Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

June 18, 2024

Magnitude Homology Equivalence

Posted by Tom Leinster

My brilliant and wonderful PhD student Adrián Doña Mateo and I have just arXived a new paper:

Adrián Doña Mateo and Tom Leinster. Magnitude homology equivalence of Euclidean sets. arXiv:2406.11722, 2024.

I’ve given talks on this work before, but I’m delighted it’s now in print.

Our paper tackles the question:

When do two metric spaces have the same magnitude homology?

We give an explicit, concrete, geometric answer for closed subsets of N\mathbb{R}^N:

Exactly when their cores are isometric.

What’s a “core”? Let me explain…

Posted at 8:46 AM UTC | Permalink | Post a Comment

June 14, 2024

100 Papers on Magnitude

Posted by Tom Leinster

A milestone! By my count, there are now 100 papers on magnitude, including several theses, by a total of 73 authors. You can find them all at the magnitude bibliography.

Here I’ll quickly do two things: tell you about some of the hotspots of current activity, then — more importantly — describe several regions of magnitude-world that haven’t received the attention they could have, and where there might even be some low-hanging fruit.

Posted at 11:04 PM UTC | Permalink | Followups (2)

June 4, 2024

3d Rotations and the 7d Cross Product (Part 2)

Posted by John Baez

On Mathstodon, Paul Schwahn raised a fascinating question connected to the octonions. Can we explicitly describe an irreducible representation of SO(3)SO(3) on 7d space that preserves the 7d cross product?

I explained this question here:

This led to an intense conversation involving Layra Idarani, Greg Egan, and Paul Schwahn himself. The result was a shocking new formula for the 7d cross product in terms of the 3d cross product.

Let me summarize.

Posted at 10:23 AM UTC | Permalink | Followups (23)