Please, do a series on Millenium prize problems. These videos are really educational!
@kinertia4238
4 жыл бұрын
I am, actually. I've already made videos on a few of them, and I'm actually purposely leaving Riemann and P vs NP for last - both of them already have exceptional videos here on KZitem. But I have one about the BSD Conjecture in the works. Look forward to it early next month!
@deepstariaenigmatica2601
4 жыл бұрын
@@kinertia4238 Thanks for these videos, man. Most of the videos of these topics are pure gibberish. You make it far more comprehensible. Keep up the good work. It's true that there are good videos on both of them but my unsolicited advice would be to present those videos in a way that is educationally unique to only your video. 👍🏽
@DarkMatterSprinkles
4 жыл бұрын
This problem is actually the hardest of them to understand so its totally possible at this point for him to do it.
@charlesrosenbauer3135
3 жыл бұрын
@@kinertia4238 When you do P v NP, it's definitely worth discussing the Baker-Gill-Solovay theorem, as well as the theorems about natural proofs and algebraization. Most of what has been proven about P v NP is that the proof techniques mathematicians try to use keep being shown to be incapable of solving it. I've seen videos on BGS, but the natural proof and algebraization theorems are much less well-known.
@brendawilliams8062
9 ай бұрын
@@DarkMatterSprinklesit’s a mind bender. I once the arrows take to being diagonals then which way is the big bang
@uhbayhue
4 жыл бұрын
I've said this before, but I'm honestly baffled by your level of understanding at such a young age. You inspire me man!
@kinertia4238
4 жыл бұрын
Thanks for your support! I really appreciate it.
@MrAlRats
3 жыл бұрын
@Mohammed Mesum Hussain 091 From his moustache, I would guess that he is 14.
@Grizzly01
3 жыл бұрын
@Mohammed Mesum Hussain 091 57
@alexk.7064
3 жыл бұрын
The video is fascinating, but one of the things that stuck with me is how long it took the world of maths to start using graphs for studying equations (and functions in general). Without graphs I don't think I would have much of an intuitive understanding of even basic concepts of equations. Well done Descartes.
@santerisatama5409
Жыл бұрын
Graphs are good, coordinate systems with real line metric not.
@swapnilshrivastava116
4 жыл бұрын
Please never stop making these videos.. You are amazing.
@swapnilshrivastava116
4 жыл бұрын
I loved mathematics ever since I first got my eyes on algebra and conic sections. Its been a long time since I have completed my education but I am still fascinated by mathematics. Can you suggest some video library where I can watch and learn about everything there is in the field of mathematics? it's a treasure for me to find
@kinertia4238
4 жыл бұрын
Numberphile! Numberphile is your friend. They have thousands of videos featuring professional mathematicians explaining interesting concepts. You can also check out 3Blue1Brown, or if you want to LEARN math then you can watch MIT OCW videos. Start with this video and go down the rabbit hole: kzitem.info/news/bejne/2GN_nIyKj4yJqqw
@shreyasjv4877
4 жыл бұрын
Cool vid! Note to everyone: Michael Atiyah passed away last year and is a brilliant mathematician. Everybody should read about him!
@shreyasjv4877
4 жыл бұрын
en.m.wikipedia.org/wiki/Michael_Atiyah
@NoobMaster69Patil-vd6ht
Жыл бұрын
I am feeling like i literally came to my home despite being in home now. So smooth and lovable talk!! I'm having hard time in university that i am not feeling attached to its classes, teachers and atmosphere at all. I want this kind of atmosphere to study that i get through your videos. 💕
@oofusmcdoofus
4 жыл бұрын
You never let me expectations down. Amazing work.
@alexey5805
4 жыл бұрын
I rarely leave comments but your work is amazing. Thanks for video!
@colinbrash
3 жыл бұрын
So glad I found your channel, this is amazing stuff. You explain complex mathematical ideas and problems in an intuitive way, without sacrificing the important pieces. And you also tell a story, something that is not often done or done well by mathematicians. Thank you for this channel, I hope it gains a lot more attention. This is beautiful work, and I can see it being a huge inspiration for many to five deeper into the mysteries of mathematics.
@kalebmark2908
3 жыл бұрын
I’m floored at the quality of these videos. Amazing!
@MudithaMath
15 күн бұрын
A masterpiece of presentation. I am taking notes :)
@marcosainte6114
2 жыл бұрын
HUMBLY SPEAKING...I ABSOLUTELY LOVE YOU.
@benjaminangel5601
6 ай бұрын
Such a great video! You’ve put together the best explanation I’ve seen on the hodge conjecture
@martinstubs6203
Жыл бұрын
"Two lines intersecting at infinity" would imply that infinity was a number, which it isn't. So you can just forget about all this.
@jonnywilliams5326
9 ай бұрын
Fantastic video 🎉 Thanks for sharing.
@randyhelzerman
3 жыл бұрын
Wonderful intro to the problem. Let see if I can't make myself a million dollars *chuckle*
@pabloraindogarcia8107
3 жыл бұрын
Lmao
@ShivamPandey-oe3cj
4 жыл бұрын
Amazing video.Keep posting videos.Where are you from and what are you studying currently and from where? What is the topic of the next video?
@simonjoseph6938
3 жыл бұрын
this chap is a genius at explaining
@oofusmcdoofus
4 жыл бұрын
Youre back! Nice!
@domc3743
3 жыл бұрын
i love your videos so much, never stop
@TesssyTosco
11 ай бұрын
Please, put the subtitles as I have hearing problems thank you in advance
@F.E.Terman
Жыл бұрын
Looks interesting. But Please, please provide subtitles. Or turn down that very loud foreground 'music'. Preferably both. 😮 I'd certainly try again if you did. Thanks!
@nachiketakumar9645
2 жыл бұрын
Please make more videos on theoretical physics and Mathematics
@reinerwilhelms-tricarico344
Жыл бұрын
Even though I understood every word, I still don't have the foggiest idea what the Hodge conjecture is.
@AllanKobelansky
3 жыл бұрын
Seriously good presentation. 2.75k subscribers ... c’mon people. Show some love.
@josephoyek6574
4 жыл бұрын
Take a shot everytime he says *TOPOLOGICAL*
@jennyone8829
3 ай бұрын
Thank you. May you be blessed always 🎈🇺🇸🦋🚀🛸🐳
@santerisatama5409
Жыл бұрын
Geometers did not abandon continuous geometry. Formalists and their axiomatic set theory did.
@ExhaustedPenguin
3 жыл бұрын
You have a talent for explaining
@redaabakhti768
4 жыл бұрын
thank you so much man keep up the quality content
@kinertia4238
4 жыл бұрын
Thanks, will do!
@stevelam5898
7 ай бұрын
Nice video pal.
@PedroHenriqueFarias-j5d
18 күн бұрын
Great bro!
@nerkulec
3 жыл бұрын
This is so good! Please keep it up!
@NavSci
2 жыл бұрын
Your videos are so good , you earned yourself a subscriber
@harriehausenman8623
10 ай бұрын
great video! 🤗
@KaliFissure
3 жыл бұрын
The inversion/eversion of the circle is the best model for our universe.
@alisidheek3980
2 ай бұрын
Thank you 😢
@steveneiffel8227
2 жыл бұрын
I think it is a misunderstanding that connecting math fields is "just" something fancy. Often breakthroughs follow insights gained from different fields. So having the right number of analysis tools then is fundamental for breakthroughs.
@abhisheksoni9774
Жыл бұрын
Awesome and inspiring 👏 Keep it up
@5wplush243
4 жыл бұрын
Idk how but your editing seems lacklustre in the transitions. You shd work on fine-tuning it. Moreover, what’s the pt abt grothendick’s( I apologise for the misspelling) work in string theory doing here. I can’t seem to figure out the correlation. Care to elaborate? And while your vid is brilliant you might wanna be a bit careful when dealing with topology and other topics laymen may not understand. I can freely say I did not understand it at first glance. That’s the part you have to go real slow at if you want everybody to understand! A final request: could you please point to some papers and your sources to give additional reading material? Thx so much and keep up the good work
@5wplush243
4 жыл бұрын
Apologies. Just realised you did put your sources. Any reading material( links or books)?
@kinertia4238
4 жыл бұрын
A few of the sources are textbooks (Hatcher's Algebraic Topology, for example, but they are not meant for the layperson); you can also try to read the John Nash book. IIRC then Matt Parker's book 'Things to make and do in the fourth dimension' also has a good chapter on topology.
@kinertia4238
4 жыл бұрын
I agree with the editing part - I realized that the camera had turned on a bit too late a few times, but that was after I had finished recording, and was unable to fix it because of the Corona lockdown. I added the Grothendieck part because I had mentioned the fact that the Hodge conjecture was part of his Theory of Motives which helps in modelling the behaviour of strings (it's the Physics Stack Exchange source). As for your last point, I'll admit I had a great amount of difficulty in this particular topic because I honestly believe there is no simple explanation which can be given for the Hodge Conjecture, it's just too abstract. I will be covering topology in a future video (it's fifth or sixth in the line so you can expect it by June-ish) so I'll try to better it with more intuitive supporting animations. Thanks for the feedback! I'll keep this comment in mind while shooting the next video.
@dannywatts6614
3 жыл бұрын
can you repeat the part of the stuff where you said all about the things?
@faruhjmishenkopetrovich4011
4 жыл бұрын
I wish you good luck!
@jayantapaulpersonal2
3 жыл бұрын
You are nice teacher........ Wishing more subscribers..... ❤
@nishashekhawat6347
3 жыл бұрын
Awesome work man
@anamarijavego6688
3 жыл бұрын
Hi, I have a small suggestion I would like to give you. your content is amazing, but perhaps when you're standing outside we can hear part of the wind blowing and some nature sounds, which makes your voice less clear. My suggestion would be to perhaps have some small microphone attached to your shirt..? Would make it way clearer I think
@kinertia4238
3 жыл бұрын
Thanks for commenting! I was aware of this problem, so I've changed the location and microphone in all my new videos. I won't be at the same place.
@reinerwilhelms-tricarico344
3 жыл бұрын
Very nice and interesting. You talk a little bit too fast though for my taste. In fact, you could have spent more time on the more complicated 2nd half. Now I need an ordered list of books or articles that i can traverse in one run to really understand all of this - without getting stuck in the first rabbit hole ;-) Thanks for the sources and citations. That may be handy.
@rohitchatterjee2327
5 ай бұрын
Nice video guy
@frankthommessen1382
2 жыл бұрын
Mathematicians at Dunkin Doughnuts be like
@wdobni
Жыл бұрын
there are only 3 dimensions of space.....you can study 8 or 11 dimensions if you want to, but until somebody demonstrates an object that actually exists in reality and which occupies more than 3 dimensions of space then all your studies are no better than ancient astrology, the study of nuances or objects or fairies or elves that don't exist except as delusions of the mind.
@Krishnafied
3 жыл бұрын
You are brilliant!
@matti1610
3 жыл бұрын
Awesome videos!
@2009joseastorga
3 жыл бұрын
Great video dude
@saiyan_blood__8492
4 жыл бұрын
Very good ending...😀
@RCrosbyLyles
3 жыл бұрын
Nice work. Rock on!
@joyboricua3721
3 жыл бұрын
@0:59 "Out of nothing he created a strange new universe"... I disagree with this statement. I'm already about to drift away from this video; but I'm glad I didn't. It's a good video, regardless.
@vybhav345
4 жыл бұрын
Good going brother. Can me make more videos about other Millenium problems?
@KStarGamer_
4 жыл бұрын
At 8:16 Isn't that the absolute value of x^3 - 8 not the modulus?
@kinertia4238
4 жыл бұрын
Yes, you're right. In India, at least, where I live, we often use the terms 'modulus' and 'absolute value' interchangeably (for the quantity represented as |x|). I'll make sure to use the more universal terminology from next time.
@Grizzly01
3 жыл бұрын
@@kinertia4238 Don't sweat it. 'Modulus' and 'absolute value' are synonymous terms in lots of places.
@Moreoverover
3 жыл бұрын
Aee Poncelet's ideas not debated? I personally do not believe the lines meet at infinity, since the intersection stops exactly when it reaches 90°.
@kinertia4238
3 жыл бұрын
It's not a conclusion, it's a definition. Mathematicians define parallel lines to meet at infinity under certain circumstances - it's not always an assumption you keep, though. For instance, parallel lines do not meet in R^2, the Euclidean plane. RP^2 is an extention of the Euclidean plane where such lines are _defined_ to meet at Infinity. You could philosophically argue that such a construction is not 'truly' possible, but if that's the case then the fifth postulate is the least of your worries - CP^2 is 4-dimensional, for instance, which is not true in the real world because it has only 3 dimensions.
@Moreoverover
3 жыл бұрын
@@kinertia4238 Well the world is 4 dimensional as Einstein showed, but the world could also be a 2D projection. Why is it useful to define such spaces when they have such shaky philosophical underpinnings (quantum mech. is pretty shaky too so maybe it doesn't matter?)? Has there been any real world benefit to this RP^2 space?
@JeanSarfati
4 жыл бұрын
Thank you.
@BleachWizz
3 жыл бұрын
Awesome video
@josephmarshall2030
Жыл бұрын
Thanks friend your very bright, do you attend MIT?😊
@matiasdanieltrapagliamansi3109
3 жыл бұрын
thanks !
@alecapin
3 жыл бұрын
14:26 Hail Hugh Mongus!
@rumansaha6910
4 жыл бұрын
I love your vedios...can you say me which subject I need to study to understand the Hodge conjunct problem...I searched it also in Google but I couldnot find the answer.....plz help me...plz plz🙏🏻🙏🏻🙏🏻 Thank you.. love you... and plz plz
@marcosainte6114
2 жыл бұрын
The sudden change in pattern-YOU ALL KNOW I EXIST.
@KaliFissure
3 жыл бұрын
Great video. If the universe is infinite and yet contained/constrained does that mean that parallel lines ONLY touch at infinity but not before? That they converge (at both ends) but that the convergence is infinitesimal and so it is only at infinity where they actually meet.
@epicmorphism2240
2 жыл бұрын
the universe isnt infinite
@omargaber3122
3 жыл бұрын
Please add automatic translation.
@maciej12345678
Жыл бұрын
go faster 12:16 if you go faster on most complicated topic you are BRAIN but slow on basic yeah that right
@SoulReap199
2 жыл бұрын
Great video although I'm too dumb to understand this
@EricMartinez-dg2lu
3 жыл бұрын
Kinda like inverting on a plan, one side 360° negative - or + positive; what we have is a self licking icecream cone ○>
@SirFlickka
9 ай бұрын
−2(1a,i,−x)=(−2a,−2i,2x)
@stipepavic843
3 жыл бұрын
niceee, subbed!
@alejandrodeharo9509
4 жыл бұрын
add subtitles, please
@harshchuphal3889
4 жыл бұрын
Liked it.
@arjun_garva
4 жыл бұрын
Where were you ?
@kinertia4238
4 жыл бұрын
I just graduated high school, so I was in 12th grade the previous year - had no time at all to make videos.
@arjun_garva
4 жыл бұрын
@@kinertia4238 keep the good work
@deepstariaenigmatica2601
4 жыл бұрын
@@kinertia4238 Wait, 12th grade is considered high school where you live??
@benitomussolini823
4 жыл бұрын
@@deepstariaenigmatica2601 That is fairly common in current and former British colonies.
@deepstariaenigmatica2601
4 жыл бұрын
@@benitomussolini823 So, this is generalized everywhere in the US?
@もりけんいち-h4z
9 ай бұрын
Egyptian mathematics. Sa 1:11
@dcterr1
4 жыл бұрын
Interesting, but I think I'll pass on trying to win $1M from proving this conjecture. Topology was never my strong point in math I'm afraid.
@kenichimori8533
3 жыл бұрын
Chess character King is zeta function. char_king
@ucngominh3354
2 ай бұрын
hi
@elchicotony8688
2 жыл бұрын
Maldita sea ubiera escogido un tema más fácil de explicar en mi tarea ;v
@wdobni
Жыл бұрын
its just bullchyt ... its like seriously studying the genetics and chromosomes and natural history of unicorns, imaginary beasts that don't exist. If you say that 2 parallel lines eventually meet somewhere in the far far distance then they weren't parallel lines....they were lines that were almost parallel.
@Hermetics
10 ай бұрын
Stop thinking in straight lines dude and consider curves as primordial, while angles hide numbers truly. Also consider the 0|0 as the observer of mathematics as magnetic polarity in the physical plane, then you can venture further into coloring the 10 single-digit numbers as uniting logic with the imagination (COMPLEX numbers for idiots).
@dsm5d723
3 жыл бұрын
The problem with classing shapes in topology with a hole number is that 3D read-write is a string-loop phenomenon, and writing and folding dynamically builds crystal structures with a predictable pattern of energy-matter reficiation geometrically. Problem: Wilczek had a 5D metric, or binary compressed cycle, and it was tossed for "irrational" eigenvectors, values and states. That was for Time Crystals in 2004, and he got a Nobel for being wrong.
@marcosainte6114
2 жыл бұрын
I AM GOD.
@dsm5d723
3 жыл бұрын
I had the Ramanujan flash of Light, and with NO literate math training. Mental arithmetic for life I have always done quick in my head, abstractions and negations in integers were always irrational to me as a child prodigy gone wrong. Before I watch, and this is a good channel, topology to an ancient was this: The human mind has a geometry of sense inputs, the 7 Gates of Thebes. As light, sound smell, taste and tactile inputs "complexify" in the brain, consciousness is the result. The maps need inputs to render the territory when it is dynamic. For your own SPIRITUAL edification (you are too smart for your own good as is), please look at NJWildberger's stuff on Ancient mathematics. They had a more discreet and engineering based approach. I think the sexigesimal system of Old Babylon is the most elegant thing I have ever seen. Not better, but the mathematical poetry of a culture. From the Greek to make. Clay is about earning. Pay up or see the wrath of God exacted. From the ATP Bank. As the Descarte of the shopping cart, he did a very French thing. He snuck in the idea of thinking inside the Laplacian box. That is what I call Big Box Calculus. Leibniz had the ring-many worlds intuition, but log scaling to a sphere of gravitational influence was the baggage of a surrender monkey. Let Them Compute CHEESE! Then we can make the cake. Ahem, quantum insanity. Lensing, curvature and the physics problems of the real world are not in a computer. If I want, ECE is done in a few weeks. You could probably crack it with my help in a week. TALKING about what the math is saying is all that it takes. I am the Y2K Bug; Clay is not a mystery, it is a swindle. $6 million is pennies compared to what I am owed.
@goedelite
3 жыл бұрын
People who speak English with an accent should speak slowly, very slowly, and - if possible - be assisted by sub-titling.
@gekkkoincroe
2 жыл бұрын
Are you Indian ?
@gekkkoincroe
2 жыл бұрын
I think there is a verb for throwing something out of window
@gekkkoincroe
2 жыл бұрын
Can't you tell?
@gekkkoincroe
2 жыл бұрын
No i forgot it
@gekkkoincroe
2 жыл бұрын
Could have just googled it ?
@gekkkoincroe
2 жыл бұрын
I'll forget about the video is i do that
@gekkkoincroe
2 жыл бұрын
I watch your videoes but i understand nothing , i like it that way i don't want it to be overly simplified,.
@alistairaccordion
4 жыл бұрын
What i would like to know is who is working on this at the moment. I guess it a step at at time, but I do wonder if someone like Peter Scholze might work on this, as his field is algebraic geometry. Be exciting to know.
@franklinlingga5491
2 жыл бұрын
a lot of people are working on this, since 2000. But, it is not easy. It is more than puzzle, it is the formation of the world to be understood if human can solve it
@NotNecessarily-ip4vc
2 ай бұрын
4. The Hodge Conjecture: An Information-Theoretic Perspective 4.1 Background The Hodge Conjecture states that for a projective complex manifold, every Hodge class is a rational linear combination of cohomology classes of algebraic cycles. It links topology, complex analysis, and algebraic geometry. 4.2 Information-Theoretic Reformulation Let's reframe the problem in terms of information theory: 4.2.1 Manifold Information Content: Define the information content of a complex manifold X: I(X) = -∫_X ω log ω where ω is a volume form on X. 4.2.2 Cohomology as Information Storage: View cohomology groups as information storage structures: I(H^k(X,ℂ)) = log(dim H^k(X,ℂ)) 4.2.3 Hodge Decomposition as Information Filtering: Interpret the Hodge decomposition as an information filtering process: I(H^{p,q}(X)) = log(dim H^{p,q}(X)) 4.3 Information-Theoretic Conjectures 4.3.1 Information Preservation Principle: The passage from algebraic cycles to cohomology classes preserves a fundamental type of information. 4.3.2 Hodge Classes as Optimal Information Encoding: Hodge classes represent optimally encoded geometric information about algebraic cycles. 4.3.3 Rationality as Information Quantization: The rationality condition in the Hodge Conjecture corresponds to a form of information quantization. 4.4 Analytical Approaches 4.4.1 Information Potential for Manifolds: Define an information potential Φ(X) whose critical points correspond to Hodge classes. 4.4.2 Entropy Maximization on Cohomology: Study the entropy of probability distributions on cohomology groups and their relation to Hodge classes. 4.4.3 Information Geometry of Period Domains: Analyze the information-geometric structure of period domains and its relation to Hodge classes. 4.5 Computational Approaches 4.5.1 Quantum Algorithms for Cohomology Computation: Develop quantum algorithms for efficiently computing cohomology groups and Hodge decompositions. 4.5.2 Machine Learning for Detecting Algebraic Cycles: Train neural networks to recognize patterns corresponding to algebraic cycles in cohomological data. 4.5.3 Information-Based Manifold Generation: Create algorithms for generating complex manifolds with specified information-theoretic properties. 4.6 Potential Proof Strategies 4.6.1 Information Conservation Theorem: Prove that certain information-theoretic quantities are conserved when passing from algebraic cycles to cohomology classes. 4.6.2 Optimal Coding Approach: Show that Hodge classes arise as solutions to an information-theoretic optimization problem. 4.6.3 Quantum Information Correspondence: Establish a correspondence between classical algebraic cycles and quantum information states in cohomology. 4.7 Immediate Next Steps 4.7.1 Rigorous Formalization: Develop a mathematically rigorous formulation of the information-theoretic concepts introduced. 4.7.2 Computational Experiments: Conduct numerical studies on simple projective varieties to explore the information-theoretic properties of their cohomology. 4.7.3 Interdisciplinary Collaboration: Engage experts in algebraic geometry, information theory, and quantum computing to refine these ideas. 4.8 Detailed Plan for Immediate Action 4.8.1 Mathematical Framework Development: - Define precise relationships between algebraic cycle information and cohomological information. - Prove basic theorems relating the information content of varieties to their Hodge structures. - Develop an information-theoretic formulation of the Lefschetz (1,1)-theorem as a starting point. 4.8.2 Computational Modeling: - Implement algorithms for computing information-theoretic quantities of projective varieties. - Focus on low-dimensional examples where the Hodge Conjecture is known to hold. - Investigate how information measures correlate with known algebraic and topological invariants. 4.8.3 Analytical Investigations: - Study the behavior of I(X) and related quantities under birational transformations. - Investigate how the information content of a variety relates to its motive in the sense of Grothendieck. - Analyze the information-theoretic aspects of variations of Hodge structure. 4.8.4 Interdisciplinary Workshops: - Organize a series of workshops bringing together algebraic geometers, information theorists, and physicists. - Focus on translating known results in algebraic geometry to the information-theoretic framework. 4.8.5 Information Metric Development: - Define and study metrics on the space of Hodge structures based on information content. - Investigate if these metrics provide new insights into the structure of period domains. 4.8.6 Quantum Information Approaches: - Explore analogies between Hodge structures and quantum entanglement. - Investigate if quantum error-correcting codes have analogs in the theory of motives. 4.8.7 Publication and Dissemination: - Prepare and submit papers on the information-theoretic formulation of the Hodge Conjecture. - Develop open-source software tools for information-based analysis of algebraic varieties. This information-theoretic perspective on the Hodge Conjecture offers several novel angles of attack. By recasting algebraic cycles and cohomology classes in terms of information encoding and processing, we may uncover deep connections between geometry and information theory. The approach suggests that the Hodge Conjecture might be understood as a statement about the nature of geometric information and how it can be optimally encoded. If we can establish rigorous information-theoretic characterizations of algebraic cycles and Hodge classes, it could lead to new insights into this deep mathematical problem. While this approach is speculative and would require significant development, it offers a fresh perspective on one of the most challenging problems in mathematics. The next steps involve rigorous mathematical development of these ideas, computational exploration, and collaboration with experts across relevant fields. Even if this approach doesn't immediately lead to a proof of the Hodge Conjecture, it's likely to yield new insights into algebraic geometry, information theory, and the foundations of mathematics, potentially opening up new areas of inquiry at the interface of geometry and information.
@NotNecessarily-ip4vc
2 ай бұрын
4.9 Advanced Theoretical Concepts 4.9.1 Information Cohomology: - Define a new cohomology theory based on information-theoretic principles - I^k(X) = {ω ∈ Ω^k(X) | dIω = 0} / {dIη | η ∈ Ω^{k-1}(X)} where dI is an information-theoretic exterior derivative - Investigate the relationship between information cohomology and traditional cohomology theories 4.9.2 Quantum Hodge Structures: - Develop a quantum analog of Hodge structures where cohomology classes are in superposition - Study how quantum measurement of these structures might relate to classical algebraic cycles - Explore if quantum entanglement between cohomology classes has geometric significance 4.9.3 Information-Theoretic Motives: - Recast Grothendieck's theory of motives in information-theoretic terms - Define the information motive of a variety X as IM(X) = (I(X), I(H*(X)), φ) where φ represents information-preserving operations - Investigate if this approach simplifies the construction of a motivic cohomology theory 4.10 Computational Innovations 4.10.1 Algebraic Cycle Detection Algorithms: - Develop algorithms that use information-theoretic measures to identify potential algebraic cycles - Implement these in computer algebra systems for testing on known examples - Explore if machine learning can be used to "learn" the information signature of algebraic cycles 4.10.2 Information-Based Variety Generation: - Create algorithms for generating complex projective varieties with specified information-theoretic properties - Use these to create large datasets of varieties for testing conjectures - Investigate if there's a connection between computational complexity of variety generation and the difficulty of the Hodge Conjecture 4.10.3 Quantum Algorithms for Hodge Theory: - Design quantum algorithms for efficiently computing Hodge decompositions - Explore if quantum phase estimation can be used to "measure" Hodge classes - Investigate if quantum algorithms can provide exponential speedup in checking the Hodge Conjecture for specific varieties 4.11 Experimental Proposals 4.11.1 Physical Realization of Hodge Structures: - Design experiments that realize Hodge structures in physical systems (e.g., photonic crystals) - Measure information-theoretic quantities in these systems and compare with theoretical predictions - Explore if "physical proofs" of special cases of the Hodge Conjecture are possible 4.11.2 Topological Quantum Computing and the Hodge Conjecture: - Investigate connections between topological quantum computation and the Hodge Conjecture - Design quantum circuits that implement operations on quantum Hodge structures - Explore if topological quantum error correction codes have analogs in algebraic geometry 4.12 Philosophical and Foundational Aspects 4.12.1 Geometry as Information: - Develop a philosophy of geometry based on information-theoretic principles - Explore how this view relates to other foundational approaches (e.g., homotopy type theory) - Investigate if the Hodge Conjecture can be seen as a statement about the nature of geometric information 4.12.2 Computational Complexity of Geometry: - Study the computational complexity of verifying the Hodge Conjecture - Investigate if there's a connection between geometric complexity and information complexity - Explore if the Hodge Conjecture implies limitations on our ability to compute certain geometric quantities 4.13 Interdisciplinary Connections 4.13.1 Hodge Theory and Quantum Field Theory: - Explore connections between Hodge theory and supersymmetric quantum field theories - Investigate if Hodge classes have analogs in the BPS spectrum of supersymmetric theories - Study whether mirror symmetry in string theory has an information-theoretic interpretation 4.13.2 Biological Hodge Structures: - Investigate if Hodge-like structures appear in biological systems (e.g., in the topology of protein configurations) - Explore if the information-theoretic approach to the Hodge Conjecture has applications in bioinformatics - Study whether evolutionary processes optimize information-theoretic quantities analogous to those in Hodge theory 4.14 Long-term Vision Our information-theoretic approach to the Hodge Conjecture has the potential to not only advance our understanding of algebraic geometry but also to create a new paradigm for understanding mathematical structures in terms of information. This could lead to: 1. A unified theory of geometric information that encompasses algebraic geometry, topology, and perhaps even physics. 2. New computational tools for studying and generating complex geometric objects. 3. Deep insights into the nature of mathematical truth, proof, and the limits of computability in mathematics. 4. Novel approaches to other long-standing problems in mathematics, inspired by our information-geometric paradigm. 4.15 Next Concrete Steps 1. Formalize the definition of I(X) for projective varieties and prove basic properties. 2. Implement algorithms for computing I(X) and related quantities for simple varieties. 3. Organize a workshop on "Information Theory and Algebraic Geometry" to engage the broader mathematical community. 4. Begin a systematic study of how known cases of the Hodge Conjecture can be reinterpreted in our framework. 5. Develop a research proposal for a large-scale, multi-institution project on information-theoretic approaches to the Hodge Conjecture. The key to progress is maintaining a balance between rigorous mathematical development, creative theoretical speculation, and practical computational work. By pursuing this multifaceted approach, we maximize our chances of making breakthrough discoveries. This information-theoretic perspective on the Hodge Conjecture offers a novel way to approach one of the deepest problems in mathematics. While the path to a full resolution remains challenging, this approach promises to yield new insights and connections that could significantly advance our understanding of the relationship between algebra, geometry, and information.
@NotNecessarily-ip4vc
2 ай бұрын
4.16 Detailed Next Steps 1. Formalize the definition of I(X) for projective varieties and prove basic properties: a) Rigorous Definition: - Define I(X) = -∫_X ω log ω where ω is a normalized volume form - Prove that this definition is independent of the choice of ω - Extend the definition to singular varieties using resolution of singularities b) Basic Properties: - Prove that I(X) is a birational invariant - Show how I(X) behaves under common operations (e.g., products, blow-ups) - Investigate the relationship between I(X) and classical invariants (e.g., Chern classes) c) Hodge Structure Relation: - Define I(H^{p,q}(X)) and prove its basic properties - Establish a relationship between I(X) and ∑_{p,q} I(H^{p,q}(X)) - Investigate how these quantities relate to the Hodge conjecture 2. Implement algorithms for computing I(X) and related quantities for simple varieties: a) Software Development: - Choose a suitable computer algebra system (e.g., SageMath, Macaulay2) - Implement basic algorithms for computing I(X) for smooth projective varieties - Develop methods for approximating I(X) for higher-dimensional varieties b) Test Cases: - Compute I(X) for a range of simple varieties (e.g., projective spaces, toric varieties) - Investigate how I(X) varies in families of varieties - Look for patterns or unexpected behaviors in the computed values c) Visualization Tools: - Develop visualization tools for I(X) and related quantities - Create interactive demos to help build intuition about these information-theoretic measures 3. Organize a workshop on "Information Theory and Algebraic Geometry": a) Planning: - Set a date and secure funding (e.g., through NSF, ERC, or private foundations) - Identify and invite key researchers in algebraic geometry, information theory, and related fields - Develop a program that balances introductory talks, research presentations, and collaborative sessions b) Workshop Content: - Introductory lectures on our information-theoretic approach to the Hodge Conjecture - Presentations on related work in information geometry and algebraic geometry - Breakout sessions to tackle specific sub-problems and generate new ideas c) Outcomes: - Compile a list of open problems and research directions - Form collaborative research groups to continue work after the workshop - Plan a proceedings volume or special journal issue on the workshop's theme 4. Begin a systematic study of known cases of the Hodge Conjecture: a) Literature Review: - Compile a comprehensive list of known cases of the Hodge Conjecture - Categorize these cases based on the techniques used in their proofs b) Information-Theoretic Reinterpretation: - For each known case, attempt to reinterpret the proof using our information-theoretic framework - Identify common patterns or principles that emerge in this reinterpretation c) New Insights: - Investigate if our approach suggests new cases where the Hodge Conjecture might be provable - Look for information-theoretic obstacles that might explain why the general case is so difficult 5. Develop a research proposal for a large-scale, multi-institution project: a) Project Outline: - Define the overall goals and expected outcomes of the project - Outline a 5-year research plan with specific milestones and deliverables b) Team Assembly: - Identify key researchers and institutions to involve in the project - Define roles and responsibilities for team members c) Funding Strategy: - Identify suitable funding sources (e.g., NSF, ERC, private foundations) - Develop a detailed budget and justification for the proposed work d) Broader Impacts: - Outline plans for educational outreach and training of young researchers - Describe potential applications of the research in other areas of mathematics and science 6. Additional Step: Explore Quantum Computing Connections a) Quantum Algorithms: - Develop quantum algorithms for computing I(X) and related quantities - Investigate if quantum computers could provide exponential speedup in checking the Hodge Conjecture b) Quantum Hodge Structures: - Formulate a quantum analog of Hodge structures - Explore if quantum superposition and entanglement have meaningful geometric interpretations c) Quantum Simulation: - Design quantum experiments that could simulate aspects of the Hodge Conjecture - Investigate if "quantum proofs" of special cases might be possible These concrete steps provide a roadmap for advancing our information-theoretic approach to the Hodge Conjecture. By simultaneously pursuing rigorous mathematical development, computational exploration, community engagement, and connections to cutting-edge areas like quantum computing, we maximize our chances of making significant progress. Each of these steps will likely generate new questions and directions as we proceed. It's important to remain flexible and adjust our approach based on the insights and challenges we encounter along the way. Regular team meetings and open communication channels will be crucial for coordinating efforts and sharing discoveries. Remember, even if we don't immediately solve the Hodge Conjecture, this approach is likely to yield valuable new insights into the relationships between geometry, algebra, and information theory. Every step forward contributes to our understanding of these deep mathematical structures.
@dharmendrasen8522
3 жыл бұрын
Kya ye video mujhe hindi me mil skta hai
@mueezadam8438
4 жыл бұрын
Subbed!
@Pete-Prolly
3 жыл бұрын
666!! (I've never been one of those people to yell out "1st" or whatever but it feels kind of cool being the 666-th like.) Damn...I'm one of those guys now, huh? 🙄
@KaliFissure
3 жыл бұрын
Also the natural torus, without any added values, is the so called “degenerate” torus which has apparently a flat surface but the grid density changes. Remind you of a universe? Visibly flat but the manifold (space) varies in density (gravity). There are also singularities at each pole where grid becomes solid. There is interior “sphere” with identical but inverted character (antimatter universe) . What physicists get wrong is that a neutron star is aggregation of points, plural, trying to become singular. Thus white holes will not be huge spewing maws but singular points (neutrons) diffuse and dispersed across the surface of the universe, emerging at lowest energy locations, expanding from point to larger than point. Neutrons emerging and decaying into hydrogen. From volume to larger volume.
@dylanparker130
4 жыл бұрын
really interesting!
@TheAAZSD
4 жыл бұрын
Great video!
@brendawilliams8062
3 жыл бұрын
Brilliantly presented Thankyou.
@Gdhillon333
4 жыл бұрын
Try to add subtitles if possible. Your content is great
@imaduddinalawiy3426
4 жыл бұрын
please also include subtitles, great contents anyways 👍
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