What is Multifractal Analysis? Modern Fractal Geometry and my research
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- Опубліковано 25 лип 2024
- Hausdorff dimension is not a fine enough analysis for many objects of interest in modern fractal geometry. Luckily Hausdorff dimension can be adapted to the study of multifractals or a multifractal decomposition via multifractal analysis. This area of modern fractal geometry is core to my math phd research area for my dissertation, and so I wanted to spend a bit of time trying to communicate a bit about it here. In the process we'll connect fractal constructions and measure theory to generate a "irregular" fractal where such a multifractal analysis is required and then give an overview of what such an analysis would involve.
00:00 Intro
00:14 What is studied in fractal geometry?
03:10 A fractal with "irregularity"
07:11 Nathan's mid video math comm crisis
07:58 A multifractal analysis with respect to local dimension
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Great video! Thanks for sharing this :)
Good video! Have you considered making them longer?
Interesting stuff you get to spend your time on!
This is very interesting! Have you done this type of analysis for other dimensions? Immediately come to mind the generalised Hausdorff dimension (defined with gauge functions), box, Assouad, etc
Another field I didn't know about but sounds cool!
Cool video! I've been thinking of making a video explaining my research too. Do you have any advice about this? In particular, who is your target audience (what background knowledge are you happy to assume?) and what do you do about stuff that seems too technical to be covered well? I find myself in a particularly abstract, categorical corner of representation theory and, to be honest, I don't even know how I'm supposed to motivate what I'm doing to someone who doesn't already at least buy into representation theory as a worthwhile thing to study
I think that it is normal to want to talk about some of the technical results/ideas that you are interested in, but I usually assume that no one knows anything about what I do when I give a talk (by consequence of being the only dynamics PhD student in my department this is sometimes necessary unless I am presenting to my advisor.) Talks are not proofs (usually) so it is okay to leave some of the technical points out as long as you can address them appropriately when asked, or even talk about them at length after the talk is over. This response is to be continued, I have more thoughts, but have to run to a research meeting!
Doesn't Mendlebrodt define a fractal explicitly as a set where Dh>Dt?