H2020Индивидуална стипендия2015–2017

ExplicitDarmonProg · Explicit Methods for the Darmon Programme

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2015-09-01 → 2017-09-30
Финансиране от ЕС: 195 455 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Елиптичните криви и т.нар. точки на Дармон се изследват чрез създаване на нови алгоритми и софтуер. Тези математически обекти са важни за криптографията и сигурността на онлайн банковите транзакции.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Explicit Methods for the Darmon Programme

Elliptic curves are mathematical objects that have been studied since Diophantus in the third century AD. They are now of vital importance in cryptography and cyber-security and are used for securing online bank transactions. In the 1960s, Birch and Swinnerton-Dyer performed massive experimental computations on elliptic curves using one of the earliest available computers (EDSAC II in Cambridge). On the basis of these computations they formulated what is now known as the Birch and Swinnerton-Dyer conjecture, one of seven very famous open problems in Mathematics, for which there is a $1M prize for resolving. All special cases of the Birch and Swinnerton-Dyer conjecture that have been proved rely on a construction due to German mathematician Heegner. In 2000, Canadian mathematician Darmon proposed a wide-ranging generalization of the Heegner construction, which has led to the concepts of Darmon points and cycles, and also to new proofs of special cases of the Birch and Swinnerton-Dyer conjecture. This project is concerned with Darmon points and cycles from an algorithmic point-of-view: it aims to develop constructions and software for these objects.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

Elliptic curves are the heart of contemporary research in number theory. For example, their modularity played a crucial role in Wiles' proof of Fermat's Last Theorem. Understanding their rational points is the subject of the Birch and Swinnerton-Dyer Conjecture (BSD), one of the seven $1M Clay Mathematics Institute Millennium Prizes. This proposal is concerned with rational points on elliptic curves and linked to both BSD and modularity.The Darmon Programme is an ambitious project initiated by Henri Darmon, aiming to provide constructions of points on elliptic curves over number fields, and to prove new cases of BSD. The proposed action is an initiative to extend the Darmon Programme, make known constructions explicit and algorithmic to provide extensive data that will be invaluable to researchers in the field, and guide further theoretical work.Warwick has one of the largest and most active explicit number theory groups in the world, consisting of 2 professors, 3 lecturers, 8 research fellows and 12 PhD students, making it a natural host for the project. The supervisor Siksek, is a leading expert on elliptic curves, rational points and modularity, with considerable experience in supervising research including 5 Marie Curie fellows.Masdeu did his undergraduate studies in Barcelona. He completed his PhD (McGill) under the supervision of Darmon, and therefore has intimate understanding of the Darmon Programme. Masdeu then worked as Ritt Assistant Professor at Columbia University before joining Warwick as a postdoc in 2014. Therefore his research experience has almost entirely been confined to North America, even though his ambition is to establish himself as an independent researcher at a prestigious European university. Masdeu has substantial research results with 8 papers accepted in strong journals. This project will integrate him into the European research environment, and give him new skills and research directions to enable him to realize his goal.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITY OF WARWICK · COVENTRYКоординаторОбединеното кралство