O-minimality and diophantine geometry
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
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Organisations
People |
ORCID iD |
| Jonathan Pila (Principal Investigator) |
Publications
Alex WILKIE
(2020)
Uniform parameterization of subanalytic sets and diophantine applications
in Annales scientifiques de l'École normale supérieure
Capuano L
(2016)
Rational points on Grassmannians and unlikely intersections in tori
in Bulletin of the London Mathematical Society
J. Pila
(2014)
O-minimality and diophantine geometry
Pila J
(2014)
Ax-Schanuel for the j-function
Pila J
(2016)
O-minimality and certain atypical intersections
in Annales scientifiques de l'École normale supérieure
Pila J
(2016)
Ax-Schanuel for the j-function
in Duke Mathematical Journal
Pila J
(2017)
On a modular Fermat equation
in Commentarii Mathematici Helvetici
Pila Jonathan
(2014)
Multiplicative relations among singular moduli
in arXiv e-prints
| Description | A primary objective of the project was the study of certain uniformization maps which generalize the exponential function. These uniformizations relate two different objects each of quite simple algebraic type, but the map represents a quotient by the action of an infinite discrete group action, and is not algebraic: it is transcendental. A classic theorem of Ax, about the exponential function, asserts that it is roughly speaking as transcendental as possible, avoiding interacting with algebraic subvarieties except when this is necessitated by its (well-known) functional equation: exp(x+y)=\exp(x).\exp(y). A major desideratum was to prove analogues of this result for more general maps of a similar type, and a key finding of the project was to prove such an analogue for the elliptic modular function. This result appears in a joint paper with Jacob Tsimerman (Toronto). A further objective was to use such results to generalize to further cases the approach of using o-minimality and point-counting to prove cases of the Zilber-Pink conjecture. A key finding in this direction is contained in a paper written jointly with Philipp Habegger (Basel), which gives, first, a result in the setting of abelian varieties which is the analogue of a well-known result of Bombieri-Masser-Zannier in the multiplicative case, and, second, a full conditional proof of Zilber-Pink in products of modular curves based on the abovementioned theorem for the elliptic modular function, though conditional on some conjectures for Galois orbits (which seem to be very difficult). |
| Exploitation Route | The results can (and have) been used by others in proving further results in these areas. |
| Sectors | Other |
| URL | https://www.maths.ox.ac.uk/people/jonathan.pila |
| Description | Model theory, functional transcendence, and diophantine geometry |
| Amount | £379,672 (GBP) |
| Funding ID | EP/N008359/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 02/2016 |
| End | 04/2019 |
| Description | A Colloquium and a seminar talk in Lyon |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Professional Practitioners |
| Results and Impact | A colloquium talk and (next day) a seminar talk to the Logic group in Lyon. |
| Year(s) Of Engagement Activity | 2015 |
| Description | A Colloquium in Basel |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | A colloquium (to a general mathematical audience) in Basel, which led to some questions and discussion. |
| Year(s) Of Engagement Activity | 2015 |
| Description | Hahn Lectures at Yale University |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Series of 3 lectures. |
| Year(s) Of Engagement Activity | 2018 |
| Description | Invited talk IHP Workshop March 2018 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Talk at the workshop ``Model Theory and applications'' associated with a semester programme at IHP Paris. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://www.youtube.com/watch?v=zsPouwyp0B8 |
| Description | Invited talk at Analytic Number Theory and Diophantine Geometry, Hanover |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | An invited talk at the meeting, which sparked some questions. |
| Year(s) Of Engagement Activity | 2015 |
| URL | https://sites.google.com/site/diophantine2015/ |
| Description | Invited talk at ICMS meeting on Recent developments in the applications of model theory...(Edinburgh) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A talk at this international meeting in honour of Angus Macintyre. |
| Year(s) Of Engagement Activity | 2014 |
| URL | http://www.icms.org.uk/workshops/macintyre |
| Description | Invited talk at ``Analysis, spectra, and number theory'', Princeton |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | An invited talk at this meeting in honour of Peter Sarnak, which sparked some questions. |
| Year(s) Of Engagement Activity | 2014 |
| URL | https://sites.google.com/site/asnt2014/ |
| Description | Invited talk at ``Moduli spaces and arithmetic geometry'', Leiden |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This talk was part of series of talks on recent progress on the Andre-Oort conjecture. |
| Year(s) Of Engagement Activity | 2015 |
| URL | https://www.lorentzcenter.nl/lc/web/2015/736/info.php3?wsid=736 |
| Description | Lecture in `Arithmetics and Geometry' (ETH) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A research talk at a meeting in arithmetic geometry celebrating 25 years of the `arithmetics and geometry' seminar at ETH (Zurich). |
| Year(s) Of Engagement Activity | 2013 |
| URL | https://www.math.ethz.ch/fim/conferences/past-conferences/2013/25-years-number-theory-seminar-eth-zu... |
| Description | Mathematics Colloquium at Princeton University |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A Colloquium talks in the mathematics department at Princeton University. |
| Year(s) Of Engagement Activity | 2018 |
| Description | Minicourse lecturer in LMS-EPSRC minicourse |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | Series of 5 lectures, 8-12 July 2013, Manchester University. Nothing notable. |
| Year(s) Of Engagement Activity | 2013 |
| URL | http://www.cl.eps.manchester.ac.uk/medialand/maths/archived-events/workshops/www.mims.manchester.ac.... |
| Description | Number Theory Seminar, EPFL, Lausanne |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Other academic audiences (collaborators, peers etc.) |
| Results and Impact | Seminar talk. None |
| Year(s) Of Engagement Activity | 2014 |
| Description | Number Theory Seminar, York |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Other academic audiences (collaborators, peers etc.) |
| Results and Impact | Seminar talk. Nothing notable. |
| Year(s) Of Engagement Activity | 2013 |
| Description | Periods and Motives, 8-12 July 2019, Berlin |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A one-week meeting in Berlin on topics around periods and motives. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www2.mathematik.hu-berlin.de/~klingleb/periods2019/main |
| Description | Pleanry talk at Clay Research Conference 2014, Oxford |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Policymakers/politicians |
| Results and Impact | A Plenary talk at the CRC 2014 (Oxford), video available online. |
| Year(s) Of Engagement Activity | 2014 |
| URL | http://www.claymath.org/events/2014-clay-research-conference-and-workshops |
| Description | Pleanry talk at ICM 2014 (Seoul) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a Plenary talk at the ICM 2014 (Seoul), with video available online. |
| Year(s) Of Engagement Activity | 2014 |
| URL | http://www.icm2014.org/ |
| Description | Ritt Lectures, Spring 2013 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Other academic audiences (collaborators, peers etc.) |
| Results and Impact | Ritt Lectures Spring 2013 at Columbia University, New York, series of 2 lectures. Nothing notable. |
| Year(s) Of Engagement Activity | 2013 |
| URL | http://www.math.columbia.edu/2013/04/23/spring-2013-ritt-lectures-by-prof-jonathan-pila/ |
| Description | Seminar talk, Hebrew University, Jerusalem |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A seminar talk. |
| Year(s) Of Engagement Activity | 2013 |
| Description | Talk at Final ERC meeting in Diophantine geometry, Rome |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | An invited talk at this ERC meeting, which sparked some questions. |
| Year(s) Of Engagement Activity | 2015 |
| URL | http://diophantinegeometry2015.sns.it/ |
| Description | Talk at `L-functions, spectra, and equidistribution' meeting at Hebrew University (Jerusalem) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | An invited talk at the meeting, which sparked some questions. |
| Year(s) Of Engagement Activity | 2013 |
| URL | http://www.as.huji.ac.il/schools/math17 |
| Description | Talk in number theory seminar at Princeton University |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Seminar talk |
| Year(s) Of Engagement Activity | 2018 |
| Description | Tarski Lectures 2013 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Other academic audiences (collaborators, peers etc.) |
| Results and Impact | Delivered the Tarski Lectures 2013 at University of California, Berkeley, series of 3 lectures. Nothing notable. |
| Year(s) Of Engagement Activity | 2013 |
| URL | http://math.berkeley.edu/2013-tarski-lectures |
| Description | Weyl Lectures at IAS Princeton |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Series of 3 lectures. URL link is to first lecture. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://www.youtube.com/watch?v=SizCK6Yk9sA |
| Description | Workshop on Functional Transcendence around Ax-Schanuel |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | A workshop associated with the Clay Research Conference 2014. |
| Year(s) Of Engagement Activity | 2014 |
| URL | http://www.claymath.org/events/functional-transcendence-around-ax-schanuel |