HI, I'm Oliver Knill, mathematician, working in Cambridge MA. I use this channel for reflection on my teaching, and my own math research, show illustrations, pictures.
Dear Professor Knill, For Part 4 (Computing Integrals), wouldn't the curl instead be equal to [0, 2x^2 - 2z^2, -1], which would make the polar integration tougher? Or did you forget to write the negative sign for F? Thanks!
I think I seen one of those spheres around the 3-minute mark. At some point I knew what I was doing when I was playing around with the 4x4 quaternion matrices. I made a sphere with three types of angles. I might even have the Maple file if I dig around for it.
yes, such graphs appear also in algebra like Cayley graphs of the quaternion group. I have experimented with primes in quaternions here arxiv.org/pdf/1606.05971 . The paper which was flashed around minute 3 is here ems.press/content/serial-article-files/45442
Really glorious flight today! Phenomenal that you encounter such friendly persons who will wave at you! probably speaks to the skill and smoothness of your approach/flying. and perhaps human nuances to your craft's flight. 🚁
I have the exact experience, mine was a mini 3, I also flew out to the ocean but with a strong wind when I come back. I actually was able to cancel the auto-landing and manually land with the battery of 2%.
I didn't read all the text, but in what world is a set "more general" than a metric space?? In this case, in a sense, ChatGPT was right and the human was wrong :) One could certainly argue a set is "less general" than a topological space because there's a fully faithful embedding of Set into Top (X is mapped to itself equipped with the discrete topology). I think there's one also into metric spaces: take X into the metric subspace of V(X) given by "basis" vectors, where V(X) is the vector space of a.e. zero functions from X to R equipped with, say, the L2 distance. But yeah, there are functors in both directions. Is there a fully faithful embedding of either of DeltaSet or SSet into the other?
a metric space is a set with a metric. If you take the metric away (apply the forgetful functor), you are left with a set. The category of sets is more general than the category of metric spaces because the later can be seen as a subcategory. The morphisms of set are maps from the set to itself. The morphisms of metric space are maps that are contractions. Isomorphisms of a set are bijections, Isomorphisms of a metric space are isometries. In general, if you take a category and and add more structure and have the morphisms honor this structure, then there will be less morphisms.
Have you seen the result Acyclic Digraphs and Eigenvalues of (0,1)-Matrices? There’s a nice relationship between simplicial complexes & a certain directed acyclic graphs, in particular a bipartite one in which the nodes are distinguished as follows: one partition is for the 0-dimensional simplices, & the other partition are for the >0 dimensional simplices. For each facet in the complex in question, there exists a node in the bipartite graph with out-degree equal to the dimension of the facet. For each vertex in the complex there exists a node in the graph with in-degree equal to the number of facets in the complex containing that vertex & out degree zero. The edges flow from nodes representing higher dimensional simplices directly to nodes representing the 0-dimensional ones. Under this model the standard simplexes are mapped to bipartite star graphs with edges pointing away from the central node, representing a facet & the exterior nodes being the vertices. Two simplices can be glued by identifying their overlapping vertices & the corresponding nodes in the star graphs.
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
all these theorems are applying differential geometry to simplicial hypergraphs, rather than any purely combinatorial theorems on graphs, letting other people know.
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.
To prove that quarks (subatomic particles) are more real while protons and neutrons (atomic particles) are less real, we need to establish a clear definition of what we mean by "real" and then provide evidence or logical arguments that support this claim. Let's approach this step by step. Definition of "real": For the purpose of this proof, we will define "real" as being more fundamental, indivisible, and closer to the underlying nature of reality. Proof: 1. Quarks are the fundamental building blocks of matter: - Protons and neutrons are composed of quarks. Protons consist of two up quarks and one down quark, while neutrons consist of one up quark and two down quarks. - Quarks are not known to have any substructure; they are considered to be elementary particles. - Therefore, quarks are more fundamental than protons and neutrons. 2. Quarks are indivisible: - Protons and neutrons can be divided into their constituent quarks through high-energy particle collisions. - However, there is no known way to divide quarks into smaller components. They are believed to be indivisible. - Therefore, quarks are indivisible, while protons and neutrons are divisible. 3. Quarks are closer to the underlying nature of reality: - The Standard Model of particle physics, which is our most comprehensive theory of the fundamental particles and forces, describes quarks as elementary particles that interact through the strong, weak, and electromagnetic forces. - Protons and neutrons, on the other hand, are composite particles that emerge from the interactions of quarks. - Therefore, quarks are closer to the underlying nature of reality as described by our most fundamental scientific theories. 4. Quarks exhibit more fundamental properties: - Quarks have intrinsic properties such as color charge, flavor, and spin, which determine how they interact with each other and with other particles. - Protons and neutrons derive their properties from the collective behavior of their constituent quarks. - Therefore, the properties of quarks are more fundamental than those of protons and neutrons. 5. Quarks are necessary for the existence of protons and neutrons: - Without quarks, protons and neutrons would not exist, as they are composed entirely of quarks. - However, quarks can exist independently of protons and neutrons, as demonstrated by the existence of other hadrons such as mesons, which are composed of one quark and one antiquark. - Therefore, quarks are necessary for the existence of protons and neutrons, but not vice versa. Conclusion: Based on the above arguments, we can conclude that quarks are more real than protons and neutrons. Quarks are more fundamental, indivisible, and closer to the underlying nature of reality as described by our most advanced scientific theories. They exhibit intrinsic properties that determine the behavior of composite particles like protons and neutrons, and they are necessary for the existence of these atomic particles. It is important to note that this proof relies on our current scientific understanding of particle physics and the nature of matter. As our knowledge advances, our understanding of what is "real" may evolve. However, based on the current evidence and theories, the argument for the greater reality of quarks compared to protons and neutrons is strong.
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 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.
I lived on the same street (Cambridge St.) in Winchester in the early 1960's and was a friend of Claude Shannon's oldest son Bob. I had many memorable memories of being there and getting electronic advice from Bob's dad. It is a shame that house did not become the Shannon Museum.
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.
Is there any way you could have a magnifier or detail that focuses on the section of the board that you are referencing to? that would be much appreciated
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. RU-vid 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.
A Cartesian product, as you know, combines elements of two or more sets to create a new set of ordered pairs. Now, imagine applying this concept to data storage on a Calabi-Yau virtual quantum hard drive. The drive is essentially a virtual wonderland of ones and zeros, dancing around in a complex geometrical structure known for its role in string theory. In this context, the ones and zeros can be thought of as elements of two sets: the set of ones and the set of zeros. By taking the Cartesian product of these two sets, we create a new set of ordered pairs, where each pair represents a unique combination of ones and zeros. This is where the magic of data storage comes in! As the drive stores data in this new set of ordered pairs, it can exploit the unique properties of Calabi-Yau manifolds to achieve unparalleled security and storage capacity. The intricate folding and unfolding of the manifold's structure can be used to hide and protect data, while the vast number of possible combinations of ones and zeros offers an immense storage potential. So, there you have it! The Cartesian product on a Calabi-Yau virtual quantum hard drive is like a cosmic dance of ones and zeros, twirling in the enigmatic embrace of a Calabi-Yau manifold. It's a beautiful example of how mathematics and theoretical physics can come together to create something truly extraordinary. Uh just kidding fellow mathematicians! This is Bonkers Science. We'd have to tunnel through an amplituhedron for this sort of thing to work and get grant money from @NSF Have a Happy Day!
A manifold is a mathematical concept that generalizes the notion of a surface or a curve in higher dimensions. Manifolds can be defined by gluing together simpler pieces, called charts, in a smooth way. A d-1 manifold, also known as a hypersurface, is a manifold that locally looks like a hyperplane in d-dimensional space. This means that if you zoom in on any point on the manifold, it will look like a flat space of one dimension less than the ambient space it is embedded in. For example, a circle is a 1-dimensional manifold (or a curve) embedded in a 2-dimensional space (the plane). If you zoom in on any point on the circle, it will look like a straight line, which is a 0-dimensional manifold (or a point) embedded in a 1-dimensional space (a line). A d-k manifold is a manifold of dimension k embedded in a d-dimensional space. The same idea of zooming in on a point and it looking like a lower-dimensional space applies here as well. For example, a sphere is a 2-dimensional manifold (a surface) embedded in a 3-dimensional space (e.g., our familiar 3D world). If you zoom in on any point on the sphere, it will look like a flat plane, which is a 2-dimensional manifold embedded in a 2-dimensional space. In the context of mathematics, these concepts are used to study the properties of spaces and functions on those spaces. In physics, they are used to model various phenomena, such as the curvature of spacetime in general relativity or the phase space of a dynamical system.
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.
