Exploring the Depths of Elliptic Curves and Their Ranks

Head on over to https://cell.ver.so/TOE and use coupon code TOE at checkout to save 15% on your first order. Get ready to witness a turning point in mathematical history: in this episode, we dive into the AI breakthroughs that stunned number theorists worldwide. Join us as Professor Yang-Hue Hi discusses the murmuration conjecture, shows how DeepMind, OpenAI, and EpochAI are rewriting the rules of pure math, and reveals what happens when machines start making research-level discoveries faster than any human could. AI is taking us beyond proof straight into the future of discovery. As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Timestamps: 00:00 Introduction to a New Paradigm 01:34 The Changing Landscape of Research 03:30 Categories of Machine Learning in Mathematics 06:53 Researchers: Birds vs. Hedgehogs 09:36 Personal Experiences with AI in Research 11:44 The Future Role of Academics 14:08 Presentation on the AI Mathematician 16:14 The Role of Intuition in Discovery 18:00 AI's Assistance in Vague Problem Solving 18:48 Newton and AI: A Historical Perspective 20:59 Literature Processing with AI 24:34 Acknowledging Modern Mathematicians 26:54 The Influence of Data on Mathematical Discovery 30:22 The Riemann Hypothesis and Its Implications 31:55 The BST Conjecture and Data Evolution 33:29 Collaborations and AI Limitations 36:04 The Future of Mathematics and AI 38:31 Image Processing and Mathematical Intuition 41:57 Visual Thinking in Mathematics 49:24 AI-Assisted Discovery in Mathematics 51:34 The Murmuration Conjecture and AI Interaction 57:05 Hierarchies of Difficulty 58:43 The Memoration Breakthrough 1:00:28 Understanding the BSD Conjecture 1:01:45 Diophantine Equations Explained 1:03:39 The Cubic Complexity 1:19:03 Neural Networks and Predictions 1:21:36 Breaking the Birch Test 1:24:44 The BSD Conjecture Clarified 1:26:21 The Role of AI in Discovery 1:30:29 The Memoration Phenomenon 1:32:59 PCA Analysis Insights 1:35:50 The Emergence of Memoration 1:38:35 Conjectures and AI's Role 1:41:29 Generalizing Biases in Mathematics 1:44:55 The Future of AI in Mathematics 1:49:28 The Brave New World of Discovery Links Mentioned: - Topology and Physics (book): https://amzn.to/3ZoneEn - Machine Learning in Pure Mathematics and Theoretical Physics (book): https://amzn.to/4k8SXC6 - The Calabi-Yau Landscape (book): https://amzn.to/43DO7H0 - Yang-Hui’s bio and published papers: https://www.researchgate.net/profile/Yang-Hui-He - A Triumvirate of AI-Driven Theoretical Discovery (paper): https://arxiv.org/abs/2405.19973 - Edward Frenkel explains the Geometric Langlands Correspondence on TOE: https://www.youtube.com/watch?v=RX1tZv_Nv4Y - Stone Duality (Wiki): https://en.wikipedia.org/wiki/Stone_duality - Summer of Math Exposition: https://some.3b1b.co/ - Machine Learning meets Number Theory: The Data Science of Birch–Swinnerton-Dyer (paper): https://arxiv.org/pdf/1911.02008 - The L-functions and modular forms database: https://www.lmfdb.org/ - Epoch AI FrontierMath: https://epoch.ai/frontiermath/the-benchmark - Mathematical Beauty (article): https://www.quantamagazine.org/mathematical-beauty-truth-and-proof-in-the-age-of-ai-20250430/ SUPPORT: - Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Support me on Patreon: https://patreon.com/curtjaimungal - Support me on Crypto: https://commerce.coinbase.com/checkout/de803625-87d3-4300-ab6d-85d4258834a9 - Support me on PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 SOCIALS: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs #science

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