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Maths BSc/MSc student at University of Helsinki. Writing a BSc thesis on measure theory. Alarmingly interested in compilers. Blogs maths in Finnish at www.nollakohta.fi. He/him.

Joined Nov 2017

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I did promise to tell about my BSc thesis topic one day. So get ready for a thread on... fractals! Along the way we'll also meet measure theory, a bit of function theory and several bad jokes.

I'll try to keep this as accessible as possible, even though I'll use some maths terminology. Feel free to ask if something is unclear! If the LaTeX equations do not render correctly for you, try opening the thread on https://mathstodon.xyz.

This is going to be a long thread, so let's get started!

Hi! I'm a maths student at Helsinki University. For the past year or so I've been writing a popular (not as in visitors) maths blog, www.nollakohta.fi, in Finnish. In addition to human languages (FIN/ENG/SWE/FRE) I respond to quite a few computer ones (esp. C#).

Interested in seeing if toots > tweets!

Software is like an onion. If you open up the inner layers you start crying

BSc thesis, +

It was the final one! Actually, I still fixed one minor typo... but now it is all done! Only some paperwork remains between me and the BSc degree!

Those 22 pages were not easy, but in the end absolutely worth the sweat. The grade is 5/5 too, so I’m pretty satisfied! 🥳

BSc thesis

I submitted the fourth draft to my advisor. Fingers crossed that it’ll be the final one!

The tough lemma wasn’t that bad after all. I finally found a proof for a similar result, and it confirmed that I had been on the right track after all. Knowing that my method was valid, I could then figure out a justification for it 😄

(My issue was whether you could pass simultaneously to supremum over y and infimum over z in one expression. Rearranging revealed that yes, you could.)

authors, mathematics

Mathematicians: "We are the most precise of them all."

Mathematician authors: "As we see in 2.3 ..."

And their book has

* a section

* a theorem

* an equation

* a table

* a figure

* some random list item

labeled "2.3".

BSc thesis, -

Regarding the previous thread (sorry for filling your timelines!):

I haven't submitted the final version yet. There's one lemma whose proof has been surprisingly hard, and all my references have just stated it without proof. It's so frustrating that the whole thesis now hinges on a single, apparently trivial lemma that I cannot prove, nor cite a proof for. I hope my advisor responds soon.

🧵 end

So, the stuff I did in my BSc thesis:

• Defined the Hausdorff measure and dimension, and showed some basic properties

• Stated the way to calculate the dimension of self-similar sets

• Defined quasisymmetry and conformal dimension

• Presented the definitions required for the big theorem

• Stated and proved that theorem (based on two existing proofs, filling in details)

This could be a prequel for an MSc thesis on conformal dimension, but I might have had enough of geometry for now...

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Can we improve these bounds from 1.630 < Cdim SC < 1.893?

Jun Kigami has proved a slightly better upper bound (1.859), using what looks like dark magic. I mean, looking at his paper (https://dx.doi.org/10.1007/978-3-662-43920-3_9) I think I understand what my thesis looks like to non-mathematical people.

For the lower bound, there are only numerical conjectures. The study of conformal dimension is only thirty-ish years old, and much remains to be understood.

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For each point in the Cantor set: define a curve with x coordinate equal to the point, and y coordinates from 0 to 1. You end up with an uncountable family of curves, all of which are contained in the Sierpinski carpet.

What's better, this curve family does have a positive modulus.

This gives a lower bound of \( 1 + \log(2)/\log(3) \) for the Cdim. The Hausdorff dimension of the carpet is an upper bound.

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There was a striking similarity between the Cantor set and the Sierpinski carpet. In fact, the Cantor set is a cross-section of the carpet.

*cue dramatic music*

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The interesting theorem for my thesis and this thread: If a set contains a curve family with positive modulus, then there is a lower bound for the conformal dimension of the set.

...Oh crud, one more definition. The modulus of a curve family is something you do in geometric function theory, a branch of complex analysis. Again, I know absolutely nothing about this stuff. The good thing is that you don't need to, and still can use the theorem!

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I should clarify that we are no longer working in \( \mathbb{R}^n \). Instead, we are working in general metric measure spaces. If the measure has some nice properties (it is doubling, and maybe Ahlfors regular), this is sufficiently close to \( \mathbb{R}^n \).

Most of the books I've seen about this are from around 2000. As I said, this is an active research topic, and I only know the bare minimum required for my thesis.

🧵

What a mouthful. As an example, \( x \mapsto x^2 \) is a quasisymmetric homeomorphism of the unit interval onto itself. This function is not bi-Lipschitz, a classic exercise on a calculus course.

Now, the conformal gauge of a set is the collection of all sets quasisymmetrically homeomorphic to it. The infimum of their Hausdorff dimensions is called the conformal dimension.

It turns out that the Cdim of Cantor set is 0. I present this without proof. What about the Sierpinski carpet?

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Mathematical definition, avert your eyes:

A homeomorphism \( f \) between metric spaces \( (X, d) \) and \( (X', d') \) is \(\eta\)-quasisymmetric, if for all distinct \( x, y, z \in X \) we have

\[ \frac{d'(f(x), f(y)))}{d'(f(x), f(z))} \leq \eta \left( \frac{d(x, y)}{d(x, z)} \right). \]

Here, \( \eta \colon {[{0},{\infty})} \to {[{0},{\infty})} \) is a strictly increasing bijection.

🧵

Instead, we consider quasisymmetric mappings. Quasisymmetry is a slightly weaker condition: such functions still map balls to roughly ellipsoids, but unlike bi-Lipschitz maps, you don't know the size of those ellipsoids. Relative distances have upper bounds, absolute distances do not.

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How do transformations affect the Hausdorff dimension of a set? If two sets are homeomorphic, what can we say about their relative dimensions?

There's a simple case: if two sets are related by a bi-Lipschitz function, they have the same dimension. Bi-Lipschitz functions distort distances only within a certain factor. Hence it is easy to derive upper and lower bounds for the Hausdorff measure of the image.

🧵

This would be a very fine point to end this thread. However, I was too ambitious for my own good. This was only the first part of my thesis.

The level of mathematics ramps up a bit now. We're going to talk about an active research topic!

🧵

As one last note, Hausdorff dimension matches the classical dimension where the latter is well-defined. If you have something with a positive, finite area, it has Hausdorff dimension 2. Thus we have, in a sense, generalized the Lebesgue measure to more complex sets.

This is not the only way to do it, but the other generalizations for dimension (e.g. box dimension) have some issues that make them less useful in practice.

🧵

As an aside, we don't really care about the Hausdorff measure here, just the dimension.

When we calculated the dimension of the Sierpinski carpet, we only got an upper bound. To get equality, we also need a lower bound. In general this is difficult, but there is a theorem for these kinds of self-similar sets with regular structure. Don't ask me about the proof, though. The measure theory involved is at high MSc level.

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Coastlines are usually 1.1 to 1.3 -dimensional, and the Cantor set is about 0.63-dimensional (you can work this out by modifying the calculation above).

Wikipedia has an excellent collection of examples at https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

🧵

This "raise diameters to n, then sum them" measure is called the Hausdorff measure. It turns out that the Hausdorff measure of a set is infinite when the dimension is too small – think about the area of a cube – and zero when too large – think about the volume of a square. The threshold is called the Hausdorff dimension.

The Sierpinski carpet is not "dense" enough to have area, but it's still "almost" two-dimensional. Hausdorff dimension captures this nuance.

- Location
- Helsinki, Finland

- Languages
- Finnish, English, Swedish, French

- More languages
- C#, C++, Rust, x64 Assembly, more

- Pronouns
- he/him

Maths BSc/MSc student at University of Helsinki. Writing a BSc thesis on measure theory. Alarmingly interested in compilers. Blogs maths in Finnish at www.nollakohta.fi. He/him.

Joined Nov 2017

A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes.
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