Some Slides from a presentation done on July 3rd 2024.
00:00 Welcome: Drone footage of Bentley college
00:18 The main Object: (G,D,R) delta set
00:27 Simplicity Clarity Generality
00:37 Short code as Algorithmic Poetry
00:43 Code example computing cohomology
00:53 Code example constructing manifolds
01:07 Brevity: Brainfuck Programming language
01:29 Running Brainfuck code online
01:38 Can AI generate a Brainfuck interpreter
01:46 Formatted version of Brainfuck interpreter
01:53 Is there a Brainfuck to an ultimate theory?
01:57 Dirac, Pauli and Bohr fooling around
02:02 Harmonices Mundi by Kepler
02:07 Physics Theories GR-QM-GUT-TOE-CFT-SM
02:17 Examples of mathematical theories
02:27 Example of physical theories
02:37 Examples of theorems in mathematics
02:42 But doing it in the finite is blasphemy
02:58 Why finite geometry?
03:07 Foundations: a constructive approach
03:17 Herman Weyl: What is a sphere
03:38 Bernhard Riemann: quanta as parts of a manifold
03:48 Gino Fano: finite geometries
03:53 Vietoris-Rips complex
04:11 Heisenberg and Jordan: quantization of space
04:17 Arthur Ruark: discrete number of discrete data
04:27 Hartland Snyder: Quantized Space-Time
04:33 David Finkelstein: points of space time in assembly
04:44 Richard Feynman and Marvin Minsky in 1982
04:48 Wolfram Physics approach
04:52 Finite geometries: Graphs, Quivers, Delta Sets
05:02 Hypergraphs have no derivatives in general
05:12 Simple graphs define canonically hypergraphs
05:24 Whitney functor
05:32 From scratch recursive implementation
05:37 Quivers: graphs allowing multiple edges and loops
05:48 Quivers have a calculus
05:52 Quivers define spectra
05:56 There are upper bounds on all eigenvalues
06:00 And a lower bound on all eigenvalues
06:04 An illustration of these bounds
06:08 Illustration code available on ArXiv
06:13 Eilenberg-Zilber Delta sets in 1950
06:23 Abstract delta sets: computer science approach
06:33 Delta sets allow operations on them
06:40 They also carry a nice topology
06:48 There are theorems: like on Euler characteristic
06:53 Some folks behind these theorems
06:57 Four special things about finite geometry
07:13 Curvature constraints: Victor Eberhard
07:23 Stability of the vacuum problem in math
07:37 Discrete Sard theorem: No singularities
07:43 Coordinates can become non-commutative
07:49 Apropos: Descartes on coordinates
07:53 Is related to the problem of Cartesian products
08:02 Discrete Sard theorem
08:15 Discrete Level sets
08:18 Discrete level sets animation
08:26 Discrete manifolds in the finite
08:43 How does calculus look like in 50 years?
08:53 We currently teach standard calculus
09:03 A future syllabus might look different
09:14 A wave front problem in the discrete
09:20 A wave front on a cube: expected to get dense
09:42 A wave front on a torus: expected to get dense
09:54 A wave front on a Riemannian manifold
10:07 Quadratic cohomology: currently under investigation
10:28 Hoping to get invariants for knots
10:38 Drone footage: time to get back to Arlington
10:50 The end