Non-commutative geometry / Panel discussion, incl. G Longo, FW Lawvere, A Peruzzi

0:00 Thank you very much indeed. And for this, can I recopy the things that I've... Okay, I'll try and do that. Do they have a photocopy machine that we can use in the building? If not, I'll get it done for you. It's really something I did not ask for. I'm interested in it. Oh, no, it looks quite interesting, actually. It looks very interesting. No, I'll recopy it so that I can... No, no, no, I'll do it, I'll do it. As a matter of fact, I very much want to get hold of... Oh, hello, Henry. Did you... Did you... Did you get a chance to copy those? Oh, brilliant. Good man. Thank you very much indeed. You know, just further to the point that you were making about the very general question of the bill, you know, how far can we actually probe? ...guiding intuitions of the kind that Hilbert was speaking about. And his response about Hilbert's Ancelica Geometry... Whose response? Bill's? No, Bill's response to the other question about Hilbert's Ancelica Geometry. Hilbert has been, you know, defamed as a formalist. The... I mean, the... ...which this calls out, I think, to me, is... Well, for instance, Bill makes this... I mean, I'm sure, absolutely correct. Cohesion and variation as blocking indicators of constants for generalised continuum hypothesis or a choice of going from X to 2X in real-world cases. Why do we have the Banach-Tarski paradigm? Precisely because of this distrust of geometric intuition, the idea that the Banach-Tarski paradox proves that geometric intuition is useless, it has to be, because we're crucified out of existence, because it clearly gives us, in the context of, of course, this global and absolute membership in the relation-based notion of consistency...

2:30 Yeah, of course, of course. What time, by the way, what time does it start this afternoon? Now what you do is you trisect each of these angles, so all the little A angles which are here are equal, and you intersect consecutively trisecting. You do that, you obtain three points alpha, beta, and gamma. You know, the theorem of Pauli is that the smaller triangle, alpha, beta, and gamma, is always incorrect. That's very beautiful because it's like symmetry out of nowhere. You start with something which is not symmetrical, which is an arbitrary triangle. And out of nowhere, you obtain symmetry, you obtain the Euclid algorithm. Now, there are simple proofs of this theorem. There are all sorts of standards. Do you start with the Euclid algorithm, form the Euclid algorithm, and leave maybe other fields with lengths?

10:00 So, obviously, if you want the fifth postulate of Euclid, it's not that. So it cannot be deduced. In quantum spaces, even in geometry, you can extend it.

12:30 You change it by adding to it a Newtonian potential.

15:00 Now you compute. You compute this equation. You find that strange lines are exactly the motion...

20:00 Mathematical physicists are able to recognize any chemical and the ones that are difficult to see is also an exercise to understand that then the observable quantities, which has been the convolutional algebra of the group, I mean you know you take functions of the group, which have non-convective examples, to spell out the difference between left and right, there is a sort of called given one parameter group of automorphisms, it's defined modulo the automorphisms which are trivial, it's inner automorphisms, and it's canonical, and you cannot change team to get whatever you want.

37:30 These are new facts, quite amazing facts. An example is that this non-communicative fluoride, there is an alarm of the Jacobi server, but now it involves odd, but it also involves Mb, U, T, Z, and because now it's no longer graded but filtered, you see that since more involved, so also divide the target into less humanly related, are not concrete objects. So it left me a very strong feeling of frustration, and in fact it took me years.

42:30 To understand that there was a beautiful answer, and that in order to get this beautiful answer, you just had to browse through the dictionary between classical and quantum mechanics. In this dictionary of dictionaries, you find that you are practical and quantum mechanics. That in quantum mechanics, a variable, a real variable, is replaced by a self-adjacent variable. You should perceive that in a different propagator. You can check its inverse as a dimension.

50:00 You see what happens? Instead of measuring distances by trying to take a thesis to what is a basic...

52:30 Of course, you have the Hilbert space, you have the algebra of coordinates, you have the square root, you'll find out that it's extinct. You can define the manifold by homology cycles, modifications of the nature of spacetime that came.

55:00 The Maxwell equation led, and it's sort of crazy to say, a slightly new community. What you find, however, is that the full standard model theory, including gravity, becomes a pure gravity theory. The group that replaces different morphisms is the non-massive group of inert morphisms.

57:30 Even for GL1. GL1, when you do the first one, involves the zeros by themselves, when there are factor and normalization physics, that there is, if you want to, which is not, like, you are depending on a scale, on something. There's exactly the right relation.

1:02:30 There's suddenly a need for something bigger. You just take the quantum curve numbers theory of, let's say, Darfur-Town equations, and just try to be logical with it, systematic, I mean, in a strong sense, and then you discover that...

1:20:00 When you find the zeros of the Riemann zeta function, or a combination of an emission spectrum is less, the fluctuations of the number of zeros of the Riemann zeta function, which are less than z in the imaginary part, you find a certain very oscillating function. When you compute a quantum mechanical system and you look at the function which is a fluctuation of the number of eigenvalues, you find a very oscillating function. But the basic fact is that when you look at the oscillating function for the Riemann zeta function, you find that it is minus. The function that you get in the case of quantum systems. And there was no way to figure and transform this minus sign into a plus sign.

1:22:30 And the way it is resolved is that the spectral rays, which come from the magnetic waveform, are in fact subtracted from white light. So when you will compute the trace, you find the trace of the white light, which is the regular representation, to white light.

1:25:00 Thank you very much for your attention and I hope to see you again in the future.

1:27:30 Thank you for your attention. Thank you for your attention. Thank you for your attention.