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February 20, 2024

Spans and the Categorified Heisenberg Algebra

Posted by John Baez

I’m giving this talk at the category theory seminar at U. C. Riverside, as a kind of followup to one by Peter Samuelson on the same subject. My talk will not be recorded, but here are the slides:

Abstract. Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from ‘spans’, where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a ‘categorified’ Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious, at least to me. However, Jeffery Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans.

Posted at 10:51 PM UTC | Permalink | Followups (4)

February 14, 2024

Cartesian versus Symmetric Monoidal

Posted by John Baez

James Dolan and Chris Grossack and I had a fun conversation on Monday. We came up some ideas loosely connected to things Chris and Todd Trimble have been working on… but also connected to the difference between classical and quantum information.

Posted at 6:46 AM UTC | Permalink | Followups (12)

February 4, 2024

The Atom of Kirnberger

Posted by John Baez

The 12th root of 2 times the 7th root of 5 is

1.333333192495 1.333333192495\dots

And since the numbers 5, 7, and 12 show up in scales, this weird fact has implications for music! It leads to a remarkable meta-meta-glitch in tuning systems. Let’s check it out.

Posted at 8:09 PM UTC | Permalink | Followups (3)