H2020Индивидуална стипендия2020–2022

MotivicCircleMethod · A motivic circle method

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

Период: 2020-07-01 → 2022-10-24
Финансиране от ЕС: 186 167 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Методът на Харди-Литълвуд се адаптира към геометричен контекст, за да се анализират пространства от рационални криви върху хиперповърхности. Това помага за разбирането на структурата им и връзката между кубичните четиримерни многообразия и повърхностите K3 в алгебричната геометрия.

Този кратък обзор е генериран от изкуствен интелект

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

Резултати накратко

A motivic circle method

The Hardy–Littlewood circle method is a well-known technique of analytic number theory that has been used to solve several important number theory problems such as the ternary Goldbach conjecture. It has also benefitted other fields, including quantum computing and algebraic geometry. The aim of the project is to adapt the method to the so-called motivic setting, giving it a more geometric flavour. The proposed applications concern spaces of rational curves on hypersurfaces. These spaces are of great interest in theoretical physics and understanding their structure can shed light on several open questions in algebraic geometry and number theory related to homological stability, weak approximation and equidistribution. Moreover, in the case of cubic fourfolds the motivic circle method is expected to provide a gateway to a series of heuristics on the relationship of these fourfolds with K3 surfaces, and thus possibly allow a new angle of attack for the major open question of the rationality of cubic fourfolds.

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

Цел на проекта

The Hardy-Littlewood circle method is a well-known technique coming from analytic number theory, which has been very successful, throughout the last century, in solving many major number-theoretic problems, among which the ternary Goldbach conjecture. In recent years, it has proven even more versatile, finding several applications beyond number theory, for example in quantum computing and algebraic geometry. The aim of this proposal is to adapt the circle method to the so-called motivic setting and apply it to a range of geometric problems. The possibility of a motivic circle method has been raised as an open question by several renowned researchers, as an increasing number of mathematicians are becoming interested in motivic methods because of their numerous applications to other areas of mathematics. The goal is to create a clear and convenient setup for the motivic circle method accessible to mathematicians less used to the motivic setting, highlighting the analogies with the usual circle method. The proposed applications concern spaces of rational curves on hypersurfaces. Describing such spaces is of interest in theoretical physics and is a very active field of study in mathematics, as understanding their structure can shed light on several major open questions in algebraic geometry and number theory. The candidate plans to carry out a thorough investigation of the geometry of such spaces using the motivic circle method, greatly generalising previous results. This will initiate a systematic approach to the study of spaces of rational curves in the motivic framework, and will in particular extend previous work by the proposed supervisor Prof. Tim Browning and his coauthors. The choice of supervisor and host institution is a natural one: while the proposed supervisor Prof. Tim Browning is a leading expert of the circle method, the candidate is very experienced with the motivic setting, and both have previously worked with spaces of rational curves.

Оригинален текст от CORDIS (на английски).

Участници

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA · KlosterneuburgКоординаторАвстрия