Category Archives: Mathematics

Fractals 6 Slides

On Saturday, I will be presenting some of my recent work, which extends Michel Lapidus’s theory of complex dimensions in \(\mathbb{R}^n\) to a larger class of homogeneous metric measure spaces. In particular, I will be discussing the complex dimensions of … Continue reading

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Seminar Talk in FRG

I gave a talk last week before the Fractal Research Group at UCR. The goal was to introduce my colleagues to the Assouad dimension, and to share some of the more interesting and/or surprising results. My notes (typos and all) … Continue reading

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The Sphere is Weakly Dense in the Ball

Another qual question, this one dealing with Hilbert spaces and the weak topology: Exercise: Let \(\mathcal{H}\) be an infinite dimensional Hilbert space. Show that the unit sphere \(S := \{x\in\mathcal{H} : \|x\| = 1\}\) is weakly dense in the unit … Continue reading

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Nowhere Differentiable Functions

In an undergraduate analysis class, one of the first results that is generally proved after the definition of differentiability is given is the fact that differentiable functions are continuous. We can justifiably ask if the converse holds—are there examples of … Continue reading

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An Elementary Duality Result

The following problem is a fairly straightforward exercise that seems to come up fairly often on qualifying exams at UCR, and I think that I finally have a nicely argued proof. I am going to put it up here mostly … Continue reading

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MMM LXXII

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MMM LXXI

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LEGO Fractals

My daughter has recently started playing with DUPLO blocks pretty consistently, which has me thinking about LEGO—the toy of my youth (and hopefully her’s in a couple of years, too!). In some ways, LEGO is an ideal medium for exploring … Continue reading

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MMM LXX

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MMM LXIX

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