Miniworkshop

Oxford, July 5-6, 2016

Future Directions in Model Theory and Analytic Functions
MIMS, Manchester, 7-10 July, 2015

Functional Transcendence around Ax-Schanuel
CMI workshop, Oxford, 29 September - 3 October, 2014

Model Theory in Geometry and Arithmetic
MSRI, 12-16 May, 2014

L-functions, Spectra and Equidistribution
IAS, Hebrew University of Jerusalem, 11-19 December, 2013

Synergies And Vistas in Analytic Number Theory
Oxford, 3-7 September, 2012

Miniworkshop in O-minimality and Diophantine Geometry
Oxford, 24-25 August, 2012

Unlikely intersections in algebraic groups and Shimura varieties
AIM workshop held at Centro De Giorgi, Scuola Normale Pisa, March 28 -April 1, 2011

Workshop in Computational Number Theory
July 7-9, 2005, University of Cantabria, Santander, Spain, part of FoCM'05

Far Hills Workshop on L-functions and p-adic cohomology: computational perspectives
CICMA-CRM workshop, 2-4 January 2004.

Future directions in algorithmic number theory
AIM workshop, Palo Alto, March 24-28, 2003

Lecture notes and surveys

Clay Research Conference 2011
``Diophantine geometry via model theory'', video here.

LMS-EPSRC Shortcourse on O-minimality and Diophantine Geometry
Manchester, July 8-12, 2013
Functional transcendence via o-minimality (5 lectures), pdf notes here.
The copyright in these lecture notes belongs to CUP. They will appear in final form
in a volume published by CUP. These notes may only be used for non-commercial purposes.

ICM, Seoul 2014
``O-minimality and Diophantine geometry'',
Expanded version of talk as preprint here, slides here, and the talk itself can be viewed here.

Clay Research Conference 2014
``The Schanuel paradigm'', slides here, video on Clay Mathematics Institute website here.

Graduate students

My thesis was supervised by Peter Sarnak, so I belong to a large academic family; see here.

My students:

Andrew Potter, PhD (Bristol), 2010

Lee Butler, PhD (Bristol), 2011

Adam Harris, DPhil (Oxford), 2013, joint supervision with Boris Zilber

Vahagn Aslanyan, DPhil (Oxford), 2017, joint supervision with Boris Zilber

Haden Spence, DPhil (Oxford), 2018

Sebastian Eterovic, DPhil (Oxford), 2019

John Armitage, DPhil (Oxford), 2021

Guy Fowler, DPhil (Oxford), 2021

Francesco Ballini, current

Some more papers (preprints may differ from final published form)

Frobenius maps of abelian varieties and finding roots of unity in finite fields,
Math. Comp. 55 (1990), 745--763, preprint.

Geometric postulation of a smooth function and the number of rational points,
Duke Math. J. 63 (1991), 449--463, preprint version by permission of the publisher.

Geometric and arithmetic postulation of the exponential function,
J. Austral. Math. Soc. Ser. A 54 (1993), 111--127, preprint.

Density of integer points on plane algebraic curves,
IMRN 1996 (1996), 903--912, preprint.

(with F. R. Villegas) Concordant sequences and integral-values entire functions,
Acta Arith. 88 (1999), 239--268, preprint.

(with H. W. Lenstra and C. Pomerance) A hyperelliptic smoothness tes, II,
Proc. London Math. Soc. (3) 84 (2002), 105--146, preprint.

Entire functions having a concordant value sequence,
Israel J. Math. 134 (2003), 317--343, preprint.

Integer points on the dilation of a subanalytic set,
Quarterly J. Math. 55 (2004), 207--223, preprint.

Note on Carlson's Theorem,
Rocky Mountain Math. J. 35 (2005), 2107--2111, preprint.

Rational points on a subanalytic surface,
Ann. Inst. Fourier 55 (2005), 1501--1516, preprint.

The density of rational points on a Pfaff curve,
Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), 635--645, preprint.

Entire functions sharing arguments of integrality, II,
Publ. Math. Debrecen 73 (2008), 215--231, preprint.

Entire functions sharing arguments of integrality, I,
Int. J. Number Theory 5 (2009), 339--353, preprint.

On the algebraic points of a definable set,
Selecta Math. (N.S.) 15 (2009), 151--170, preprint.
The final published version is available at link.springer.com, here.

Rational points of definable sets and results of Andre-Oort--Manin-Mumford type,
IMRN, 2009, 2476--2507, available here.

Counting rational points on a certain exponential-algebraic surface,
Ann. Inst. Fourier 60 (2010), 489--514, preprint. And correction (all theorems remain true).

Modular Ax-Lindemann-Weierstrass with derivatives
Notre Dame J. Formal Logic 54 (2013), 553--565 (Oleron volume in honour of Pillay).
Preprint version here.

Special point problems with elliptic modular surfaces,
Mathematika 60 (2014), 1-31, doi:10.1112/S0025579313000168, copyright UCL.
Preprint version here.

(with P. Habegger) Some unlikely intersections beyond Andre-Oort,
Compositio 148 (2012), 1-27. Preprint version here.

(with J. Tsimerman) The Andre-Oort conjecture for the moduli space of Abelian surfaces,
Compositio 149 (2013), 204-216, arXiv version here.

(with L. Capuano, D. Masser, and U. Zannier)
Rational points on Grassmannians and unlikely intersections in tori,
Bull. LMS 48 (2016), 141-154. Published version here.

(with J. Tsimerman)
Multiplicative relations among singular moduli,
arXiv preprint here, Annali della Scuola Normale Superiore di Pisa, to appear.

On a modular Fermat equation, arXiv preprint here,
Commentarii Mathematici Helvetici 92 (2017), 85-103.