## Publications

### Pure mathematics

29. (with K. Juschenko) Soficity, short cycles and the Higman group, preprint.28. (with P. Candela) On the dimension of additive sets,

*Acta Arith.*167 (2015), no. 1, 91-100.

27. Random generators of the symmetric group: diameter, mixing time and spectral gap (with Á. Seress and A. Zuk),

*J. of Algebra.*421 (2015), 349-368.

26. The ternary Goldbach conjecture is true, preprint.

25. Major arcs for Goldbach's problem, preprint.

24. (with D. Platt) Numerical verification of the ternary Goldbach conjecture up to 8.875e30,

*Experiment. Math.*22 (2013), no. 4, 406-409.

23. Growth in groups: ideas and perspectives,

*Bull. Am. Math. Soc.*52 (2015), no. 3, 357-413.

22. (with N. Gill and M. Rudnev) On growth in an abstract plane,

*Proc. Am. Math. Soc.*143 (2015), no. 8, 3593-3602

21. Minor arcs for Goldbach's problem, preprint.

20. Bounds on the diameter of Cayley graphs of the symmetric group (with J. Bamberg, N. Gill, T. Hayes, Á. Seress and P. Spiga),

*J. Algebraic Combin.*

19. Square-free values of f(p), f cubic,

*Acta Math.*213 (2014), no. 1, 107-135.

Th3. Groupes, courbes et croissance, habilitation thesis, Paris-Sud (Orsay).

18. On the diameter of permutation groups (with Á. Seress),

*Annals of Math.*179 (2014), no. 2, . 1, 107-135. 611-658.

17. Deterministic methods to find primes (as

*D. H. J. Polymath*, with T. Tao and E. Croot),

*Math. Comp.*81 (2012), no. 278, 1233-1246.

16. Growth in solvable subgroups of GL

_{r}(Z/pZ) (with N. Gill),

*Math. Annalen*360 (2014), no. 1-2, 157-208.

15. Growth of small generating sets in SL

_{n}(Z/pZ) (with N. Gill),

*Int. Math. Res. Notices.*, Vol. 2011, 4226--4251.

14. An explicit incidence theorem in F

_{p}(with M. Rudnev),

*Mathematika*, 57 (2011), no. 1, 135--145.

13. Improving Roth's theorem in the primes (with A. de Roton), Int. Math. Res. Notices. (2011), Vol. 2011, 767--783.

12. Growth in SL

_{3}(Z/pZ), J. Eur. Math. Soc. (JEMS), vol. 13, no. 3, pp. 761--851.

11. Power-free values, repulsion between points, differing beliefs and the existence of error, CRM Proceedings and Lecture Notes, v. 46 (2008), 81-88.

10. How small must ill-distributed sets be? (with A. Venkatesh), in: Chen, W. W. L. (ed.) et al.,

*Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday.*Cambridge University Press, 2009, pp. 224-234.

9. Growth and generation in SL

_{2}(Z/pZ),

*Ann. of Math.*167 (2008), 601-623.

8. The parity problem for irreducible polynomials, submitted.

7. Power-free values, large deviations and integer points on irrational curves,

*J. Théor. Nombres Bordeaux*19 (2007), 433-472.

6. The parity problem for reducible polynomials,

*J. London Math. Soc. (2)*73 (2006), 415-435.

5. Integral points on elliptic curves and 3-torsion in class groups (with A. Venkatesh),

*J. Amer. Math. Soc.*19 (2006), 527--550.

4. Root numbers and ranks over global function fields (with B. Conrad and K. Conrad),

*Adv. Math.*198 (2005), 684--731.

3. On the behaviour of root numbers in families of elliptic curves, submitted.

2. On the square-free sieve,

*Acta Arith.*115 (2004), 349-402.

Th2. Root numbers and the parity problem, Ph.D. thesis, Princeton University, April 2003, math.NT/0305435.

1. Enumeration of tilings of diamonds and hexagons with defects (with I. M. Gessel),

*Electron. J. Combin.*6 (1999), no. 1, R16.

Th1. Edge effects on local statistics in lattice dimers: a study of the Aztec diamond (finite case), senior thesis, Brandeis University, May 1998, math.CO/0007136.

### Book - Pure mathematics

B1. The ternary Goldbach problem, to appear in*Ann. of Math. Studies*.

*Note: B1 supersedes preprints 26, 25 and 21 above.*

### Monographs - Pure mathematics

M2. Isomorphismes de graphes en temps quasi-polynomial (d'apres Babai et Luks, Weisfeiler-Leman,. . . ). Exposé no 1125 du séminaire Bourbaki. To appear in*Astérisque*.

M1. Azar y aritmética,

*Monografías del Instituto de Matemática y Ciencias Afines*, v. 50, IMCA, Lima, Perú.

### Popular articles - pure mathematics

GP3. The ternary Goldbach problem, to appear in*Proc. ICM (Korea, 2014).*

GP2. La conjecture de Goldbach ternaire, to appear in

*Gaz. Math.*

GP1. La conjetura débil de Goldbach,

*Gac. R. Soc. Mat. Esp.*16 (2013), no. 4.

*Note: GP1, GP2 and (in part) GP3 are based on my exposition in English on the subject.*

### History and pedagogy

HP3. A modern vision of the work of Cardano and Ferrari on quartics (with M. Helfgott),*CONVERGENCE*, an online journal of the Mathematical Association of America, July 2009.

HP2. A noncalculus proof that Fermat's principle of least time implies the law of refraction (with M. Helfgott),

*Am. J. Phys.*70 (2002), no. 12, 1224-1225.

HP1. Maxima and minima before Calculus (with M. Helfgott),

*Pro Mathematica*XII (1998), nos. 23-24, 135-158.