# DIOPHANTINE PROBLEMS Streszczenie raportu

Project ID:
267273

Źródło dofinansowania:
FP7-IDEAS-ERC

Kraj:
Italy

## Final Report Summary - DIOPHANTINE PROBLEMS (Integral and Algebraic Points on Varieties, Diophantine Problems on Number Fields and Function Fields)

Events organized within the project and funded by it:

1- Pisa meeting September 8th - October 8th 2012. Over seventy participants from various countries attended this meeting (in different periods), delivering talks and collaborating among themselves. All main themes of the project were touched (Unlikely Intersections and relations with o-minimality, Algebraic dynamics, Integral points, Diophantine equations).

2- Cetraro meeting, July 13-28th 2014: a continuation of the Pisa meeting with a smaller duration. There have been around fourty confirmed participants, including well-known distinguished mathematicians and directors of celebrated mathematical institutions.

3- Roma meeting 2015 at Accademia dei Lincei : This has been almost a conclusive step of the project, withvery few emblematic talks on the achievements of the project.

4 – Meeting of Cetraro 2015: this short meeting involved especially PI and team members Corvaja and Mantova, Canci, ERC post-docs Capuano and Barroero, and a few other participants. It was mainly inteded to go ahead with some collaborations and to explain in a few talks some of the results.

Post-doc grants:

After a previous post-doc grant assigned to Martin Widmer (who interrupted earlier then expected beacause of a permanent position), we have assigned three other grants; two have been declined, one for health reason and a second for a permanent position, whereas Dr. Lars Kuehne (PhD ETH Zurich) has started in March 2013, for about two years.

PhD V. Mantova too has collaborated and is still collaborating with the PI on themes of the project and similarly for SNS PhD students L. Capuano and O. Merkert. Also some students of other team members (Corvaja, Masser, Pila, Habegger) are working on themes of the project.

Participation to international events:

Several talks have been delivered on the topics of the project by the PI and team members, for instance at the Hindry school in Paris (June 2012), Oxford University, Cetraro meeting on logic, CNTA 12 (Canadian international number theory meeting), Szpiro conference (New York 2012), Wuestholz conference (ETH 2013), Chennai 2013, Princeton 2013, Journees Arithmetiques 2013, Thue conference in Bordeaux September 2013 (Corvaja, Masser, PI, Evertse, Levin, Habegger), Austro-German biennial meeting (PI) September 2013, ICM Seoul, CNTA XII, Schmidt conference and ESI meeting in Vienna (PI, Corvaja, Masser, Habegger).

Publications:

At least 8 papers on unlikely intersections (sometimes joint of the PI with Masser and Corvaja and one paper with Capuano and Pila) have been published (JEMS, Math. Annalen, Advances in Math., Ens. Math., Edinburgh J Math., J. Bordeaux, Proc. Cambridge Math. Soc., Bull. London Math. Soc.), or submitted. Several other papers on integral points and related topics have been also published (on JEMS, J. Diff. Geometry, Annales Inst. Fourier, Proc. A.M.S.). Several further papers are in the course of being written or already submitted. A book of the PI published by the prestigeous Princeton UP, Annals of Math. Studies, has appeared in 2012.

1- Pisa meeting September 8th - October 8th 2012. Over seventy participants from various countries attended this meeting (in different periods), delivering talks and collaborating among themselves. All main themes of the project were touched (Unlikely Intersections and relations with o-minimality, Algebraic dynamics, Integral points, Diophantine equations).

2- Cetraro meeting, July 13-28th 2014: a continuation of the Pisa meeting with a smaller duration. There have been around fourty confirmed participants, including well-known distinguished mathematicians and directors of celebrated mathematical institutions.

3- Roma meeting 2015 at Accademia dei Lincei : This has been almost a conclusive step of the project, withvery few emblematic talks on the achievements of the project.

4 – Meeting of Cetraro 2015: this short meeting involved especially PI and team members Corvaja and Mantova, Canci, ERC post-docs Capuano and Barroero, and a few other participants. It was mainly inteded to go ahead with some collaborations and to explain in a few talks some of the results.

Post-doc grants:

After a previous post-doc grant assigned to Martin Widmer (who interrupted earlier then expected beacause of a permanent position), we have assigned three other grants; two have been declined, one for health reason and a second for a permanent position, whereas Dr. Lars Kuehne (PhD ETH Zurich) has started in March 2013, for about two years.

PhD V. Mantova too has collaborated and is still collaborating with the PI on themes of the project and similarly for SNS PhD students L. Capuano and O. Merkert. Also some students of other team members (Corvaja, Masser, Pila, Habegger) are working on themes of the project.

Participation to international events:

Several talks have been delivered on the topics of the project by the PI and team members, for instance at the Hindry school in Paris (June 2012), Oxford University, Cetraro meeting on logic, CNTA 12 (Canadian international number theory meeting), Szpiro conference (New York 2012), Wuestholz conference (ETH 2013), Chennai 2013, Princeton 2013, Journees Arithmetiques 2013, Thue conference in Bordeaux September 2013 (Corvaja, Masser, PI, Evertse, Levin, Habegger), Austro-German biennial meeting (PI) September 2013, ICM Seoul, CNTA XII, Schmidt conference and ESI meeting in Vienna (PI, Corvaja, Masser, Habegger).

Publications:

At least 8 papers on unlikely intersections (sometimes joint of the PI with Masser and Corvaja and one paper with Capuano and Pila) have been published (JEMS, Math. Annalen, Advances in Math., Ens. Math., Edinburgh J Math., J. Bordeaux, Proc. Cambridge Math. Soc., Bull. London Math. Soc.), or submitted. Several other papers on integral points and related topics have been also published (on JEMS, J. Diff. Geometry, Annales Inst. Fourier, Proc. A.M.S.). Several further papers are in the course of being written or already submitted. A book of the PI published by the prestigeous Princeton UP, Annals of Math. Studies, has appeared in 2012.