# Prof. Alex Wilkie FRS

- Room 2.137
- School of Mathematics
- University of Manchester
- Oxford Road, Manchester, M13 9PL, UK
- alex.wilkie[at]manchester.ac.uk
- Tel: +44 (0) 161 275 5858
- Fax: +44 (0) 161 275 5819

### School Responsibilities:

**Alex Wilkie ** FRS is a mathematician known for his contributions to Model theory and logic . Previously professor of mathematics at the University of Oxford , he was appointed to the Fielden Chair of Pure Mathematics at the University of Manchester in 2007 .

Wilkie gained his PhD from the University of London in 1973 under the supervision of Wilfrid Hodges with a dissertation entitled *Models of Number Theory *. He was elected a Fellow of the Royal Society in 2001.

## Page contents:

## Sequences and Series

## Model Theory

Comments on the 2014 exam performance

## Fourier Analysis and Lebesgue Integration

Chapter 2: Countability and Cantor Sets

Chapter 3: Measure and Lebesgue Integration

Chapter 4: Fourier Series and the Space of Square-integrable Functions

Extra Reading Material for Levels 4 and 6

Lecture Notes Week 1Solution Sheet 1Lecture Notes Week 2Lecture Notes Week 3Solution Sheet 2Lecture Notes Week 4Lecture Notes Week 5Coursework Test## Publications

1. On models of arithmetic-answers to two problems raised by H. Gaifman, J Symb Logic, 40 (1975) (1), 41-47.

2. A note on products of finite structures with an application to graphs, J Lond Math Soc (2), 14 (1976), 383-384.

3. On the theory of end-extensions of models of arithmetic, in: Set Theory and Hierarchy Theory V, SLNM 619, Springer-Verlag, 1997, 305-310.

4. On models of arithmetic having non-modular substructure lattices, Fund. Math., XCV (1977), 223-237.

5. Reconstruction theorems for families of sets (with R Rado), J Lond Math Soc (2), 17 (1978), 5-9.

6. Applications of complexity theory to sigma-zero definability problems in arithmetic, in: Model Theory of Algebra and Arithmetic, SLNM 834, Springer-Verlag, 1980, 363-369.

7. Some results and problems on weak systems of arithmetic, in: Logic Colloquium '77, North-Holland, 1980, 285-296.

8. Models of arithmetic and the rudimentary sets (with J B Paris), Bull Soc Math Belg 33 (1981), 1, 157-169.

9. On discretely ordered rings in which every definable ideal is principal, in: Model Theory and Arithmetic, SLNM 890, Springer-Verlag, 1981, 297-303.

10. On core structures for Peano arithmetic, in: Logic Colloquium '80, North-Holland, 1982, 311-314.

11. Delta-zero sets and induction (with J B Paris), in: Open Days in Model Theory and Set Theory, Leeds University, 1983, 237-248.

12. Gromov's theorem on groups of polynomial growth and elementary logic (with L van den Dries), J Algebra, 89 (1984), 391-396.

13. An effective bound for groups of linear growth (with L van den Dries), Arch Math, 42 (1984), 391-396.

14. Counting problems in bounded arithmetic (with J B Paris), in: Methods in Mathematical logic, SLNM 1130, Springer-Verlag, 1985, 317-340.

15. Characterizing some low arithmetic classes (with J B Paris and W G Handley), Coll Math Soc Janos Bolyai, 44 (1984), 353-364.

16. Modeles non-standard en arithmetique et theorie des ensembles (with J-P Ressayre), Pub Math de l'universite Paris VII,1986, (147 pages).

17. On sentences interpretable in systems of arithmetic, in: Logic Colloquium '84, North-Holland, 1986, 329-342.

18. Classification of quantifier prefixes over exponential diophantine equations (with J P Jones and H Levitz), Z fur Math Log und Grund der Math, 32 (1986), 388-406.

19. Counting delta-zero sets (with J B Paris), Fund Math,127 (1) (1986), 67-76.

20. On the scheme of induction for bounded arithmetic formulas (with J B Paris), Annals of Pure and Applied Logic, 35 (1987), 261-302.

21. On schemes axiomatizing arithmetic, in: Proc of ICM, Berkeley, Ca, USA, 1986 (1988), 331-337.

22. Provability of the Pigeonhole principle and the existence of infinitely many primes (with J B Paris and A R Woods), J Symb Logic, 53, (1988), 12355-1244.

23. On the theory of the real exponential field,Illinois J Math, 33, 3, (1989), 384-408.

24. On the existence of end extensions of models of bounded induction, in: Logic, Methodology and Philosophy of Science, VIII (Moscow 1987), Stud Logic and Found Math, 126, North Holland, 1989, 143-161.

25. On defining C-infinity,J Symb Logic, 59, (1994), (1), 344.

26. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J Amer Math Soc, 9, (4), 1996, 1051-1094.

27. On the decidability of the real exponential field (with A J Macintyre), in: Kreiseliana.About and Around Georg Kreisel', A K Peters, 1996, 441-467.

28. Schanuel's conjecture and the decidability of the real exponential field, in: Algebraic Model Theory, 1997, Kluwer, 223-230.

29. O-minimality, in: Proceedings of the ICM, Berlin, 1998, Vol 1, Documenta Mathematica, J.DMV, 1998, 633-636.

30. Model theory of analytic and smooth functions,in:Models and Computability, LMS Lecture Notes Series 259, CUP, 1999, 407-419.

31. A theorem of the complement and some new O-minimal structures, Selecta Mathematica, New Ser.5 (1999), 397-421.

32. On exponentiation - a solution to Tarski's High School Algebra Problem, in: Connections between Model Theory and Algebraic and Analytic Geometry, Quaderni di Matematica, vol. 6 (ed. by Angus Macintyre), Naples, 2000, 107-129. ( dvi file , postscript file )

33. Quasianalytic Denjoy-Carleman classes and *o *-minimality (with J-P Rolin and P Speissegger), J Amer Math Soc,16, (2003), (4), 751-777. (See http://www.math.wisc.edu/~speisseg/preprints/quasi.ps.)

34. The laws of integer divisibility and solution sets of linear divisibilty conditions (with L van den Dries), J Symb Logic, 68, (2003), (2), 503-526. ( postscript file )

35. Diophantine properties of sets definable in *o *-minimal structures, J Symb Logic, 69, (2004), (3), 851-861. ( dvi file )

36. Fusing *o *-minimal structures, J Symb Logic, 70, Number 1, March 2005, 271-281 ( dvi file )

37. Covering definable open sets by open cells, Proceedings of the RAAG Summer School Lisbon 2003: O-minimal structures, Eds M. Edmundo, D. Richardson, A.J. Wilkie (2005), 77-103 ( dvi file )

38. Liouville functions, Lecture Notes in Logic 19, Logic Colloquium 2000, Eds R. Cori, A. Razborov, S. Tudorcevic, C. Wood (2005), 383-391 ( dvi file )

39. The rational points of a definable set. (with J Pila.) Duke Mathematical J,Vol.133, No.3 (2006), 591-616.

40. Some local definability theory for holomorphic functions. Model Theory with Applications to Algebra and Analysis, Vol 1 (2008) LMS Lecture Note Series 349, CUP, 197-213.

41. Locally polynomially bounded structures (with G. O. Jones), Bull. London Math. Soc. 40 (2008) 239-248.

42. O-minimal structures, Expose no.985, Seminaire Bourbaki, Volume 2007/2008, Asterisque (2009).

43. A Schanuel property for exponentially transcendental powers, (with Martin Bays and Jonathan Kirby), Bull. London Math. Soc. 42 (2010) 917-922.