Daniel Tubbenhauer's Algebraic Geometry Syllabus

I'm really enjoying this, since his most recent video which is a huge motivational shove to understand why Algebraic Geometry is an interesting thing to learn about:

What is Computer Algebra [for]?

So you are wondering what varieties are now, ...

And then you find out that this dates back to Poincaré in the early 20th Century, when Algebraic Geometry started to really connect with topology:

And here is one of the key results: the Zariski "Topology"

It seems to me to be linked through the closure processes. See Kuratowski closure axioms and Terence Tao on Mathematics and Artificial Intelligence.

Then he suggests going back even earlier to the Coordinate Ring

Subscribe to Visual Maths.

Let's not forget Norman Wildberger on the CA/AU front:

Taking it down to semi-rings:

And here is the famous series of lectures on Box Arithmetic.

Subscribe to Insights into Mathematics.

My gut feeling is that this will turn out to be useful for understanding Quantum Mechanics and the Dirac Bra-Ket Formalism and how it connects with Cosmology. See  Jade on The Shape of Space and Curt Jaimungal Talking With Roger Penrose:

If you're confused about the matching circles argument, see the paper Promise of Future Searches for Cosmic Topology in PhysRevLett, 132, 171501 (2024):

Standard cosmology combines general relativity and quantum mechanics to produce a simple model accounting for the distribution of matter in the observable Universe. The average spatial curvature of this model is observationally constrained to be flat, or nearly so [1]. However, general relativity concerns only the local geometry of the spacetime manifold, not its topology. Quantum processes in the very early Universe may induce “nontrivial” (multiply connected) topology of spacetime [2,3] that remains present today on very large physical scales, even if inflation occurs [4]. Indeed, the temperature variations in the cosmic microwave background (CMB) suggest the presence of statistically anisotropic correlations, much as would result from nontrivial topology of comoving spatial sections. ...

If all of this makes you feel like a dog watching TV, then join the club!  PBS Space Time - What If The Universe Did NOT Start With The Big Bang?

Subscribe to UpAndAtom.

In words, the intuition I have is that the problems of observation and observables in Quantum Mechanics come from the experimental setups we use to verify the predictions of quantum mechanics. We are constructing a theory of observations as a greatest fixed-point under which the exceptions appear as falsifications. So what we are doing in physics is constructing a global picture from a patchwork of local observations knitted together in such a way that we can describe the experiments in terms of local coordinate systems, i.e. in some sort of vector space which is a quotient of all equivalent observations: equivalent in the sense that the descriptions of our experiments can be translated consistently from one space to another. So what holds this global structure together is the consistency of the rules that we use to translate languages each into the others, respecting the underlying models. So it is a kind of global congruence. And then what we are looking for to explain these bizarre aspects of non-local quantum states is something to do with the topology of the whole global quotient space.

At 20:33 I said "He's very careful about specifying the domain as an open epsilon-ball" but I should have said "an open set U". See the Wikipedia Introduction to gauge theory which is what sparked that insight about how gauge theories such as Yang-Mills are essentially phenomenological rather than ontological. See The Bright Side of Mathematics video on Lipshitz continuity here and the subsequent lecture on Uniqueness of Solutions here. It's part of a rather good series of lectures on Ordinary Differential Equations.

This is a video I made for Jade after I went to talk to Hugh Osborn at DAMTP about her Norton's Dome video script. If Hugh's home page looks old-fashioned it's because it's essentially the one I made as a default for all members of the research group sometime around 1996!

See Amanda Gefter and Curt Jaimungal Talk Physics and Philosophy for more about the participatory Universe.

And for what all this has to do with Logic, see Question About the Alan Turing Archives at King's College.