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i added a transcript of the conversation in people.math.harvard.edu/~knill/pedagogy/argument/. it was chat GPT4

Thanks. By the way, the counting of positive eigenvalues 0-1 matrices and acylic digraphs are both 1-line computations in Mathematica M[n_]:=Length[Select[Map[Eigenvalues,Map[(Partition[#,n])&,Tuples[{0,1},n^2]]],(#==Re[#]&&Min[Re[#]]>0)&]]; R[0_]:=1;R[n_]:=Sum[(-1)^(k+1)*Binomial[n,k] 2^(k(n-k))R[n-k],{k,n}]; R[3] ==M[3] The equivalence of R and M follows because an acyclic digraph has an adjacency matrix with all eigenvalues 0 and because if a 0-1 matrix has all positive eigenvalues positive, they all must be 1. Yes, simplicial complexes can define various graphs, also bipartite graphs. I had not seen what you stated about these bipartite graphs. Apropos simplicial complexes. I proved something related: take a simplicial complex G and look at the intersection matrix, a 0-1 matrix. This matrix is always unimodular and the number of positive eigenvalues minus the number of negative eigenvalues is the Euler characteristic. arxiv.org/abs/1907.03369

@@OliverKnill ah that’s a nice way to look at the bijection! I don’t speak Mathematica though… but your argument is clear. I forgot to mention an important property of the graph representation for simplicial complexes I tried to explain: that the adjacency matrix squares to the 0 matrix. Here’s a picture that shows an example: drive.google.com/file/d/19w3h9H2pVN8Z16CflgiYK3SkWbANCyp4/view?usp=drivesdk I’ll take a look at your paper later today

Thanks for this comment. There is indeed a bit of a cultural divide between graph theorists and combinatorial topologists. Many graph theorists look at graphs as one dimensional simplicial complexes. One can also have the point of view that a graph is a vessel for other structures, like simplicial complexes or topological spaces or order structures or sheave theoretical structures. This is similar than when we look at Euclidean space as a set at first. It can carry many topologies, the most natural one the topology coming from the Euclidean distance. For graphs, the most natural simplicial structure is the one obtained from the complete subgraphs. There are many others, like looking at the zero and one dimensional simplices V and E only which is very limiting.

In this talk, the word "atom" was only used figuratively for the smallest open sets in a finite topological space. Nobody "really" knows what elementary particles are. We have models and some of them, like the Standard model are very successful. Whether there is any physics related to the mathematics I look at is not clear. One can speculate but I prefer to look for mathematical theorems involving the structures. What is nice about math is that if you establish a theorem and prove it, then it remains true and real for all times. Our models for physics might change and our interests in certain subjects of math change too but the theorem remains a theorem.

thanks! Greetings to Turkiye.

@@OliverKnill Thank you again

thanks for watching. You look at an interesting point which always confuses students. I look at this point at the level surface of a function of 3 variables. It is f(x,y,z) = z-(x^2+y^2). We can write the level surface also as z=x^2+y^2+c. This is a paraboloid as drawn and not upside down. You could also look at a paraboloid that is upside down. Then you would look at a level surface of z+(x^2+y^2). THere is a way that you can see this upside down, if you would go into 4 dimensions and look at the graph of the function f(x,y,z) but we can not see this 4 dimensional graph.

Wow, that is exciting. I feel the same. Claude Shannon had been one of the most influencial figures in the last century. I visited the Shannon beach on the other side of the Mystic yesterday. It should have been named after Claude Shannon and not after the politician Charles Shannon. The Entropy house of shannon is featured in the movie "The Bitplayer" from 2018. I have a clip here: people.math.harvard.edu/~knill/various/bitplayer

immer heikel hier Ratschlaege zu geben, weil die Situation laufend wechselt. Als ich dort flog gab es schon Restriktionen um das Schaffhauser Gefaengnis (geofenced). Ich selbst habe registrierte Dronen und eine Fluglizenz.

@@OliverKnill danke für deine Info 😉

thanks. Was lucky to have seen the 3-body episode the day before ...

I know, because my dad owns your dad.

good suggestion. It might well be that the technology evolves that such a zoom could be done by the user (point your mouse on a part of the video and it shows that part in higher resolution. This is not impossible as my videos are all 4K). At the moment, I have to do these videos in a limited amount of time: I think about what to talk about the night before, let my dreams work over night on it, write the board and then talk for about 10-15 minutes. The production of one talk uses maybe 20 hours of work, where the bulk of the work goes into the thinking and the write up early in the morning to the board. The videos are recorded in 4K. You can in youtube change the resolution of the video to 4K (click on the wheel icon in the tools bar at the bottom of the video). With 4K and a good monitor, you can read every part of the board. I can read the board on my phone if I tune up the resolution. ON the phone one can with the hand zoom in. Your point also brings up something interesting about learning and presentation. When presenting something live in a talk or lecture, the paradigms of splitting things up into small parts (e.g. no more than 8 lines in each slide, very few information on each part), definitely should be followed and I do that myself. CZcams has enabled other ways: I can in one picture get the entire content together. This allows somebody to see in a few seconds, whether they want to see it and not waste any time. If it should interest, one can then read the texts I wrote about it. One of the major problems these days is information overflow and time limitations.

I agree with the time constraint & information over load, I can read up from the generous description Do you happen to know of any context in which there’s an inverse to barycentric subdivision? I’ve been playing with the matrix that performs barycentric subdivision & noticed that the k-fold edgewise subdivision matrix has a factorization involving the inverse of the barycentric subdivision. I don’t have a clue what that means geometrically, but it works!

@@danielprovder Barycentric refinement can not be reversed within the class of simplicial complexes. The refinement matrix A telling how the f vector of the refinement depends on the f vector of the complex is upper triangular and invertible but the inverse is a rational matrix. You can of course invert on the image of the refinement operation by just going back, but the image is a small class of geometries.

The notion of manifolds took a long time to evolve. What I'm interested in is in finite notions of manifolds that have the same content and topological meaning than what the continuum provides. No geometric realization of course is allowed as this would betray the assumption of staying in a finite setting. A lot of things work beautifully like cohomology or curvature. I measure the success of a definition in what theorems one can prove in it. Gauss-Bonnet is a great example where one has to work pretty hard to get to a theorem that works in arbitrary dimensions. It was finished in the 40ies by Chern. The curvature involves Phaffians of the Riemann curvature tensor and its pretty heavy to compute, even numerically. Even when doing that for simple manifolds it is hard to compute. In the discrete (for any network, not only for manifolds), it is 2 lines of code. And it gets to the Gauss-Bonnet-Chern integrand in the continuum. This talk is about submanifolds of manifolds obtained by looking at level surfaces or intersections of several level surfaces. Also here, the mathematics in the finite is much, much simpler. There are no singularities like in algebraic geometry for example. The question of whether space or time or space time are discrete will probably never be answered as we are 25 order of magnitudes off (compare the size of a quark and the size of the Planck length). But we might see more and more that the mathematics of finite space is much more elegant and beautiful than the mathematics in the continuum where we have to do sheaf theoretical hacks to express things like curvature. Computing codimension k level sets in d manifolds needs only a few lines of code.

Thanks. The Cartesian product used in graph theory is useless for my purposes. It treats graphs as one dimensional objects and gets back a one dimensional object. If you take the product of two one dimensional objects, we want to have a dimensional object. If we take the product of two circles we want to get a torus. We essentially want to have two dimensional cells. The concept of CW complex allows to do that by treating the "squares" which have been produced as cells. But now, we are in a different category and outside graph theory. Graph theory is a great way to describe higher dimensional objects by taking the complete subgraphs as "cells". If you take the "usual graph theoretical Cartesian product" the dimensions do not add up. I myself have worked on other products like the "Shannon product= Strong product" which is nice in that Kuenneth works and curvature multiplies but the dimension does not work. The Shannon product of two one dimensional graphs is already 3 dimensional because it contains K4 graphs. The Barycentric refineed product I mention in the talk is also called Stanley Raisner product. It has almost all properties we want to have like that the product of two manifolds is a manifold, that dimensionsadd up and Kunneth works but it is not associative. The construction of G x H is by taking as the new vertices pairs (x,y) of simplices, where x is a simplex in G and y is a simplex in Y. Now connect two such pairs if one is contained in the other coordinate wise. If you take the product with 1 you get the Barycentric refinement because the new vertices are the complete subgraphs and two are connected if one is contained in the other. The product is not associative. By the way, if you look it up, mathematicians call the category of graphs not cartesian closed for a silly reason: there is no terminal object. While 0 (the empty graph) is a good candidate, it is not since in the category of graphs, one takes graphs homomorphisms as morphisms and there is no morphism from a graph to the empty graph. Morphisms have to map vertices to vertices and edges to edges.

polish space is the jargon in the biz instea of "separable, completely metrizable metric space"

can unfortunately only be built if we can tunnel through an amplituhedron.

@@OliverKnill I'll let grok know, thank you so much!

@@OliverKnill I'm gonna contact Elon's Boring company and see if they can Tunnel through an Amplituhedron and What are the steps we need to do to tunnel through one? Redefine an axiom or find some equation why not just assume that we cannot tunnel through one and just do it in a quantum simulation

@@WorldRecordRapper tough luck. The boring company is not doing that well. Maybe in an other place of the string theory landscape .

​@@OliverKnill I just froze Elon's Grok AI by asking it about ist (sqrt[-1]) dimensions then it spouted out [link](#tweet=n) where n is anything from 0 to at least the prime number 80000000000000000000000000000000000000000000000001 by time of this study. My prompt then turned into a thinking "..." which is characteristic of what grok was getting frozen as when asked about the squareroot of the negative first dimension.

What would be good problems not involving prime numbers? The list on this video has the Ulam conjecture, the Euler brick conjecture and the Beals conjecture. It would be easy to replace them with conjectures involving primes. On the other hand finding problems in number theory that 1) can be understood by a general audience and 2) do not involve primes and 3) is widely known, is harder. Any suggestions are welcome. I myself like problems about Pythagorean triples arxiv.org/abs/2205.13285

Thank you. Does Collatz conjecture involve primes too?

@@user-pj5yw6rz2u in some sense yes as the decision is involved whether the number is even or odd. If n is even you divide by 2, if n is odd you produce 3n+1. I would not call this a conjecture about primes. It is a great example of a problem that can be understood by anybody and which is believed to be very, very hard.

thanks.

Povray itself is written in C++ and the source is available www.povray.org/beta/source/. But Povray itself is a complete programming language. I myself have not used in from Racket but used C programs oder Mathematica programs write Povray code.

​​@@OliverKnill sprichst du deutsch? Ich spreche auch deutsch. Ich würde gern lernen wie ich eine andere Programmiersprache wie Racket auf Pov-Ray nutze. Ich mag visuelles oder akustisches lernen. Ich habe kürzlich entdeckt, dass es für audio engineering z.b. Sonic Pi gibt.

@@mariobroselli3642 ja, ich bin aus der Schweiz.

@@OliverKnill hast du schon mal das Processing Environment probiert?

@@mariobroselli3642 ja, aber das war schon lange her, vielleicht 15 Jahre her, als es hier in Mode war in CS kursen.

thanks! It is addictive, especially the FPV. It is like being inside a plane.

Thanks for the suggestion. I just edited the subtitles. By the way, the handout to this lecture is here: people.math.harvard.edu/~knill/teaching/math1a2024/handouts/lecture13.pdf

thanks. That video was done just before the pandemic hit early in 2020. Seems like an eternity since.

Better say that Primes behave probabilistically. There was once a nice talk of Terrence Tao at MIT which I had attended more than 15 years ago which had the title "structure and randomness in the prime numbers". One way to quantify this to look athe Moebius mu function. If this function was sufficiently random (so that the iterated law of logarithms holds), then this would imply the Riemann hypothesis.

thanks. It looks like an eternity since i had looked at this. The open questions still all stand.

thanks monsieur. I will try. Note however that this is always a tough cutting game to bring a slide show of essentially one hour content to 57 seconds. You should try once. And there would be a lot more to say about separable systems. By the way, one of the ways to bring it down to one minute in the shorts format is to do microcuts, removing fractions of a second here and there. Producing such a video needs a couple of hours. To do it perfectly would take a day or more. But thanks for watching anyway and the suggestion.

​@@OliverKnillsomeone who can follow at such a pace probably doesn't need the video. You can try working with parts, it can maybe give you a bit more breathing room

@@monsieurLDN Valid point and I have heard it before. There are different styles and tastes and approaches to teaching. I myself like brevity and the shorts format. If one does not like something, one at least one has not wasted time. There is also the concept of review. If one has seen something already and wants a refresher, the shorts format is great. Also, expositions about classical topics like this (I teach this since 40 years) are abundant. I would not see the point of doing something that already exists. There are thick books and long lectures which dwell on this in detail. Even youtube has hundreds if not thousands of entries allowing to learn this in due time. A minute version of an entire lecture does not replace the actual lecture, but it allows to revisit a few points in a very short time. In the current case, the video very close to a lecture given to class. And it puts also a stamp on something important: the question of existence of solutions. The Toricelli example makes this memorable. I learned and taught this already as an undergrad and it is a hard sell. Usually only mathematicians appreciate the question of existence.

A good program (especially for 3D fractals) is "mandelbulber". I like also the program xaos. There is even an online javascript version xaos-project.github.io/XaoSjs/

About curvature: Calabi yau are indeed Ricci flat