RWHG · Random walks on hyperbolic groups
7РП — „Хора“ (Действия „Мария Кюри“)
- Период
- 2009-10-01 → 2010-11-30
- Финансиране от ЕС
- 156 713 €
- Участници
- 2
- Схема
- MC-IEF
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
Случайните разходки върху хиперболични групи изследват връзката между вероятностите, алгебрата и геометрията, като например чрез теореми за централното ограничение. Това помага да се разберат алгебричните и геометричните свойства на дадени структури чрез поведението на динамичните процеси върху тях.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Random walks on hyperbolic groups
The project lay at the confluent of the following 'different' fields of mathematics that may appear quite isolated at the first glance: probability theory, algebra, and geometry. Here random walk theory, a branch of probability theory, plays a major role. In general there are two points of view to look at the relation between probability theory on the one side and algebra and geometry on the other side. The probabilistic viewpoint concerns all questions regarding the impact of the underlying structure on the behaviour of the corresponding random walk. Typically one is interested in transience / recurrence, spectral radius, asymptotic behaviour of the transition probabilities, rate of escape, central limit theorems, harmonic functions and convergence to a boundary at infinity. On the other hand, random walks are a useful tool to describe the structure that underlies the dynamic. In particular, algebraic and geometric properties can be classified due to the behaviour of the corresponding random walks. One specific aim of the project was to find a classification on which algebraic structures an analogue of the central limit theorems holds true. A first step towards this ambitious objective was to consider hyperbolic groups where we successfully proved a central limit theorem for the important subclass of co-compact Fuchsian groups. Even better, our proof strategy seems to apply for large classes of groups that fulfil two main assumptions that we conjecture to hold for large classes of finitely generated groups. The project ended prior to maturity since the host institute offered Sebastian Müller an assistant professorship. This situation is best possible in order to pursue the proposed research topics and underlines the good collaboration of the researchers involved.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
The project lies at the confluent of different mathematical fields: probability theory, algebra, and geometry. There is a contribution of A. V. Vershik in a special Springer volume about the future of mathematics in the 21st century that points out the prospects and challenges that are comprised in the interplay of probability theory and algebra. Here random walk theory, a branch of probability theory, plays a mayor role. There are two points of view to look at the relation between probability theory, algebra, and geometry. The probabilistic viewpoint concerns all questions regarding the impact of the underlying structure on the behavior of the corresponding random walk. Typically one is interested in transience/recurrence, spectral radius, rate of escape, and central limit theorems. On the other hand, random walks are a useful tool to describe the structure that underlies the random walk. In particular, algebraic and geometric properties can be classified due to the behaviour of the corresponding random walks. The project falls exactly into this topic: we will study random walks on hyperbolic groups. The objectives are to prove a central limit theorem for random walks on hyperbolic groups and provide geometric interpretations of the asymptotic variance. This will arise from a geometric perspective in the flavour of the interpretation for the rate of escape in terms of entropy and requires deeper knowledge of hyperbolic geometry together with inspiration and new ideas. The project will settle the ground for future collaboration, not only between France and Germany but also on an European level, since the host institute and the applicant have strong European contacts. Furthermore, the project can be seen as a continuation and complement of the existing Marie Curie contract ``European Training Courses and Conferences in Group Theory''.
Оригинален текст от CORDIS (на английски).
Участници
Връзки
Данни: CORDIS, © Европейски съюз