# 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 TestSolution Sheet 3Lecture Notes Week 72012 exam2013 examLecture Notes Week 8Solution Sheet 4Lecture Notes Week 9Extra Reading Solutions 1-8Extra Reading Solutions 9-17## 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.