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New perspectives on the box counting dimension of sets and measures

Project description

Exploring fractal geometry’s enigmas

In the realm of fractal geometry, measuring the ‘size’ of sets with various dimensions presents a profound challenge. Backed by Marie Skłodowska-Curie Actions, the NewBoxDim aims to decipher this enigma. Anchored in the box-counting dimension (a core mathematical concept with wide-ranging implications), the project embarks on an innovative journey. By harnessing thermodynamic formalism, it seeks to unravel the elusive box dimension of self-affine sets. It fuses dimension theory with percolation methods to explore intricate fractal percolation models' connectivity. Additionally, NewBoxDim delves into the convergence rate of the chaos game, intertwining with the L^q spectrum. These revelations promise to reshape dimension exploration and could influence genomic sequence analysis.

Objective

Fractal geometry is a young, very active and rapidly growing field of research that is connected to different areas of mathematics and the natural sciences. Roughly speaking, dimension theory is concerned with measuring the ‘size’ of sets by defining different notions of dimension. At the center of NewBoxDim is the box counting dimension which is a fundamental concept within mathematics that has found applications outside of mathematics as well. The main objective of NewBoxDim is to open new perspectives on the study of the box dimension of sets and measures by exploring three very recent alternative approaches. In the first approach, the goal is to develop unifying machinery based on thermodynamic formalism in order to tackle a long-standing open problem about the existence of the box dimension of self-affine sets. In the second approach, we aim to fuse ideas from dimension theory and methods from percolation theory in a novel way to study the fine scale structure and connectivity properties of different fractal percolation models. The third and final objective of NewBoxDim is to advance our understanding about the time evolution of the chaos game whose convergence rate is determined by the box dimension of the measure driving this stochastic point process. We aim to obtain more nuanced information by making an interesting connection to the L^q spectrum (also known as generalized fractal dimensions) of the measure and characterizing the measure which minimizes its box dimension. As an interdisciplinary application, our results on the chaos game could have a direct impact on the Chaos Game Representation (and its variants) which is a method that was developed to analyze genomic sequences in bioinformatics.

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HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships

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Call for proposal

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(opens in new window) HORIZON-MSCA-2022-PF-01

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Coordinator

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Net EU contribution

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€ 141 782,40
Address
REALTANODA STREET 13-15
1053 Budapest
Hungary

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Region
Közép-Magyarország Budapest Budapest
Activity type
Other
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