6 жыл бұрын
6 жыл бұрын
6 жыл бұрын
6 жыл бұрын
Пікірлер
@alfellati 10 күн бұрын
This channel is so underrated ❤
@MudithaMath 10 күн бұрын
A masterpiece of presentation. I am taking notes :)
@PedroHenriqueFarias-j5d 13 күн бұрын
Great bro!
@RanaprathapPrathap-u8h Ай бұрын
bro u are underrated
@ucngominh3354 Ай бұрын
hi
@Misteribel 2 ай бұрын
The million dollar price has been used to finance the Poincaré Chair for young, promising mathematicians.
@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.
@alisidheek3980 2 ай бұрын
Thank you 😢
@zhess4096 2 ай бұрын
11:18 - Is it a spiralling looping line across the torus? Turns it into a slinky
@amihartz 2 ай бұрын
Flipping your state doesn't even flip the other person's state when they are entangled. If you have two quantum coins in a superposiiton of HH and TT, if you flip the second one before measuring it then you change the correlation to HT and TH. It is a common misconception that if you flip the second one then you've change it to TT and HH, which is not what happens. It's not different in this sense than having two real coins in an envelope where you flip yours upsidedown before opening it.
@monoman4083 2 ай бұрын
As a professional mathematician, I should say that i am very visually interesting, but contain many mistakes: -x
@jennyone8829 3 ай бұрын
Thank you. May you be blessed always 🎈🇺🇸🦋🚀🛸🐳
@samuel_El_188 3 ай бұрын
У много рядов и 512 бит
@TymexComputing 3 ай бұрын
17:17 how precisely is 1.03313660856 equal to 1.0013660856 and if not equal which one is L(1) for equals(. , . ) 5 plus(. , . ) 3rd_power(X) 3rd_power(Y) ? Thank you.
@itsRAWRtime007 4 ай бұрын
great analogy
@BELLAROSE21212 4 ай бұрын
Solved
@saraswati999 4 ай бұрын
You are brilliant thank you for the videos
@rohitchatterjee2327 5 ай бұрын
Nice video guy
@MinMax-kc8uj 5 ай бұрын
I don't know any of this stuff, but I do know that Quantum Electrodynamics graph. That graph is correct. It has polynomial solutions in it to.
@benjaminangel5601 6 ай бұрын
Such a great video! You’ve put together the best explanation I’ve seen on the hodge conjecture
@calicoesblue4703 6 ай бұрын
Nice😎👍
@jaeimp 6 ай бұрын
In "Slight error at 14:14, the equation should be y^3, not y^2." I think you meant to type "Slight error at 14:14, the equation should be x^3, not x^2."
@stevelam5898 7 ай бұрын
Nice video pal.
@comic4relief 8 ай бұрын
13:27 What is a "distant corner of the galaxy"? Why call the sun insignificant?
@comic4relief 8 ай бұрын
9:38 An arcsecond is a thirty-six hundredth of a degree.
@もりけんいち-h4z 8 ай бұрын
Egyptian mathematics. Sa 1:11
@SirFlickka 9 ай бұрын
−2(1a,i,−x)=(−2a,−2i,2x)
@jonnywilliams5326 9 ай бұрын
Fantastic video 🎉 Thanks for sharing.
@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).
@SanjaySingh-oh7hv 10 ай бұрын
Let me echo what others have said. This video is so very excellent for explaining and making accessible a complex and difficult and abstract topic. The creator and host of this video is a gifted teacher and orator. So glad I found this video. Adding it to my playlist for future reference!
@harriehausenman8623 10 ай бұрын
great video! 🤗
@TesssyTosco 11 ай бұрын
Please, put the subtitles as I have hearing problems thank you in advance
@williamzame3708 11 ай бұрын
Smale came *before* Stallings, not after - but their work was independent of each other.
@aymantimjicht173 11 ай бұрын
Vous avez mit les note que Vous voulez une correction au natonal, aracher 30000 Dh de Les pauches de ma famille et me tuer (Grâce à Vous une maladie cronique j arrive pas a dormire son medic ) Bravo
@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!
@nyckhusan2634 Жыл бұрын
In 2010 Gregory Perelman proved that we live in 4-D Universe ( X^2+Y^2+Z^2+t^2=1). Now he is working on the problem of " Holes ". Possibly, will prove how this Universe was born.
@uncmac 2 ай бұрын
He quit math
@Jon.B.geez. Жыл бұрын
The final comment represents a complete misunderstanding of physics and lack of appreciation for mathematics. It is widely believed that solving Yang Mills equations can lead into novel insights about quark confinement on the physical side, and of both instantons and monopoles on the mathematical side. So don’t let this video discourage you everyone, solving the Yang mills equations can indeed lead to great insights furthering our understanding of the universe
@harriehausenman8623 Жыл бұрын
Great content, really fine production. Thanks!
@RSLT Жыл бұрын
👍 Just watched the video and I loved it! Hit that like button and subscribed to your channel. Can't wait for more amazing content like this! Keep up the great work! 👊😄
@polfosol Жыл бұрын
The content was pretty cool and nicely presented, but the background music is SUPER-ANNOYING. God I got a headache. It's a miracle that I managed to survive through the end.
@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.
@martinstubs6203 Жыл бұрын
Gravity isn't a force!
@r1a933 Жыл бұрын
Bro took the whole explanation to another level 💯
@The-KP Жыл бұрын
One of the best, clearest videos on any topic, let alone maths or topology. So nice I watched it twice!
@ND62511 Жыл бұрын
A bit of a correction on your explanation of Fermat’s Last Theorem; the theorem states that there are no NON-TRIVIAL INTEGER solutions to the equation a^n + b^n = c^n where n > 2. It’s really easy to get solutions to the equation if a, b & c are allowed to be real numbers. Infinitely many, actually.