5. Talk about examples.
  6. Prove only tautologies.
  7. Put the theorem in context.
  8. Pay attention to the audience.
  9. Don't introduce too many ideas.
  10. Write an abstract.
  11. Find out in advance how long the colloquium is, and prepare accordingly.
  12. Don't use an overhead projector.

Model-theory

I work in that branch of mathematics and mathematical logic known as model-theory, and especially in the model-theory of fields.

Though I know nobody else who does so, I hyphenate “model-theory” on the principle that two nouns functioning as one should be symbolically linked. The hyphen seems particularly useful here because:

In Turkish, modeller—that is, “models”—plus teori gives modeller teorisi: the link between the nouns is indicated by the ending on teori.

In his encyclopedic book Model Theory (Cambridge, 1993), Wilfred Hodges proposes to define model-theory as “the study of the construction and classification of structures within specified classes of structures.” A “structure” might be an abelian group, say, or a Banach algebra. Hodges makes no requirement on how classes of structures are specified: in particular, the classes need not be given by first-order logical axioms. Still, presumably the classes will have some formal logical specification.

I have been considering defining model-theory as the study of structures quâ models of theories, where a “theory” is a set of formal logical sentences, not necessarily first-order.

In A Shorter Model Theory (Cambridge, 1997), Hodges says model-theory is “algebraic geometry without fields.” This definition should prevent any confusion of model-theory with mathematical modelling. It does however suggest—wrongly—that the model-theory of fields is algebraic geometry. (On the latter point, see Carol Wood's article in The Emissary for June 1998 (but the article is not on the MSRI website; the author provided me with the pdf file.)

The confusion with mathematical modelling will also be prevented if model-theory is called “definability-theory”; in a paper referred to below, Angus Macintyre suggests that it will come to be so called.

I like the conception of model-theory as “the geography of tame mathematics,” proposed by Lou van den Dries in a talk at MSRI in 1998. (I transcribed his slides with LaTeX: pdf, ps, dvi, tex.) Fields provide good examples of tame behavior (as van den Dries's talk suggests). One task of model-theory is to explain why various fields should be tame; in this way, the model-theory of fields is a sort of converse to algebraic geometry.

My work

Research articles

  1. With Pillay, Anand. A note on the axioms for differentially closed fields of characteristic zero. J. Algebra 204 (1998), no. 1, 108--115. MR1623945 (99g:12006)
  2. Function fields and elementary equivalence. Bull. London Math. Soc. 31 (1999), no. 4, 431--440. MR1687564 (2001a:03080)
  3. Differential forms in the model theory of differential fields. J. Symbolic Logic 68 (2003), no. 3, 923--945. MR2000487 (2004h:03080)
  4. Geometric characterizations of existentially closed fields with operators. Illinois J. Math. 48 (2004), no. 4, 1321--1343. MR2114160 (2006e:03053)
  5. Model-theory of vector-spaces over unspecified fields. Archive for Mathematical Logic, vol. 48, no 5 (2009), p. 421, DOI 10.1007/s00153-009-0130-x
  6. Fields with several commuting derivations. Submitted.
  7. Numbers (a review of various mathematical understandings and misunderstandings of the natural numbers, and a generalization of the class of ordinal numbers that is to an arbitrary algebra as ON is to (ℕ, 1, x↦x+1); draft).
  8. Representation theorems for rings (an investigation of what makes associative rings and Lie rings special among rings; draft).

I once arranged summaries of the earlier of the papers above, with links to additional notes, slides, drafts and so forth, under three heads:

Expository

Talks

I gathered slides of some talks into one directory. (I also made pages for my 2003 talks in Antalya, Helsinki, and Van.)

Mathematical typesetting

For typesetting with LaTeX, I now make use mainly of three books:

  1. Guide to LaTeX, Kopka and Daly, fourth edition, 2004
  2. The LaTeX Companion, Mittelbach and Goossens, second edition, 2004
  3. The LaTeX Graphics Companion, Goossens, Rahtz, and Mittelbach, 1997

In former days I made a lot of use of the following electronic documents among my files:

See also CTAN (the Comprehensive TeX Archive Network), especially the info directory.

Some other articles:

Links

Associations and institutions

People

Meetings

Archives

Logic and Model Theory

Teaching

Other

Son değişiklik: Wednesday, 05 April 2017, 18:59:19 EEST