H2020Индивидуална стипендия2021–2022

RRMAP · Riemann-Roch and motives for arithmetic problems

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

Период: 2021-01-01 → 2022-12-31
Финансиране от ЕС: 172 932 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Аритметичната геометрия прилага методи от алгебричната геометрия към проблеми в теорията на числата, като например използването на теоремата на Риман-Рох. Тези инструменти помагат за описанието на геометричните и аритметичните свойства на обектите, с които работи съвременната математика.

Този кратък обзор е генериран от изкуствен интелект

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

Резултати накратко

Riemann-Roch and motives for arithmetic problems

Mathematics was born in Greece 2500 years ago with the study of Geometry. Since then, Geometry has been at the heart of Western civilization and its Philosophy, as Plato prescribed at the Academia. Since then, our society has pursued to unveil the secrets of Geometry. The development of Geometry, following Gauss, Riemann and Grothendieck, has been instrumental in our modern understanding of our world. From Einstein's General Relativity to the expected formalizations in a unifying theory of the macro and micro study of the world we live in, we rely on geometric tools to describe the phenomena we observe. The The current research project is framed in the area of Arithmetic Geometry, the part of Mathematics which applies techniques and ideas borrowed from Algebraic Geometry to Number Theory problems, which should be considered as the direct heir to Greek reflections on Geometry. More concretely, during the nineties, arithmetic geometers of the previous century have successfully developed a satisfactory homotopy theory, called motivic homotopy theory. This new framework has been successful enough to provide the motivic analog of Grothendieck's six functors formalism, the basic formalism for satisfactory cohomology theory. These are the modern tools mathematicians use to define "shape" and Many Number Theory problems are related to cohomology theories such as higher K-theory or the Chow ring. A central theorem of the subject is the celebrated Grothendieck's Riemann-Roch theorem and Gillet's extension to higher K-theory. This theorem compares the direct image in cohomology of K-theory and the Chow ring. Combined with regulators, the Riemann-Roch theorem provides formulas describing both arithmetic and geometric properties. In this research project, I propose to develop Riemann-Roch-type theorems and motivic techniques to attack arithmetic problems.

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

Цел на проекта

Our project “Riemann-Roch and Motives for Arithmetic Problems” aims to develop techniques in the area of Motives and the Riemann-Roch to attack arithmetic problems. To be more concrete we aim to attack:- The integral Riemann-Roch: At SGA VI Grothendieck developed his landmark Riemann-Roch result stating an integral version of it as an open question. Later on, research of Fulton, MacPherson and Pappas raised Grothendieck original conjecture to a more complete statement related to traces, which is known today only in the complex geometric setting. We aim to prove this conjecture in its full generality. -The discrete Riemann-Roch: At SGA5 Grothendieck proved his wellknown Ogg-Shafarevich formula computing the Euler characteristic of a constructible sheaf over curve in terms of the genus, the Swan conductor and therank. This formula plays a central role in the original strategy to prove the Weyl conjectures. Grothendieck also conjectured that this formula would fit into a Riemann-Roch type theorem for the K-group of étale constructible sheaves and general schemes, which he called the “discrete Riemann-Roch”. We aim to attack this theorem from the motivic point of view.-Intersection theory in the arithmetic setting: A major objective of Algebraic Geometry is to define a product algebraic cycles forin the arithmetic setting. So far, this product has being defined with rational coefficients. The first definition, due to Gillet-Soulé, was achieved throughout the Adam’s operations, the Adams Riemann-Roch and theGrothendieck-Riemann-Roch. We aim to explore some of Gillet-Soulé’s ideas and the arithmetic bivariant integral version of the Riemann-Roch to explore a definition of the intersection product of cycles after killing certain torsion on the Chow groups related to the codimension of the cycle

Оригинален текст от CORDIS (на английски).

Участници

AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS · MadridКоординаторИспания