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КартаВръзкиТемиБългарияНощта (K-TRIO)Мария Склодовска-КюриЗа ученициЗа учителиМетодологияИзненадай ме
FP6Докторантска мрежа2006–2010

GTEM · Galois theory and explicit methods

6РП — Действия „Мария Кюри“

Период
2006-10-01 → 2010-09-30
Финансиране от ЕС
2 574 379 €
Участници
13
Схема
RTN

Линиите свързват координатора с партньорите. За проекти отпреди 2014 г. CORDIS не винаги дава точни координати. Тези точки са на ниво град или държава.

Накратко на български

Теорията на Галоа и аритметичната геометрия се използват за създаване на алгоритми, които реално конструират математически обекти, като например точки върху криви. Тези методи помагат за подобряване на сигурността на данните и криптографията в електронните комуникации.

Този кратък обзор е генериран от изкуствен интелект

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

Резултати накратко

Final Activity Report Summary - GTEM (Galois Theory and Explicit Methods)

The traditional focus of pure mathematics is on deciding whether certain mathematical objects exist or not. For instance, one wants to know if given equations have solutions, or whether structures with given properties exist. In the past few decades however an increasing need has become apparent for methods that actually construct these mathematical objects if they exist. This trend has not meant a shift from pure to applied mathematics, but it is rather changing the nature of pure mathematics itself: algorithmic thinking and the language of complexity theory are entering into pure mathematics. There are two forces driving this trend towards explicit methods in the area of number theory and arithmetic geometry. The first is the desire of mathematicians to understand the mathematical universe in a more thorough way than before, and to develop computer labs to allow computer experiments with mathematical objects of a greater variety than in traditional numerical computation. Secondly, the vastly increased use and complexity of electronic communication and networking has created needs in data security and coding theory that are often met by applications of unexpected branches of number theory and arithmetic geometry. For instance the subjects of lattice basis reduction and algorithms for elliptic curves over finite fields are having a profound impact on cryptology. For four years the GTEM network provided a European research platform in explicit methods in Galois theory, number theory, and arithmetic geometry, including the training of 12 PhD students, and numerous workshops and summer schools. Research achievements include improved understanding and an improved algorithmic handle on Galois representations, point counting on curves over finite fields, and on the Arakelov class group in number theory.

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

Цел на проекта

Driven by applications in data security and networking, computational techniques are gaining importance in a range of areas in number theory and arithmetic geometry. The GTEM network will unite 12 centres of expertise of international stature in this broad range in a common research project to make hitherto purely abstract parts of advanced number theory and arithmetic geometry accessible to efficient computation. This project will develop feasible computational methods in the areas with proven or expected applications in cryptology and coding theory. To illustrate the goalm recall that numerical analysis is the discipline that makes the mathematical notion of a real number, which is intrinsically infinite in nature, accessible to calculations on a computer through schemes for efficient finite precision computations. The present objective is to find methods to model more complex mathematical objects than real numbers such as elliptic curves and Galois representations with a particular focus on the objects a rising in the mathematics of data security and networking. The training goal is to educate young European mathematicians that are capable of meeting the challenges of new applications of advanced number theory and arithmetic geometry in communications, bot h for academia and industry. The network will appoint 12 ESRs that will each complete a PhD thesis on tasks in the project.

Оригинален текст от CORDIS (на английски).

Участници

Връзки

Данни: CORDIS, © Европейски съюз