this entire presentation... really has me incredibly excited.. i've always loved Dynamics, bifurcations.. and a lot of eager exploration of the Zeta function Zeros. i've only recently done a dive of Group theory and Abstract geometries. odd... intuitions and connections among different branches suddenly clicked. i mean... this took decades to sink in for me. Weierstrass... Ramanujan. it's chilling now. The fact that he actually existed.... and the very last paper he wrote.
Viewers might also like the Mathologer video, "The hardest 'What comes next?' (Euler's pentagonal formula)", since it also discusses partitions and series. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-iJ8pnCO0nTY.html
Yes! Furthermore, Mathologer took on the challenge of proving the famous Ramanujan sum to -1/12 rigorously, applying precise definitions of convergence. His only fully serious video as far as I know.
Elliptic curves are dual to modular forms -- Fermat's last theorem. Homology is dual to cohomology. Union is dual to intersection. Integration is dual to differentiation. Points are dual to lines -- the principle of duality, geometry. "Always two there are" -- Yoda.
@@theflaggeddragon9472 Subfields are dual to subgroups -- the Galois correspondence. Homology is the study of sameness, homo is dual to hetero. Absolute sameness (isomorphism) is dual to relative sameness (homomorphism). Reducing the number of dimensions or states is a syntropic process -- homology. Hypervolumes become volumes, volumes become surfaces or planes, planes become lines and lines become points, at each stage you are reducing the number of dimensions, 4D, 3D, 2D, 1D and points are zero dimensional. Increasing the number of dimensions or states is an entropic process -- co-homology. Points become lines become planes become volumes and then hypervolumes etc. Sine is dual to cosine -- the word "co" means mutual and implies duality. SINH is dual to COSH -- hyperbolic functions. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non teleological physics (entropy). The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking! Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- abstract algebra. Injective is dual to surjective synthesizes bijective or isomorphism. Integers or real numbers are self dual as they are their own conjugates (complex):- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html Entropy is dual to evolution (syntropy). If evolution has a target, goal, purpose or objective then it is a syntropic process -- teleological.
@@theflaggeddragon9472 Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Group theory is the study of duality, domains (groups) are dual to co-domains (fields) - isomorphism. The word isomorphism actually means duality -- equivalence, similarity.
@@theflaggeddragon9472 I like these series of lectures about algebraic topology:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Ap2c1dPyIVo.html You have to do the hard work if you want to get a good grasp of topology -- mobius loops, Klein bottle, the torus, the real projective plane etc. The lectures rewire your brain in the correct way. Affine projective geometry is dual to projective hyperbolic geometry.
So ive basicly no idea what im talking about, there is just an intuition that the monstergroup might be the mathematical representation of our physical universe or somehow the heart of the theory of everything. For example, is there a posibility that the four fundamental forces which are symetric is some sense (except gravity, where we havent found the counterpart yet) are just a small part of all symetries that we would observe if we could zoom further in and out? We as humans have of course only access and senses to a small part of dimensions because the rest is trivial to our survival, and so they would be unnecessary from an evolutionary standpoint. And the same evolutionary argument goes for the other fundamental forces/symetric points which we have no access to, they might only emerge in higher rank complex/energy systems as galaxy clusters(dark energy) and might even be found in lower rank complex/energy systems. And that all this "shit" is represented in the monster group. The monstergroup is finite and therefore computational, right? Sry for my bad spanish im from england.
No, the number of dimensions need not necessarily correspond to the number of physical spacetime dimensions. The number of dimensions corresponds to the number of degrees of freedom. For instance, in 3D space there are 3 translations and 3 rotations: at least 6 degrees of freedom, which is double the number of space dimensions. So yes, your first intuition was correct. And your third intuition too: the number of degrees of freedom is finite!
The monster group has some relationship to string theory but we don't know how deep it goes. The monster might just be one small part of a "landscape" of string theory possibilities that includes many other symmetry groups
A lot of people said "no" here but actually I believe they are wrong, and that you are mostly right in your deduction. If you look up Complexity Theory, it's well known that it seems most systems in the physical world are scale invariant. It's no surprise we see a periodic table of groups, and a periodic table of particles and a periodic table of elements and maybe even some periodic table of strings in string theory... the whole thing is invariant with scale in accordance with complexity theory. (Like the modular vibration of a string in string theory) The above scale invariance indicates that most phenomenon and structures in nature might actually just be emergent properties of things at smaller scales. I can't speak for the Monster Group since i don't know that much about it in particular, but if you look up Complexity Theory being pioneered by Stephan Wolfram and Leonard Susskind/Aaronson, Wolfram describes that there is a Rulial space (logic space), in which rules/logic are constant and observers creating languages to describe these rules are relative, very much like how space-time is relative to the speed of light. In the Rulial Space, there are technically a near infinite number of rules, and sets of logic that can describe the physics of a universe, and that us as observers in this universe are merely picking a reference frame to describe these rules. Now about the Monster Group, I do not know much about it, but if you think about how even the standard model could be an emergent model, of perhaps string theory, then the monster symmetry doesn't seem so outlandish, because string theory becomes a very real possibility as a more fundemental physical framework then the current standard model, if we assume that scale invariance holds true...which is like saying that 1+1 equals 2 has to hold true, which we know that it does! So if you asked me what I believe is the most fundamental theory it would go from the most fundemental : Complexity Theory -> String Theory -> Standard Model.
Zeta of -1 is not described by the original summation (it only converges for s>1), but it’s “analytic continuation” which is a functional equation that brings the zeta function to the rest of the complex plane
