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This page is obsolete; I have no links to it, as far as I know; and if you are looking for something, you should probably go through my homepage.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
- Advice to mathematicians (written by others)
- Remarks on model-theory (what I do)
- My own work (research, expositions, talks)
- TeX and other typesetting resources
- 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.
- Don't be intimidated by the audience.
- Don't try to impress the audience with your brilliance.
- The first 20 minutes should be completely understandable to graduate students.
- Carry everyone along.
- Talk about examples.
- Prove only tautologies.
- Put the theorem in context.
- Pay attention to the audience.
- Don't introduce too many ideas.
- Write an abstract.
- Find out in advance how long the colloquium is, and prepare accordingly.
- 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:
- The word “model” is sometimes an adjective, as in “model train” (no hyphen). A model train is generally not a “real” train. A model-train (written with a hyphen) would be a real train carrying models (perhaps Fabio and Iman). Model-theory is a real theory, not a model of one: it is the theory of models.
- Model-theory could be a model theory, that is, an exemplary theory; but its name should not express this.
- The phrase “model theory of fields”, with no hyphen, might be wrongly construed as “model theory-of-fields”.
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
- 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)
- Function fields and elementary equivalence. Bull. London Math. Soc. 31 (1999), no. 4, 431--440. MR1687564 (2001a:03080)
- Differential forms in the model theory of differential fields. J. Symbolic Logic 68 (2003), no. 3, 923--945. MR2000487 (2004h:03080)
- Geometric characterizations of existentially closed fields with operators. Illinois J. Math. 48 (2004), no. 4, 1321--1343. MR2114160 (2006e:03053)
- 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
- Fields with several commuting derivations. Submitted.
- 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).
- 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
- “Undecidability of a function-field in two variables” (An earlier version of this paper, distributed to some people, was “Undecidability of C(T0, T1)”.)
- Others (textbooks, notes for students or myself…)
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:
- Guide to LaTeX, Kopka and Daly, fourth edition, 2004
- The LaTeX Companion, Mittelbach and Goossens, second edition, 2004
- 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:
- The Not So Short Introduction to LaTeX 2e, Oetiker et al., version 4.13, 10 September 2003, xiv + 131 pp.
- LaTeX2e: The macro package for TeX [dvi file], by Leslie Lamport et al., edition 1.1, December 1994, ii + 78 + vi pp.; the index is on p. i
- AMS-LaTeX Version 1.2 User's Guide [dvi file], January 1995, vi + 42 pp.
- Comprehensive List of Symbols [pdf file], Scott Pakin, 29 September 2003
See also CTAN (the Comprehensive TeX Archive Network), especially the info directory.
Some other articles:
- “TeX and LaTeX 2e”, Michael Downes, Notices of the AMS 49 11, December 2002
- “Mathematical Word Processing” (p. 2 of link; letter by William C. Hoffman to the Editor of the Notices of the AMS, March 2003 in response to the last; I do not share Mr Hoffman's fondness for WYSIWYG editors and Microsoft products)
- Math Typesetting for the Internet (from the Math Forum, "An Online Math Education Community Center")
Links
Associations and institutions
- AMS (American Mathematical Society) MathSciNet
- MSRI (Mathematical Sciences Research Institute)
- Fields Institute for Research in Mathematical Sciences, Toronto
- ASL (Association for Symbolic Logic)
- UMd math department
- UIUC math department
- IMU (International Mathematical Union)
- Isaac Newton Institute for Mathematical Sciences
- EMS (European Mathematical Society), Ankara mirror
- MODNET (Marie Curie Research Training Network in Model Theory and its Applications): Introductory notes
- Federated World Directory of Mathematicians
People
- Ayşe Berkman
- Alexandre Borovik
- Zoé Chatzidakis
- Lou van den Dries
- Timothy Gowers
- Wilfred Hodges
- Angus Macintyre
- Barry Mazur
- Anand Pillay
- Thomas Scanlon
- Carol Wood
- Boris Zilber
Meetings
- Logicum Urbanae Lugduni, July 6–10, 2009
- Groups and Models: Cherlin Bayramı, Istanbul, June 8–17, 2009
- Panhellenic Logic Symposium, Volos, Greece, 5–8 July, 2007
- Differential Fields Workshop, 2007.07.08–10, Leeds
- Modnet (Research Training Network in Model Theory, 2005–2008):
- Antalya Algebra Days
- Gödel Centenary 2006, Vienna
- Congress and School on Universal Logic, 2005.3.26–4.3, Montreux
- Model Theory and Applications to Algebra and Analysis, 2005.1.17–7.15, Newton Institute
- Groups, geometry and logic (Groupes, géométrie et logique), 2004.9.27–10.1, Centre International de Rencontres Mathématiques, Marseille
- Logic Colloquium:
- Euro-Conference in Model Theory and Applications, Ravello, May 2002
- Premier Congrès Franco-Américain de Mathématiques AMS-SMF, Lyon, 2001
- WORLD MATHEMATICAL YEAR 2000
- Workshop on Hilbert's Tenth Problem and relations with arithmetic and algebraic geometry, 2-5 November 1999, Gent, Belgium
Archives
- The University of Michigan Historical Mathematics Collection
- Kolekcja Matematyczna (Polska Biblioteka Wirtualna Nauki), including Fundamenta Mathematicae
- MIT mathematics courses
- Hilbert's Tenth Problem: St Petersburg, with a mirror in Greenville
- JSTOR math journals
- Valuation Theory Home Page
- Front for the Mathematics ArXiv
Logic and Model Theory
- “Model theory: Geometrical and set-theoretic aspects and prospects” by Angus Macintyre (Bull. Symb. Logic 9 (2003), 197–212)
- “Logical Dreams” by Shaharon Shelah (Bull. Amer. Math. Soc. 40 (2003), 203–228)
- “Model Theory”, Anand Pillay, Notices of the AMS, December, 2000
- Model theory (article in the Stanford Encyclopedia of Philosophy by Wilfred Hodges)
- Model Theory page in the Open Directory Project—needs editor
- Model theory entry in the Wikipedia—needs work
Teaching
- “Telling the Truth” by Steven Zucker (an opinion piece in the Notices of the AMS, March 2003); see also the “orientation material” from the author's webpage).
- Common Errors in Undergraduate Mathematics, Eric Schechter, Vanderbilt Univ. Says the author: “This web page describes the errors that I have seen most frequently in undergraduate mathematics, the likely causes of those errors, and their remedies. I am tired of seeing these same old errors over and over again. (I would rather see new, original errors!) I show this document to my undergraduate students at the beginning of each semester.”
Other
- Mathematics Under the Microscope: Atomic objects, structures and concepts of mathematics (Alexandre Borovik)
- Zeilberg's Opinions
- Timothy Gowers
- Association des Collaborateurs de Nicolas Bourbaki
- Clay Mathematics Institute (Millennium Prize Problems)
- Connecting Mathematics (a thesaurus of some mathematical terms)
- Eric Weisstein's World of Mathematics. Use with discretion. Apparently it is a vast collection of mathematical facts, written down with an authoritative tone, but by somebody who may not know what they mean. See for example the entry on the Ax-Kochen Isomorphism Theorem, an entry which also gives no references. And how indeed does the mathematician respond to Mr Weisstein's entry for proof? Perhaps with the question of whether there is a difference between the good and the true, and in particular whether, say, number-theoretic conjectures are something good, regardless of whether anybody tries to prove them.
- Interactive Mathematics Miscellany and Puzzles "Without going into research and speculations as to what causes math anxiety I hope to create a resource that would help to learn, if not math itself, then, at least, ways to appreciate its beauty." (Alexander Bogolmolny)
- Euclid's Elements:
- English text, with Geometry Applets
- Greek text (also the full Heath edition in English)
- Hartshorne, Robin, Teaching geometry according to Euclid, Notices Amer. Math. Soc. 47 (2000), no. 4, 460--465
- Elementary Linear Algebra (Lecture Notes by Keith Matthews, 1991)