Research

Most of my work is in that part of mathematical logic called model theory. Here is a research statement (February 22, 2012; 33 pages, size A5, 10 point type):

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Articles

Oldest to newest:

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. MR2505433 (2011a:03032)
6. Fields with several commuting derivations. Journal of Symbolic Logic 79 (2014), no. 1, 1–19.
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. Induction and Recursion, The De Morgan Journal, 2 no. 1 (2012), 99–125
9. Representation theorems for rings (an investigation of what makes associative rings and Lie rings special among rings; draft).
10. Chains of theories and companionability. With Özcan Kasal. Proceedings of the American Mathematical Society 143 (2015), 4937–4949.
11. Abscissas and Ordinates. Journal of Humanistic Mathematics 5 (2015), no. 1, pages 233–264. DOI: 10.5642/jhummath.201501.14
12. On the Foundations of Arithmetic in Euclid (draft).
13. Thales and the Nine-point Conic, The De Morgan Gazette 8 no. 4 (2016) 27–78.
14. On Commensurability and Symmetry, Journal of Humanistic Mathematics, Volume 7, Issue 2 (July 2017), pages 90–148. DOI: 10.5642/jhummath.201702.06
15. Fields with automorphism and valuation. With Özlem Beyarslan, Daniel Max Hoffmann, and Gönenç Onay. arXiv:1710.07173 [math.LO]. Archive for Mathematical Logic: published online, 31 March 2020. DOI: 10.1007/s00153-020-00728-5
16. Conics in Place. Annales Universitatis Paedagogicae Cracoviensis | Studia Ad Didacticam Mathematicae Pertinentia, 13, 12–150. DOI: 10.24917/20809751.13.2
17. Conic Diagrams, Journal of Humanistic Mathematics, Volume 12 Issue 2 (July 2022), pages 378–398. DOI: 10.5642/jhummath.NFHQ4170

The following is not research, but an essay; it is not particularly about mathematics, though readers are assumed to be interested in education and mathematics:

• St John's College, The De Morgan Journal, 2 no. 2 (2012), pp. 63–73