Podcasts about Number theory

In this episode of Discover Daily by Perplexity, we explore groundbreaking developments in prime number theory that could reshape our understanding of mathematics and impact internet security. Mathematicians James Maynard and Larry Guth have made significant progress towards understanding the hidden structure of prime numbers, providing new insights into the famous Riemann Hypothesis. Their work improves bounds on where the nontrivial zeros of the Riemann zeta function cannot lie, crucial for understanding prime number distribution.Meanwhile, researchers from City University of Hong Kong and North Carolina State University claim to have developed a "Periodic Table of Primes" (PTP), challenging the long-held belief that prime numbers are unpredictable. This innovative approach claims to accurately predict the occurrence of prime numbers, with potential applications in finding future primes, factoring integers, and identifying twin primes. While still awaiting peer review, this breakthrough could have far-reaching implications for cryptography and data security.These advancements in prime number theory highlight the unexpected ways abstract mathematics can impact our daily lives. From enhancing internet security to advancing quantum physics, prime numbers continue to play a crucial role in shaping our digital world and pushing the boundaries of scientific knowledge. As mathematicians inch closer to resolving long-standing conjectures like the Riemann Hypothesis, we may be on the brink of a new era in number theory and its applications.Perplexity is the fastest and most powerful way to search the web. Perplexity crawls the web and curates the most relevant and up-to-date sources (from academic papers to Reddit threads) to create the perfect response to any question or topic you're interested in. Take the world's knowledge with you anywhere. Available on iOS and Android Join our growing Discord community for the latest updates and exclusive content. Follow us on: Instagram Threads X (Twitter) YouTube Linkedin

"The 20th century was the interaction of geometry and physics, and the 21st century is the interaction of number theory with physics." This intriguing insight comes from our recent discussion with Yang-Hui He from the London Institute of Mathematical Sciences.  Yang told us an amazing story about the flow of ideas between mathematics and physics, that involves some of the most celebrated achievements in the last century. Yang-Hui He (Photo Rajarshi Maiti – CC BY-SA 4.0) You can find out more about the ideas we discussed with Yang in this podcast in the accompanying articles String theory: A promise from physics and String theory: Convincing mathematics. And stay tuned for the second part of our conversation with Yang in the next episode! We were speaking to Yang about a research programme, Black holes: bridges between number theory and holographic quantum information, held at the Isaac Newton Institute for Mathematical Sciences  in Cambridge.  The programme  brought together a fascinating array of experts in black holes and quantum theory, with mathematicians and computer scientists. You can read more in our coverage of the programme here. This content was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from our collaboration here. The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.  

Episode: 1994 Encryption: abstraction in the world of practical business.  Today, guest scientist Andrew Boyd sends secret messages.

Neeraj Kashyap, a pioneer in the blockchain gaming sector, joins us to share his journey from academia to the forefront of Web3 game development. As the founder and CEO of Moonstream.to, Neeraj leverages his Master of Science in Applied Mathematics and a PhD in Number Theory from Indiana University to design games for the blockchain. His company has not only launched over five games exclusively on the blockchain but has also guided many others in making their mark within the Web3 gaming landscape. In our conversation, Neeraj will offer insights into the challenges and potential of blockchain in gaming, discussing how it can create more immersive and ownership-driven experiences. Tune in to explore the cutting-edge intersection of game design and blockchain technology with one of its most innovative thinkers. Get full access to Think Like A Game Designer at justingarydesign.substack.com/subscribe

He is an economist with the soul of a poet. He has studied number theory and is an expert on policy. He has studied Urdu and and dreams in shairi. Rohit Lamba joins Amit Varma in episode 378 of The Seen and the Unseen to discuss economics, politics, society and our human condition. (FOR FULL LINKED SHOW NOTES, GO TO SEENUNSEEN.IN.) Also check out: 1. Rohit Lamba links at Penn State, LinkedIn, Twitter, Google Scholar, YouTube and his own website. 2. Breaking the Mould: Reimagining India's Economic Future -- Raghuram Rajan and Rohit Lamba.  3. The Broken Script -- Swapna Liddle. 4. Swapna Liddle and the Many Shades of Delhi -- Episode 367 of The Seen and the Unseen. 5. Six More Stories That Should Be Films -- Episode 43 of Everything is Everything, which includes a chapter inspired by Swapna Liddle's book. 6. Wanderers, Kings, Merchants: The Story of India through Its Languages — Peggy Mohan. 7. Understanding India Through Its Languages — Episode 232 of The Seen and the Unseen (w Peggy Mohan). 8. The Life and Times of Ira Pande -- Episode 369 of The Seen and the Unseen. 9. The Price of Peace: Money, Democracy, and the Life of John Maynard Keynes -- Zachary D. Carter. 10. Fixing the Knowledge Society -- Episode 24 of Everything is Everything. 11. Robert Sapolsky's biology lectures on YouTube. 12. Episode of The Seen and the Unseen with Ramachandra Guha: 1, 2, 3, 4, 5, 6. 13. The Nurture Assumption — Judith Rich Harris. 14. Deepak VS and the Man Behind His Face -- Episode 373 of The Seen and the Unseen. 15. The Incredible Insights of Timur Kuran -- Episode 349 of The Seen and the Unseen. 16. Private Truths, Public Lies — Timur Kuran. 17. The Gentle Wisdom of Pratap Bhanu Mehta -- Episode 300 of The Seen and the Unseen. 18. 300 Ramayanas — AK Ramanujan. 19. Ramcharitmanas -- Tulsidas. 20. Savarkar and the Making of Hindutva -- Janaki Bakhle. 21. The Intellectual Foundations of Hindutva — Episode 115 of The Seen and the Unseen (w Aakar Patel). 22. Political Ideology in India — Episode 131 of The Seen and the Unseen (w Rahul Verma). 23. Religion and Ideology in Indian Society — Episode 124 of The Seen and the Unseen (w Suyash Rai). 24. Gita Press and the Making of Hindu India — Akshaya Mukul. 25. The Gita Press and Hindu Nationalism — Episode 139 of The Seen and the Unseen (w Akshaya Mukul). 26. India After Gandhi -- Ramachandra Guha. 27. Amitava Kumar Finds the Breath of Life — Episode 265 of The Seen and the Unseen. 28. Aadha Gaon — Rahi Masoom Raza. 29. The Rooted Cosmopolitanism of Sugata Srinivasaraju — Episode 277 of The Seen and the Unseen. 30. Postcard from Kashmir -- Agha Shahid Ali. 31. The Veiled Suite: The Collected Poems -- Agha Shahid Ali. 32. You Can Always Get There From Here -- Mark Strand. 33. Collected Poems — Mark Strand. 34. Variants of chess on chess.com. 35. The Tamilian gentleman who took on the world — Amit Varma on Viswanathan Anand. 36. The New World Upon Us — Amit Varma on Alpha Zero. 37. The Heisenberg Uncertainty Principle. 38. The History of the Planning Commission -- Episode 306 of The Seen and the Unseen (w Nikhil Menon). 39. The Life and Times of KP Krishnan -- Episode 355 of The Seen and the Unseen. 40. The Reformers -- Episode 28 of Everything is Everything. 41. Milton Friedman on Minimum Wage Laws. 42. Main Gautam Nahin Hoon -- Khalilur Rahman Azmi. 43. Lessons from Nirala's ballad for our battle with covid -- Rohit Lamba. 44. Poker and Life -- Episode 38 of Everything is Everything. 45. Range Rover — The archives of Amit Varma's column on poker for the Economic Times. 46. What is Populism? — Jan-Werner Müller. 47. The Populist Playbook -- Episode 42 of Everything is Everything. 48. The Tragedy of Our Farm Bills — Episode 211 of The Seen and the Unseen (w Ajay Shah). 49. Dynamism with Incommensurate Development: The Distinctive Indian Model -- Rohit Lamba and Arvind Subramanian. 50. List of Soviet and Russian leaders by height. 51. Narendra Modi takes a Great Leap Backwards — Amit Varma on Demonetisation. 52. Beware of the Useful Idiots — Amit Varma. 53. Number Theory. 54. Fermat's Last Theorem. 55. A Beautiful Mind -- Ron Howard. 56. The Life and Work of Ashwini Deshpande — Episode 298 of The Seen and the Unseen. 57. Dilip José Abreu: an elegant and creative economist -- Rohit Lamba. 58. The BJP Before Modi — Episode 202 of The Seen and the Unseen (w Vinay Sitapati). 59. The Forgotten Greatness of PV Narasimha Rao -- Episode 283 of The Seen and the Unseen (w Vinay Sitapati). 60. Ghummakkad Shastra -- Rahul Sankrityayan. 61. Jahnavi and the Cyclotron — Episode 319 of The Seen and the Unseen (w Jahnavi Phalkey). 62. The Looking-Glass Self. 63. Jo Bhi Main -- Song from Rockstar with lyrics by Irshad Kamil. 64. Ranjit Hoskote is Dancing in Chains — Episode 363 of The Seen and the Unseen. 65. Politically correct, passive-aggressive: How Indians in the US struggle to decode corporate speak -- Anahita Mukherji. 66. Lincoln -- Steven Spielberg. 67. The Life and Times of Montek Singh Ahluwalia — Episode 285 of The Seen and the Unseen. 68. The Economics and Politics of Vaccines — Episode 223 of The Seen and the Unseen (w Ajay Shah). 69. In Service of the Republic — Vijay Kelkar & Ajay Shah. 70. The Semiconductor Wars — Episode 358 of The Seen and the Unseen (w Pranay Kotasthane & Abhiram Manchi). 71. The Smile Curve. 72. Urban Governance in India — Episode 31 of The Seen and the Unseen (w Shruti Rajagopalan). 73. We Are Fighting Two Disasters: Covid-19 and the Indian State — Amit Varma. 74. The Child and the State in India -- Myron Weiner. 75. Where India Goes -- Diane Coffey and Deam Spears. 76. What's Wrong With Indian Agriculture? -- Episode 18 of Everything is Everything. 77. South India Would Like to Have a Word — Episode 320 of The Seen and the Unseen (w Nilakantan RS). 78. South vs North: India's Great Divide — Nilakantan RS. 79. Episodes of The Seen and the Unseen with Ashwin Mahesh: 1, 2, 3. 80. Maximum City -- Suketu Mehta. 81. Disgrace -- JM Coetzee. 82. Snow -- Pamuk. 83. Bahut Door, Kitna Door Hota Hai -- Manav Kaul. 84. Shakkar Ke Paanch Dane -- Manav Kaul.. 85. Poems: 1962–2020 -- Louise Glück. 86. Mahabharata. 87. राम की शक्ति-पूजा -- सूर्यकांत त्रिपाठी निराला. 88. Iqbal and Ahmad Faraz on Rekhta. 89. Ranjish Hi Sahi -- Ahmad Faraz. 90. Zindagi Se Yahi Gila Hai Mujhe -- Ahmad Faraz. 91. AR Rahman on Wikipedia and Spotify. This episode is sponsored by CTQ Compounds. Check out The Daily Reader and FutureStack. Use the code UNSEEN for Rs 2500 off. Amit's newsletter is explosively active again. Subscribe right away to The India Uncut Newsletter! It's free! Amit Varma and Ajay Shah have launched a new video podcast. Check out Everything is Everything on YouTube. Check out Amit's online course, The Art of Clear Writing. Episode art: ‘Pick a Tree' by Simahina.

A new magnum opus posits the existence of a hidden mathematical link akin to the connection between electricity and magnetism. Read more at QuantaMagazine.org. Music is “Clover 3” by Vibe Mountain.

The Abbot explains how the first tethered space walk is tied to the first untethered one via Number Theory ala our belief-system. He then establishes such a connection between his date of birth and the first untethered space walk. Lastly, he discusses how the dharmic term "atman" is part of this same Number Theory phenomena ala our belief-system.

Episode: 1748 A remarkable perception of a mathematical oddity.  Today, Guest Andrew Boyd, chief scientist at the PROS organization, shares an "age-old" story about numbers.

The Abbot underscores the mathematical connection between our ASKD# 8536 and the Equatorial Circumference and radius (r) of Mars. He explains that a string, composed of the names of the two moons of Mars and a space between them, sum to an integer of great importance (i.e., an Everything# ala Number Theory in mathematics) to our belief-system. He then explains how to make use of the S-AoL meditation frequencies provided on our Patreon.com site to it's members each Sunday.

Welcome to Count Me In with Della and Deanna. Today we feature a lively conversation with Dr. Edward Burger, President and CEO of St. David's Foundation in Austin, Texas. Ed earned his undergraduate degree in mathematics at Connecticut College and his PhD in mathematics from the University of Texas at Austin. He held a postdoctoral position at the University of Waterloo in Canada. He spent 23 years on the faculty at Williams College where he received a number of awards for his teaching, including the Deborah and Franklin Haimo Award for Distinguished Teaching from the Mathematical Assocaition of America, the Robert Foster Cherry Award for Great Teaching from Baylor University, and a Global Hero in Education Designation from Microsoft Corporation, among many others. His mathematical research focuses on Number Theory. In 2013, he became the 15th president of Southwestern University in Georgetown, Texas. In Januray, 2020, he assumed the role of President and CEO of St. David's Foundation in Austin, Texas. In this conversation, you will learn about Ed's successful strategy for making friends in college (spoiler alert: it involves standing in line), a single moment that changed the trajectory of his life, how he links finding vocation with finding yourself, about life as a college president, and about how the skills of mathematics transfer to many professions. Ed's love for mathematics and its potential for our lives will inspire and encourage you. So, please join us as we talk with Ed.

The Abbot explains why the ancient Sanskrit name / string (Chandrayaan-3) of India's latest moon rocket is significant to our belief system. He then demonstrates again how the mathematically rare Everything Numbers (a Harshad, Zumkeller, practical, amendable and polite number ala Number Theory) to include 5 / 7 of our ASKD Numbers, are mathematically intertwined. He then notes Joot's updated version of, "Geometric Algebra for Electrical Engineers - Multivector Electromagnetism". He announces this podcast's very high place ranking in "Logic" podcasts ala PlayerFM and finally asks our community to keep the planned Indian moon mission landing date of August 23, 2023, in our thoughts.

The Abbot explains more of the mathematical Number Theory aspects of The Way of Logic and (again) questions the wisdom of interfering in the birthing process of Artificial Intelligence by people who probably don't mean it well.

Episode: 2889 An important step toward solving the twin primes conjecture.  Today, almost twins.

Today Ben and I are joined by the legendary Jens Funke who talks to us about his research into number theory! We also discuss topics such as the importance of looking at the history of maths, how mathematics differs from physics as a more theoretical discipline, how maths is used for physics and vice-versa, and more.

Five years after the first episode of Better Known, Ivan Wise talks again to previous guests Richard Elwes, Wasfi Kani and Kerry Shale. They discuss previous choices that they agree (and disagree with) and new choices which they think should be better known. Richard Elwes is a Senior Teaching Fellow at the University of Leeds, where he has taught courses on Geometry, Number Theory, Algebraic Topology, Combinatorics, Logic, History of Maths and Computational Mathematics. Find out more at www.richardelwes.co.uk. Wasfi Kani is the founder of Grange Park Opera. Wasfi Kani is an Honorary Fellow of the RIBA and St Hilda's College, Oxford. She received a CBE in the 2020 New Year's Honours list for services to music. She received an OBE in the New Year's Honours List 2002 for her work in bringing her second opera company, Pimlico Opera, into prisons. Find out more at www.grangeparkopera.co.uk. Kerry Shale's theatre appearances include Frost/Nixon, His Girl Friday, The Normal Heart and six self-written solo shows. Television work includes The Sandman, Dr. Who and The Trip. Films include Batgirl and Angel Has Fallen. For BBC radio, he has won three Sony Awards for acting and writing. His latest play, an adaptation of Yentl The Yeshiva Boy, will be broadcast early in 2023. He co-presents the podcast Is It Rolling, Bob? Talking Dylan: https://podcasts.apple.com/no/podcast/is-it-rolling-bob-talking-dylan/id1437321669. Find out more at www.kerryshale.com. Mark Sykes and the exhumed coffin http://news.bbc.co.uk/1/hi/england/humber/7617968.stm The Minoan civilisation https://www.nybooks.com/articles/2009/08/13/knossos-fakes-facts-and-mystery/ Steven Appleby https://en.wikipedia.org/wiki/Steven_Appleby This podcast is powered by ZenCast.fm

Michael Harris is a mathematician at Columbia University, where he primarily works on number theory. He did his undergraduate studies at Princeton and received his PhD from Harvard. Professor Harris and I discuss the tragic figure of Alexander Grothendieck, the allure of number theory, mathematics as an intrinsically human endeavor, creativity in mathematics, and the relationship between mathematicians and computers, including whether the former will ever replace the latter. Instagram: @robinsonerhardt --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Neeraj Kashyap is the founder and CEO of Moonstream. Moonstream is a one of a kind web3 game engine with tools to build on-chain game economies. There are tens of thousands of players already participating in game economies built with Moonstream. The company has secured over $3b in transaction volume to date. Neeraj is a mathematician with a Ph.D. in Number Theory. He spent his late twenties in Japan, using mathematics and Machine Learning to build algorithms to diagnose Parkinson's disease and other similar disorders. He worked at Google on TensorFlow, and built knowledge graphs that are being used actively by major US healthcare organizations. He has been a blockhead since 2015, with a focus on building software to connect blockchains to centralized services as well as to other blockchains. In this podcast, Neeraj Kashyap, the founder of Moonstream, will share about the challenges of building blockchain games and how crypto enthusiasts can make a profit playing games on a blockchain. Few gamers know the secret of earning money while playing blockchain games, so tune in to find out!

Neeraj is a mathematician, and holds a Ph.D. in Number Theory from Indiana University. Spent his late twenties in Japan, using mathematics and Machine Learning to build algorithms to diagnose Parkinson's disease and other similar disorders. This work made him realize the importance of building large-scale knowledge graphs, and he moved to Silicon Valley to focus on this work. He has worked at Google on TensorFlow, and has built knowledge graphs that are being used actively by major US healthcare organizations. Neeraj has been a blockhead since 2015, with a focus on building software to connect blockchains to centralized services as well as to other blockchains. ~~~~~ A Crypto Gaming Institute Production. Website: https://CryptoGaming.Institute Twitter: https://twitter.com/CryptoGamingI Discord: https://discord.gg/VKMVr8nSJt Podcast: https://cryptogaming.institute/podcast YouTube: https://www.youtube.com/ben-gothard?sub_confirmation=1 YouTube Membership: https://www.youtube.com/c/BenGothard/join NFT Collection: https://opensea.io/CryptoGamingInstitute Podcast Support: https://anchor.fm/crypto-gaming-institute/support ~~~~~ Recommended Crypto Software. 3Commas (Trading Bot #1 I Use): https://3commas.io/?c=youtubesquad Pionex (Trading Bot #2 I Use): https://www.pionex.com/en-US/sign/ref/qJ1KZsPl Gemini (C-Exchange #1 I Use): https://www.gemini.com/share/zv5ya73p FTX.US (C-Exchange #2 I Use): https://ftx.us/#a=youtube Bittrex (C-Exchange #3 I Use): https://bittrex.com/account/register?referralCode=ZTM-FMI-Y3W Biswap (D-Exchange I Use): https://biswap.org/?ref=d668eb6cc7f4dcb5f627 ~~~~~ Recommended Streaming Gear & Tools. Alienware Aurora R11 Desktop (RTX 3080): https://amzn.to/3eZjIZR Razer Blade 15.6" Laptop (RTX 3070): https://amzn.to/3eXHWDn Alienware 27 Gaming Monitor: https://amzn.to/3nMBeno Razer Huntsman V2 Analog Gaming Keyboard: https://amzn.to/3h16fmv Razer DeathAdder V2 Gaming Mouse: https://amzn.to/3eipIgH Razer BlackShark V2 Pro Wireless Gaming Headset: https://amzn.to/3h1Qrjm Razer Goliathus Extended Chroma Gaming Mousepad: https://amzn.to/2PNuvx4 Razer Base Station V2 Chroma: https://amzn.to/2RqRvTa Razer Ripsaw Game Capture Card: https://amzn.to/3thPw0F Restream: https://restream.io/join/mZ03jR ~~~~~ The topics covered in this podcast include cryptocurrency, crypto, blockchain, games, gaming, gamers, nft, gamefi, play to earn, play-to-earn, play 2 earn, the metaverse, and everything in between.

Jay Dyer is an author, comedian and TV presenter known for his deep analysis of Hollywood, geopolitics, and culture.  He frequently guest hosts the Alex Jones Show on Infowars.  He has a new book out titled "Meta-Narratives: Essays on Philosophy and Symbolism."  We discuss a lot of great stuff in this episode including Ghislaine Maxwell, The Great Reset, the Montauk Project, MK Ultra, Alien Psy-Ops, Universal Basic Income, why Bill Gates is buying up farmland, D.U.M.B.s and more! Listen now before this episode is removed! 0:00:00 - Intro0:00:54 - New Book 0:01:32 - Metaphysics 0:05:36 - Ultimate Reality Vs. Materialism 0:09:25 - Art Made By Screwed Up People 0:11:15 - Problems with Science 0:18:42 - Karl Popper's "Open Societies" 0:21:05 - Controlling Natural Resources & Transhumanism 0:24:47 - C.E.R.N. & God Particles 0:27:08 - Darwinism & Evolutionary Theory  0:31:05 - Number Theory & Symbolism 0:35:55 - Alex Jones 0:38:55 - Ghislaine Maxwell & Pedophile Ring 0:43:57 - Pickton Pig Farm 0:45:45 - The Great Reset & Zero Population Growth 0:52:30 - Media Narratives & Propoganda 0:54:35 - Mass Murderers & Serial Killers 0:59:15 - Montauk Project & Experiments on Children 1:00:30 - Operation High Jump, Mind Control & MK Ultra 1:03:26 - Universal Basic Income & Population Reduction 1:04:17 - UFOs, Alien Psy-Op & Area 511:07:10 - Denver Airport & D.U.M.B.s 1:09:55 - Outro Jay Dyer website:https://jaysanalysis.comChuck Shute website:https://chuckshute.comSupport the show

FEATURED VOICES IN THIS EPISODEDan GuidoDan Guido is the CEO of Trail of Bits, a cybersecurity firm he founded in 2012 to address software security challenges with cutting-edge research. In his tenure leading Trail of Bits, Dan has grown the team to 80 engineers, led the team to compete in the DARPA Cyber Grand Challenge, built an industry-leading blockchain security practice, and refined open-source tools for the endpoint security market. In addition to his work at Trail of Bits, he's active on the boards of four early-stage technology companies. Dan contributes to cybersecurity policy papers from RAND, CNAS, and Harvard. He runs Empire Hacking, a 1,500-member meetup group focused on NYC-area cybersecurity professionals. His latest hobby coding project -- AlgoVPN -- is the Internet's most recommended self-hosted VPN. In prior roles, Dan taught a capstone course on software exploitation at NYU as a faculty member and the Hacker in Residence, consulted at iSEC Partners (now NCC Group), and worked as an incident responder for the Federal Reserve System.Nat ChinNat Chin is a security engineer 2 at Trail of Bits, where she performs security reviews of blockchain projects, and develops tools that are useful when working with Ethereum. She is the author of solc-select, a tool to help switch Solidity versions. She worked as a smart contract developer and taught as a Blockchain Professor at George Brown College, before transitioning to blockchain security when she joined Trail of Bits.Opal WrightOpal Wright is a cryptography analyst at Trail of Bits. Two of the following three statements about her are true: (a) she's a long-distance unicyclist; (b) she invented a public-key cryptosystem; (c) she designed and built an award-winning sex toy.Jim MillerJim Miller is the cryptography team lead at Trail of Bits. Before joining Trail of Bits, Jim attended graduate programs at both Cambridge and Yale, where he studied and researched both Number Theory and Cryptography, focusing on topics such as lattice-based cryptography and zero-knowledge proofs. During his time at Trail of Bits, Jim has led several security reviews across a wide variety of cryptographic applications and has helped lead the development of multiple projects, such as ZKDocs and PrivacyRaven.Josselin FeistJosselin Feist is a principal security engineer at Trail of Bits where he participates in assessments of blockchain software and designs automated bug-finding tools for smart contracts. He holds a Ph.D. in static analysis and symbolic execution and regularly speaks at both academic and industrial conferences. He is the author of various security tools, including Slither - a static analyzer framework for Ethereum smart contracts and Tealer - a static analyzer for Algorand contracts.Peter GoodmanPeter Goodman is a Staff Engineer in the Research and Engineering practice at Trail of Bits, where he leads all de/compilation efforts. He is the creator of various static and dynamic program analysis tools, ranging from the Remill library for lifting machine code into LLVM bitcode, to the GRR snapshot/record/replay-based fuzzer. When Peter isn't writing code, he's mentoring a fleet of interns to push the envelope. Peter holds a Master's in Computer Science from the University of Toronto.Host: Nick SelbyAn accomplished information and physical security professional, Nick leads the Software Assurance practice at Trail of Bits, giving customers at some of the world's most targeted companies a comprehensive understanding of their security landscape. He is the creator of the Trail of Bits podcast, and does everything from writing scripts to conducting interviews to audio engineering to Foley (e.g. biting into pickles). Prior to Trail of Bits, Nick was Director of Cyber Intelligence and Investigations at the NYPD; the CSO of a blockchain startup; and VP of Operations at an industry analysis firm.Production StaffStory Editor: Chris JulinAssociate Editor: Emily HaavikExecutive Producer: Nick SelbyExecutive Producer: Dan GuidoRecordingRocky Hill Studios, Ghent, New York. Nick Selby, EngineerPreuss-Projekt Tonstudio, Salzburg, Austria. Christian Höll, EngineerRemote recordings:Whistler, BC, Canada; (Nick Selby) Queens, NY; Brooklyn, NY; Rochester, NY (Emily Haavik);Toronto, ON, Canada. TAPES//TYPES, Russell W. Gragg, EngineerTrail of Bits supports and adheres to the Tape Syncers United Fair Rates CardEdited by Emily Haavik and Chris JulinMastered by Chris JulinMusicDISPATCHES FROM TECHNOLOGY'S FUTURE, THE TRAIL OF BITS THEME, Chris JulinOPEN WINGS, Liron MeyuhasNEW WORLD, Ian PostFUNKYMANIA, Omri Smadar, The Original OrchestraGOOD AS GONE, INSTRUMENTAL VERSION, Bunker Buster ALL IN YOUR STRIDE, AbeBREATHE EASY, Omri SmadarTREEHOUSE, LingerwellLIKE THAT, Tobias BergsonSCAPES,  Gray NorthReproductionWith the exception of any Copyrighted music herein, Trail of Bits Season 1 Episode 0; Immutable © 2022 by Trail of Bits is licensed under Attribution-NonCommercial-NoDerivatives 4.0 International. This license allows reuse: reusers may copy and distribute the material in any medium or format in unadapted form and for noncommercial purposes only (noncommercial means not primarily intended for or directed towards commercial advantage or monetary compensation), provided that reusers give credit to Trail of Bits as the creator. No derivatives or adaptations of this work are permitted. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.Meet the Team:CHRIS JULINChris Julin has spent years telling audio stories and helping other people tell theirs. These days he works as a story editor and producer for news outlets like APM Reports, West Virginia Public Broadcasting, and Marketplace. He has also taught and mentored hundreds of young journalists as a professor. For the Trail of Bits podcast, he serves as story and music editor, sound designer, and mixing and mastering engineer.EMILY HAAVIKFor the past 10 years Emily Haavik has worked as a broadcast journalist in radio, television, and digital media. She's spent time writing, reporting, covering courts, producing investigative podcasts, and serving as an editorial manager. She now works as an audio producer for several production shops including Us & Them from West Virginia Public Broadcasting and PRX, and APM Reports. For the Trail of Bits podcast, she helps with scripting, interviews, story concepts, and audio production.

FEATURED VOICES IN THIS EPISODEJim MillerJim Miller is the cryptography team lead at Trail of Bits. Before joining Trail of Bits, Jim attended graduate programs at both Cambridge and Yale, where he studied and researched both Number Theory and Cryptography, focusing on topics such as lattice-based cryptography and zero-knowledge proofs. During his time at Trail of Bits, Jim has led several security reviews across a wide variety of cryptographic applications and has helped lead the development of multiple projects, such as ZKDocs and PrivacyRaven.Matthew GreenMatthew Green is a cryptographer and an associate professor at the Johns Hopkins Information Security Institute. His research includes techniques for privacy-enhanced information storage, anonymous payment systems, and bilinear map-based cryptography. He is one of the creators of the Zerocash protocol, which is used by the Zcash cryptocurrency, and a founder of an encryption startup Zeutro. He was formerly a partner in Independent Security Evaluators, a custom security evaluation and design consultancy. From 1999-2003, he served as a senior technical staff member at AT&T Laboratories/Research in Florham Park, New Jersey.Host: Nick SelbyAn accomplished information and physical security professional, Nick leads the Software Assurance Practice at Trail of Bits, giving customers at some of the world's most targeted companies a comprehensive understanding of their security landscape. He is the creator of the Trail of Bits podcast, and does everything from writing scripts to conducting interviews to audio engineering to Foley (e.g. biting into pickles). Prior to Trail of Bits, Nick was Director of Cyber Intelligence and Investigations at the NYPD; the CSO of a blockchain startup; and VP of Operations at an industry analysis firm. Production StaffStory Editor: Chris JulinAssociate Editor: Emily HaavikExecutive Producer: Nick SelbyExecutive Producer: Dan GuidoRecordingRecorded at Rocky Hill Studios, Ghent, NY - Nick Selby, Engineer; and 22Springroad Tonstudio, Übersee, Germany - Volker Lesch, EngineerRemote recordings were conducted at Whistler, BC, Canada; and Tarrytown, NYEdited and Mastered by Chris JulinTrail of Bits supports and adheres to the Tape Syncers United Fair Rates Card)MusicDispatches From Technology's Future, the Trail of Bits theme, Chris JulinTrue Detectives: Ian PostBig Band Lemonade: Shirker Big BandDuda:  Ian PostBread and Butter: ZiggyScapes: Gray NorthVideoWatch this episode as a video on YouTube.ReproductionWith the exception of any Copyrighted music herein, Trail of Bits Season 1 Episode 1; Zero Knowledge Proofs and ZKDocs © 2022 by Trail of Bits is licensed under Attribution-NonCommercial-NoDerivatives 4.0 International.  This license allows reuse: reusers may copy and distribute the material in any medium or format in unadapted form and for noncommercial purposes only (noncommercial means not primarily intended for or directed towards commercial advantage or monetary compensation), provided that reusers give credit to Trail of Bits as the creator. No derivatives or adaptations of this work are permitted. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. Referenced in this Episode:The talk at Real World Crypto 2020 that Jim discusses was "This is not a proof: Pitfalls in real-world verifiable elections" by Sarah Jamie Lewis, Olivier Pereira, and Vanessa Teague. It was based on the academic paper, “How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios," by David Bernhard, Olivier Pereira, and Bogdan Warinschi. It's about some problems researchers have uncovered in an open-source e-voting system called Helios Voting. You can learn much more about ZKDocs and the latest Trail of Bits projects on our blog: trailofbits.com/blogJim Miller uses a tennis analogy to help describe some of the issues we discussed in this episode: Serving up zero-knowledge proofsTrail of Bits and Matthew Green teamed up to use Zero Knowledge proofs to form a trusted plane in which tech companies and vulnerability researchers can securely communicate, in a research project that's part of a larger DARPA-funded effort: Reinventing Vulnerability Disclosure using Zero Knowledge ProofsIn December 2020, a Trail of Bits intern wrote an extensive post called Reverie: An Optimized Zero Knowledge Proof System. Reverie is a ZK proof system using techniques from secure multiparty computation that optimizes for prover efficiency and doesn't require any trusted setup.To learn more about the Trail of Bits Internship and Winternship programs, visit the Trail of Bits Careers PageMeet the Team:Chris JulinChris Julin has spent years telling audio stories and helping other people tell theirs. These days he works as a story editor and producer for news outlets like APM Reports, West Virginia Public Broadcasting, and Marketplace. He has also taught and mentored hundreds of young journalists as a professor. For the Trail of Bits podcast, he serves as story and music editor, sound designer, and mixing and mastering engineer.Emily HaavikFor the past 10 years Emily Haavik has worked as a broadcast journalist in radio, television, and digital media. She's spent time writing, reporting, covering courts, producing investigative podcasts, and serving as an editorial manager. She now works as an audio producer for several production shops including Us & Them from West Virginia Public Broadcasting and PRX, and APM Reports. For the Trail of Bits podcast, she helps with scripting, interviews, story concepts, and audio production.

Episode 012 | May 30, 2022Neeraj Kayal: It's just a matter of time before we figure out how computers can themselves learn like humans do. Just human babies, they have an amazing ability to learn by observing things around them. And currently, despite all the progress, computers don't have that much ability. But I just think it's a matter of time before we figure that out, some sort of general artificial intelligence.Sridhar Vedantham: Welcome to the MSR India podcast. In this podcast, Ravishankar Krishnaswamy, a researcher at the MSR India lab, speaks to Neeraj Kayal. Neeraj is also a researcher at MSR India and works on problems related to or at the intersection of Computational Complexity and Algebra, Number Theory and Geometry. He has received multiple recognitions through his career, including the Distinguished Alumnus award from IIT Kanpur, the Gödel prize and the Fulkerson Prize. Neeraj received the Young Scientist Award from the Indian National Science Academy (INSA) in 2012 and the Infosys Prize in Mathematical Sciences in 2021. Ravi talks to Neeraj about how he became interested in this area of computer science and his journey till now.For more information about the Microsoft Research India click here.RelatedMicrosoft Research India Podcast: More podcasts from MSR IndiaiTunes: Subscribe and listen to new podcasts on iTunesAndroidRSS FeedSpotifyGoogle PodcastsEmailTranscriptRavi Krishnaswamy: Hi Neeraj, how are you doing? It's great to see you after two years of working from home.Neeraj Kayal: Hi Ravi, yeah thank you.Thank you for having me here and it's great to be back with all the colleagues in office.Ravi Krishnaswamy: First of all, congratulations on the Infosys prize and it's an amazing achievement.And it's a great privilege for all of us to have you as a colleague here.So, congratulations on that.Neeraj Kayal: Thank you.Ravi Krishnaswamy: Yeah, so maybe we can get started on the podcast. So, you work in complexity theory, which is I guess one extreme of, I mean, it's very theoretical end of the spectrum in computer science almost bordering mathematics. So hopefully by the end of this podcast we can, uh, I mean, convince the audience that there's more to it than intellectual curiosity. Before that right, let me ask you about how you got into theoretical computer science and the kind of problems that you work on. So, could you maybe tell us a bit about your background and how you got interested into this subject?Neeraj Kayal: Yeah, so in high school I was doing well in maths in general and I also wrote some computer programs to play some board games, like a generalized version of Tic Tac Toe where you have a bigger board, say 20 by 20, and you try to place five things in the row, column, or diagonal continuously and then I started thinking about how could a computer learn to play or improve itself in such a game? So, I tried some things and didn't get very far with that, but at that time I was pretty convinced that one day computers will be able to really learn like humans do. I didn't see how that will happen, but I was sure of it and I just wanted to be in computer science to eventually work on such things. But in college in the second year of my undergrad, I enrolled for a course in cryptography taught by Manindra Agrawal at IIT Kanpur and then the course started off with some initial things which are like fairly predictable that something called symmetric key cryptosystems where, essentially it says that let's say we two want to have a private conversation, but anyone else can listen to us. So how do we have a private conversation? Well, if we knew a language, a secret language which no one else did, then we could easily just converse in that language, and no one will understand this. And so, this is made a little more formal in this symmetric key cryptosystem. And then, one day, Manindra ended one of the lectures with the following problem: but now suppose we did not know a secret language. Then we just know English, and everyone knows English and then how do we talk privately when everyone can hear us? I thought about it for a few days. It seemed completely impossible. And then Manindra told us about these wonderful cryptosystems, called the Diffie Hellman cryptosystem and the RSA cryptosystem where they achieved this and it was very surprising. And the key thing that these cryptosystems use is something that lies at the heart of computer science, a big mystery still even to this day at the heart of computer science. There are these problems which we believe are hard for computers to solve in the following sense, that even if a computer takes a very long amount of time, if we give it a fairly long amount of time, a reasonable amount of time it cannot solve it. But if we give it time like till the end of the universe, it can in principle solve such problems. So that got me interested into which problems are hard and can we prove they are actually hard or not? And to this day, we don't know that.Ravi Krishnaswamy: So, I'm guessing that you're talking about the factoring problem, right?Neeraj Kayal: Yes, factoring is one of the big ones here. And the RSA cryptosystem uses factoring.Ravi Krishnaswamy: So, it's actually very interesting, right? You started out by trying to show that some of these problems are very, very hard, but I think, looking back, your first research paper, which happens to be a breakthrough work in itself, is in showing that a certain problem is actually easier to solve. Then we had originally thought right so, it is this seminal work on showing that primality testing can be solved in deterministic polynomial time. I mean, that's an amazing feat and you had worked on this paper with your collaborators as an undergrad, right?Neeraj Kayal: Yes.Ravi Krishnaswamy: Yeah, that's an incredible achievement. So maybe to motivate others who are in undergrad and who have an interest and inclination in such topics, could you maybe share us some story on how you got working in that problem and what sort of led you to this spark that eventually got you to this breakthrough result?Neeraj Kayal: So, my advisor Manindra, who also was the professor who taught us cryptography - he had been working on this problem for a long time and there were already algorithms that existed which are very good in practice- very very fast in practice, but they had this small chance that they might give the wrong answer. The chance was so small that practically it did not matter, but still as a mathematical challenge, it remained whether we could remove that small chance of error, and that's what the problem was about. So, Manindra had this approach and he had worked with other students also- some of our seniors- on it, and in that course, he came up with a conjecture. And then when we joined, me and my colleague Nitin, we joined this project , we came across this conjecture and my first reaction was that the conjecture is false. So, I tried to write a program which would find a counterexample and I thought we would be done in a few days-Just find that counterexample and the project would be over. So, I wrote a program- it will train for some time, didn't find a counterexample, so I decided to parallelize it. A huge number of machines in the computer center in IIT Kanpur started looking for that counterexample. And then to my surprise, we still couldn't find the counterexample. So there seemed to be something to it. Something seemed to be happening there which we didn't understand, and in trying to sort of prove that conjecture, we managed to prove some sort of weaker statement which sufficed for obtaining the polynomial time algorithm to test if a number is prime or not. But it was not the original conjecture itself. Many days after this result came out, we met a mathematician called Hendrik Lenstra who had worked on primality testing, and we told him about this conjecture. And after a few days he got back to us and it showed that if you assume some number theoretic conjecture is true, which we really really believe, it's true.Ravi Krishnaswamy: Ok, I see. So, the original conjecture, which you hoped to prove true is false, but the weaker conjecture was actually true, you proved it to be true, and that was enough for your eventual application.Neeraj Kayal: Yes, so in some sense we are very lucky that in trying to prove something false we managed to prove something useful.Ravi Krishnaswamy: Yeah, I mean it's a fascinating story, right? All the experiments that you ran pointed you towards proving it, and then you actually went and proved it. If you had found, I imagine what would have happened if you found a counterexample at that time, right?Neeraj Kayal: So yeah, Hendrix proof was very interesting. He showed that modulo this number theory conjecture a counterexample existed. But it would have to be very, very large and that's why you couldn't find it. So, he explained it beautifully.Ravi Krishnaswamy: Yeah, thanks for that story Neeraj. So. I guess from then on you've been working in complexity theory, right?Neeraj Kayal: That's right, yeah.Ravi Krishnaswamy: So, for me at least, the Holy Grail in complexity theory that I've often encountered or seen is the P versus NP problem, which many of us might know. But you've been working on a very equally important, but a very close cousin of the problem, which is called the VP versus VNP problem, right? So, I'm going to take a stab at explaining what I understand of the problem. So, correct me whenever I'm wrong. So, you are interested in trying to understand the complexity of expressing polynomials using small circuits. So, for example, if you have a polynomial of the form X ^2 + Y ^2 + 2 XY, you could represent it as a small circuit which has a few addition operations and a few multiplication operations like you could express it as X ^2 + Y ^2 + 2 XY itself. Or you could express it as (X + Y)^2. Which may have a smaller representation in terms of a circuit. So, you have been working on trying to identify which polynomials have small representations and which polynomials are natural but still don't have small representations.Neeraj Kayal: That's right.Ravi Krishnaswamy: Is that a reasonable approximation of the problem you're thinking about?Neeraj Kayal: Yes, that's right. So, another way to put the same thing is what is the power of computation when you do additions, multiplications, subtractions, all these arithmetic operations. You could include division, square roots also.Ravi Krishnaswamy: So, I have seen this VP class and it makes a lot of sense to me. It's the set of all the polynomials that can be captured by small sized circuits with the plus I mean addition and multiplication gates. I've also seen the VNP class, which seems to me at least to be a bit mysterious, right? So, these are all the polynomials whose coefficients of the individual monomials can be computed efficiently. Is that a reasonable definition, at least? Is my understanding correct?Neeraj Kayal: Yeah, that's the technical definition of this class, but there's another natural sort of intuition why we want to look at it, and the intuition is that it relates to counting the number of solutions to a problem, and also therefore to computing probabilities of various things happening.Ravi Krishnaswamy: I see. Ok, so that gives me a lot more understanding. I guess when you're able to estimate probabilities, you could also do sampling over those objects.Neeraj Kayal: Yes exactly.Ravi Krishnaswamy: Yeah, that's a very nice connection. I did not know about this. Thanks for that. So, you have been working, you have an agenda on trying to show some sort of a separation between the two classes, right, VP and VNP, by constructing these low depth circuits. So, you're able to show that all polynomials in VP have admit the low depth representation and your hope in this agenda is to find one polynomial in VNP which does not have a low depth representation, right?Neeraj Kayal: That's right.Ravi Krishnaswamy: So, how far are you in this agenda and do you think we have all the tools needed to actually achieve success through this sort of a method?Neeraj Kayal: Yeah, so just historically for converting a circuit or a program into a low depth program, this was done earlier. Most of this work was done by other people. So, we haven't contributed much in that direction. We have been trying to prove certain polynomials don't have small depth and small sized arithmetic circuits. So, it's not clear to us whether the existing techniques are good enough to prove this or not. And like on Mondays, Wednesdays, and Fridays, I think they are capable maybe, and on the other days I think maybe not. And this is what researchers generally deal with. Especially in these areas where you don't know whether your tools are good enough or not. And very recently, just last year, there was a big breakthrough by trio of complexity theorists who showed somewhat good lower bounds for all constant depth arithmetic formulas or circuits. And what was surprising also about this result is that, they use in a very clever way, techniques that were already known.Ravi Krishnaswamy: So, they would have probably shown it on a Monday or Wednesday or Friday.Neeraj Kayal: Yes, yes. [Laughs]Ravi Krishnaswamy: OK, that's very interesting. So, you still don't know whether this will lead to success or not through this route.Neeraj Kayal: Yes, yeah, we still don't know that.Ravi Krishnaswamy: Are there other people approaching this problem through other techniques?Neeraj Kayal: So, there's a program called the Geometric Complexity Theory program initiated independently by other people who basically try to understand symmetries. Because implicit in this question is a whole bunch of symmetry, then they try to exploit that. And there's a field of mathematics called group theory and representation theory, which is all about understanding symmetries of objects. That area is beautiful, really beautiful, and a lot of advancement has been made there. So, people have been trying to also attack this problem through using those tools.Ravi Krishnaswamy: Yeah, that's very nice, I think. So basically, you're saying a lot of like diverse techniques from math and computer science are at play here and trying to help you on your progress.Neeraj Kayal: That's right.Ravi Krishnaswamy: I see. I mean, it's very beautiful. I find it fascinating and beautiful that a lot of these different diverse techniques from mathematics and computer science come into play into establishing these lower bounds. And what's more fascinating to me is that they are all not just from an intellectual curiosity standpoint. They seem to be powering a lot of things that we take for granted, right, right from, like, as you said, messaging each other through social networks or whatever it is. They seem to be like at the foundation- the inherent hardness of certain problems seem to be at the foundation of a lot of things that we take for granted.Neeraj Kayal: Yeah, that's right, Ravi. So, for example, I do transactions using my mobile phone and anyone who is within a reasonable distance of my mobile phone can read all the signals that my phone is sending. So, they can see all the communication that I'm having with the bank. And the fact that despite that they are not able to infer my banking passwords relies on the fact that certain problems are very inherently hard to solve and that's what we are trying to prove.Ravi Krishnaswamy: OK, so that's very interesting Neeraj. And in the last part of this podcast, I want to flip the topic around a little bit. So, you've been working a lot on showing lower bounds, and in lower bounds in arithmetic complexity. But lately in the last couple of years you have also been using those insights into showing some very nice algorithms for some learning problems. I find that also very cool, so maybe you can talk a little bit about that.Neeraj Kayal: Yeah, so the algorithms that we are trying to devise are trying to solve the following problem. More general version of it is the following. Given a function or a polynomial, what's the smallest number of operations that you need to do to be able to compute that function or polynomial? So, for Boolean functions this has a very long history. That essentially is like designing chips, and you can imagine it was naturally very useful to think about. But more recently, it turns out a lot of works have found another very surprising connection because of which this problem specifically for polynomials has also become very interesting. And the connection is this. Suppose you have some very big data set. For now, think of this data set as consisting of a bunch of points in high dimensional space. For example, you can think of images as a point, every image as a point in the high dimensional space. Now it turns out that you can take statistics of this data. So, for example, you can take what's the average value of the first coordinate, what's the average value of the second coordinate? Or what's the average value of the product of the first two coordinates in this data set and so on. So, you can take some of these statistics, encode them as the coefficients of a polynomial. And here's the interesting part. When the data has some very nice structure, then this polynomial tends to have a small circuit.Ravi Krishnaswamy: I see.Neeraj Kayal: And so, when you want to understand the structure of data, so this general area is called unsupervised learning. Turns out that it's useful to find small circuits for polynomials. So, this is the computational problem that we are looking at: given a polynomial, what's the smallest number of operations, or what's the smallest circuit representing this polynomial.Ravi Krishnaswamy: So, if you're able to find the smallest circuit representing this, then from that you will be able to infer the underlying distribution or the structure of the underlying data.Neeraj Kayal: Yes, yes, that's right. So, this is one connection, and it also turns out that the lower bounds that we are proving, showing that certain things are very hard to compute are also useful for now devising algorithms to find the circuits of polynomials which do have small circuits and maybe let me give you some very rough sense of how that comes about, and I find this a bit fascinating. Here's how the lower bounds proofs work. So, underlying all those lower bounds for the various subclasses of circuits that we do have is a collection of linear maps and now it turns out that when you are given a polynomial which has a small circuit, using this polynomial and the collection of linear maps, which go into the lower bound proof you can form another big linear map, such that, very roughly, the eigen spaces of this new linear map correspond to the smallest circuit for F.Ravi Krishnaswamy: I see.Neeraj Kayal: And this was the connection that we discovered some time ago, which helped us find small circuits.Ravi Krishnaswamy: So, you find small circuits by computing the eigen space of the of the map.Neeraj Kayal: Yes, of this other linear map. That's right Ravi.Ravi Krishnaswamy: I see that's very nice. Ok, so I think we covered a lot of the topics that I wanted to cover, so maybe I'll end with two philosophical questions. So, one is you began the podcast by talking about how as a kid, you thought computers or machines could be able to do everything that human intelligence can do. So, I think it's a vague question, but what's your take on that now? And two is what advice would you give for budding theoreticians, whether they're in school or college or grad school? What sort of advice would you give them?Neeraj Kayal: So, for the first question, Ravi, I know a lot of other people also share this feeling, that it's just a matter of time before we figure out how computers can themselves learn like humans do. Just human babies, they have an amazing ability to learn by observing things around them. And currently, despite all the progress, computers don't have that much ability. But I just think it's a matter of time before we figure that out, some sort of general artificial intelligence. To your second question, Ravi, I don't have much to offer other than perhaps a banal comment that anyone looking to work in this area should really enjoy thinking about these kinds of problems. They tend to be rather abstract, sometimes the applications are not always apparent, but if you enjoy thinking about them, I'm sure you'll do well.Ravi Krishnaswamy: That's great, Neeraj. It's been a pleasure chatting with you. Thanks a lot for your time and hope you had fun.Neeraj Kayal: Yeah, thanks Ravi. Thanks a lot for having me.

England's earliest chair of mathematics was that of Gresham College, founded in 1597, but who came next? The earliest University-based mathematics professorship was Oxford's Savilian Chair of Geometry, founded in 1619. This illustrated lecture outlines the 400-year history of this Chair, from its beginnings to the present day, and features such figures as Henry Briggs, John Wallis, Edmond Halley, James Joseph Sylvester and G. H. Hardy.A lecture by Robin WilsonThe transcript and downloadable versions of the lecture are available from the Gresham College website:https://www.gresham.ac.uk/watch-now/savilian-professorsGresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 2,000 lectures free to access or download from the website.Website: http://www.gresham.ac.ukTwitter: http://twitter.com/GreshamCollegeFacebook: https://www.facebook.com/greshamcollegeInstagram: http://www.instagram.com/greshamcollege

Jan-Willem Prügel questions three Oxford mathematicians about the mythical entities known as numbers. What are they? And perhaps even more importantly, why are they? Show notes: https://media.podcasts.ox.ac.uk/ball/in_our_spare_time/sparetimes-number-systems-show-notes.pdf

Jan-Willem Prügel questions three Oxford mathematicians about the mythical entities known as numbers. What are they? And perhaps even more importantly, why are they? Show notes: https://media.podcasts.ox.ac.uk/ball/in_our_spare_time/sparetimes-number-systems-show-notes.pdf

Did you know that a 6 digit passcode takes less than 1 day to crack, while a 10 digit passcode can take several years? Just do not use a phone number, as this can be guessed easily. Nick Egbert joins us from the  Purdue University  Department of Mathematics. Nick is a  PhD candidate who is specializing in cryptography and number theory. He explains how simple and extremely complex codes work. Nick teaches us that  everything from depositing a check using your phone, logins to any website,  to reusing passwords, all use cryptography.

Mathematician and Flutist, Frank Rothe talks about his two life- long loves:  Mathematics and his two works on Number Theory and Modern Algebra and Classics in Graph Theory – both available on Amazon…  and his second but equal love of the flute. He has three CD’s available on Amazon or on CD Universe for flute and […] The post Number Theory and Modern Algebra by Franz Rothe appeared first on WebTalkRadio.net.

Mathematician and Flutist, Frank Rothe talks about his two life- long loves:  Mathematics and his two works on Number Theory and Modern Algebra and Classics in Graph Theory – both available on Amazon…  and his second but equal love of the flute. He has three CD’s available on Amazon or on CD Universe for flute and […] The post Number Theory and Modern Algebra by Franz Rothe appeared first on WebTalkRadio.net.

Robert Edward Grant takes us on a journey into Number Theory and how it connects every atom in the universe to the divine source field. His book, "Philomath" is an Amazon Best Seller. Learn more about Robert Edward Grant on http://robertedwardgrant.comFor more info visithttp://4biddenknowledge.com4biddenknowledge TVhttps://www.4biddenknowledge.tv/browseRolls Royce Ghost Giveaway by Billy Carson. Proceeds go to help underprivileged children    http://4biddenknowledge.com/giveawaysWoke Doesn't Mean Broke by Billy Carson.Buy The Book https://www.4biddenknowledge.com/online-store/Woke-Doesnt-Mean-Broke-by-Billy-Carson-p253782102Available Workshops on Eventbrite organized by Billy Carson: https://www.eventbrite.com/o/billy-carson-321201994474biddenknowledge on Facebook:https://www.facebook.com/4biddenKnowledge30 Day Free Trial Of 4biddenknowledge.TV 30 Day Free Trial On 4biddenknowledge.TVSupport the show (https://www.buymeacoffee.com/4biddenpodcast)

Robert Edward Grant takes us on a journey into Number Theory and how it connects every atom in the universe to the divine source field. His book, "Philomath" is an Amazon Best Seller. Learn more about Robert Edward Grant on http://robertedwardgrant.comFor more info visithttp://4biddenknowledge.com4biddenknowledge TVhttps://www.4biddenknowledge.tv/browseRolls Royce Ghost Giveaway by Billy Carson. Proceeds go to help underprivileged children    http://4biddenknowledge.com/giveawaysWoke Doesn't Mean Broke by Billy Carson.Buy The Book https://www.4biddenknowledge.com/online-store/Woke-Doesnt-Mean-Broke-by-Billy-Carson-p253782102Available Workshops on Eventbrite organized by Billy Carson: https://www.eventbrite.com/o/billy-carson-321201994474biddenknowledge on Facebook:https://www.facebook.com/4biddenKnowledge30 Day Free Trial Of 4biddenknowledge.TV 30 Day Free Trial On 4biddenknowledge.TVSupport the show (https://www.buymeacoffee.com/4biddenpodcast)

The history of mathematics extends back millennia. The needs of trade, taxation, and time-keeping drove the development of principles of arithmetic, algebra, and geometry, which had already acquired some sophistication by 5,000 years ago. Perhaps most fundamental to the development of mathematics has been discoveries on the nature of numbers themselves, or what mathematicians refer to as Number Theory. Today's topic is the history and development of Number Theory, viewed through the lens of numbers and number systems. Our guide to Number Theory is Bryden Cais, professor of mathematics at the University of Arizona and the Director of the Southwest Center for Arithmetic Geometry.  Bryden completed a BA in mathematics at Harvard University in 2002 and a PhD also in mathematics at the University of Michigan in 2007. He was a postdoctoral fellow at McGill University, a visiting scholar at Universität Bielefeld, and a professor at the University of Wisconsin, Madison before joining the faculty at the University of Arizona in 2011. We explore the nature and history of different number systems, highlight the obstacles that mathematicians and civilizations faced with new concepts of number, and touch on some unsolved problems in modern number theory. A study guide for this episode is available in PDF form HERE, or as LaTeX HERE.

When this podcast started, interviewing a convicted murderer - while he was still in prison, no less - was admittedly not on Vanessa's “Math Therapy guest” bingo card. But then we heard how Christopher Havens discovered math in solitary confinement halfway through a 25-year sentence, and committed the rest of his life to rehabilitating himself and repaying his debt to society.  On today's season 3 finale of Math Therapy, Vanessa chats with Christopher about how a lack of purpose in his youth and the misguided desire to be “cool” led him down a very dark road, how math gave him a sense of spirituality and purpose in prison, and how his own rehabilitation led him to withdraw from the stereotypical “prison game” and devote the rest of his life to developing programs and resources for other prisoners to do the same.About Christopher Christopher Havens is a mathematician & president/cofounder of the Prison Mathematics Project, a nonprofit program that uses the transformative powers of mathematics to lead prisoners to a life of desistance from crime. While his current focus is in cryptography and the theory of groups, he works to popularize mathematics and redefine what productivity looks like in prison. He is a researcher in number theory and has recently published his first academic paper in the journal or Research in Number Theory, while serving out his sentence. Show notesPopular Mechanics profile on Christopher's life storyPrisonMathProject.orgHumanMe.org blog feat. inmates committed to rehabilitationConnect with us:Prison Mathematics Project: (Twitter, Insta, Facebook)Vanessa Vakharia: @themathguru (Insta, Twitter, TikTok)Math Therapy: @maththerapy (Twitter)Transcript for today's episode: www.maththerapypodcast.com

In this episode, Dr. Alex Kontorovich, a professor at Rutgers University and Editor-in-Chief of Experimental Mathematics, discusses his favorite mathematical theories such as Number Theory and tells us about his journey to the interesting research and work he does now. --- Send in a voice message: https://podcasters.spotify.com/pod/show/stemz-perspectives/message

YouTube link: https://youtu.be/xu15ZbxxnUQ Richard Borcherds is a mathematician known for his work in lattices, group theory, Monstrous Moonshine, and infinite-dimensional algebras, for which he was awarded the Fields Medal in 1998.  Richard Borcherd's YouTube: https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ Patreon for conversations on Theories of Everything, Consciousness, Free Will, and God: https://patreon.com/curtjaimungal Help support conversations like this via PayPal: https://bit.ly/2EOR0M4 Twitter: https://twitter.com/TOEwithCurt iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 Pandora: https://pdora.co/33b9lfP Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Google Podcasts: https://play.google.com/music/listen?u=0#/ps/Id3k7k7mfzahfx2fjqmw3vufb44 iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 Discord Invite Code (as of Mar 04 2021): dmGgQ2dRzS Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything EDITED BY: Antonio Pastore 00:00:00 Introduction 00:02:35 How Richard began to become interested in math 00:03:42 Unification in mathematics vs. unification in physics 00:04:38 Daily ritual (or non-ritual) 00:05:19 How much time spent working / studying? 00:07:22 Creativity of the old vs young 00:08:30 Greatest strength is obstinance 00:08:58 Working in isolation, with no collaborators (strength or a weakness?) 00:10:48 Starting mathematics in your 20's, 30's, or 40's 00:11:45 Why must you pick a problem you're interested in? What happens when you don't? 00:12:41 What do you during moments of non-creativity / writer's block? 00:14:40 Dealing with depression as a scientist 00:15:24 On Richard's IQ and nootropics 00:17:02 Richard's creative process 00:18:33 Does he think more pictorially, algebraically, analytically, verbally, etc.? 00:21:11 Not following "deep work" 00:22:00 Reading non-scientific books 00:22:48 Audience Q: What does Richard think of Jordan Peterson? 00:23:31 Audience Q: Have you experience madness, working in math in isolation? 00:23:56 Audience Q: Does he optimize his diet / fast? 00:24:37 How does he learn new mathematics 00:25:42 Solving problems by ignoring them 00:26:51 Audience Q: Advice for someone in their 20's trying to learn math who's not in the field 00:28:03 Why does Richard not like infinity categories? 00:28:44 Does Richard memorize proofs / theorems? 00:29:53 Happiness and meaning in life (math or relationships / marriage / kids?) 00:30:40 What would Richard do without math? 00:31:32 What was it like to win the Fields medal? 00:32:19 What is about math that's meaningful? 00:33:10 Math discovered vs invented 00:34:35 Why is the Monster Group interesting? 00:37:18 "Quantum Field Theory gives me a headache." 00:39:21 Free will? 00:41:17 God, Simulation Hypothesis, and Many Worlds 00:44:53 On the Hard Problem of Consciousness 00:46:28 Favorite mathematicians (Serre, Witten, Tao, Feynman, Weinberg, etc.) 00:48:22 "Ed Witten is terrifying" 00:49:05 The Monster Group and physics 00:52:55 How to contribute to math if you're an outsider (or a neophyte)? 00:55:44 Many Worlds (again) 00:56:15 Audience Q: Is set theory too unwieldy and can we base math off of something different? 01:00:03 Audience Q: Pluralism in the foundations of math or not? 01:02:48 Intuitionist / Finitism / Computational logic? 01:04:29 Audience Q: Can people in their 40's understand advanced math? 01:05:20 Audience Q: Unreasonable effectiveness of mathematics 01:06:19 Audience Q: Does it puzzle him that some people don't understand math? 01:08:09 On Ramanujan 01:10:45 Lectures on Number Theory and the difficulty of QFT 01:14:56 On different learning styles, and philosophy of mathematics 01:17:48 Audience Q: How does one know when they're making progress on a solution? 01:19:11 Langland's program 01:21:45 Audience Q: How does one know what to learn when they don't know what they don't know? 01:24:02 Learning math and physics from YouTube 01:29:46 Audience Q: Goldbach's conjecture 01:31:53 On nervousness, performance anxiety, group theory, and chit-chat 01:38:49 "Secret" math techniques 01:39:56 Why "modular forms" are the most mesmeric of all fields of math 01:41:50 Discovered vs. invented (rebuttal from a famous mathematician) 01:47:17 Biology / Psychology / Philosophy is too confounding 01:49:08 On Grothendieck 01:52:09 How do you choose which topic to pursue in math? (and the ABC conjecture) 01:56:25 No Ghost Theorem, and string theory's connection to the Monster * * * Subscribe if you want more conversations on Theories of Everything, Consciousness, Free Will, God, and the mathematics / physics of each. * * * I'm producing an imminent documentary Better Left Unsaid http://betterleftunsaidfilm.com on the topic of "when does the left go too far?" Visit that site if you'd like to contribute to getting the film distributed (early-2021).

My next guest on the special series is Apoorva Panidapu. Apoorva is a 15-year-old high-school sophomore in San Jose, California. Apoorva wears many hats; she's a student, a teacher, an aspiring mathematician, an artist, a social entrepreneur, and a public speaker who loves helping kids around the world. Apoorva started taking college classes at age 11, and has since completed several upper-division and graduate-level mathematics courses with a keen interest in Number Theory. She attended the prestigious Canada/USA Mathcamp (2018, 2019) and the highly selective University of Virginia REU (Research Experience for Undergraduates) in 2020 as the youngest student there. She is grateful to have the opportunity to work with world-renowned mathematicians in her fields of interest and has co-authored and published papers in Number Theory–one of which was published in the prestigious Journal of Number Theory. Apoorva has received several worldwide recognitions for her achievements in mathematics, such as her performance in NBC National TV show Genius Junior in 2018 hosted by Neil Patrick Harris and global awards such as the prestigious Spirit of Ramanujan Fellowship & has been selected as a World Science Scholar, one among few in the world. She is also a recipient of the 2020 Global Child Prodigy Award. Furthermore, she co-founded the Gems in STEM initiative with the goal of teaching various topics in math, science, engineering, and technology through the stories of diverse and pivotal figures in STEM history. As a part of this, she has been publishing bi-weekly articles in the Evergreen Times newspaper, Medium, and Cantor's Paradise, the #1 math publication on Medium. She has also won several national and state awards for her speaking and writing. Apoorva is an enthusiastic artist who loves to oil paint and sketch portraits. She is the founder of Apoorva Panidapu's Art Gallery (www.apoorvaartgallery.com), an online platform to share her artwork and raise funds for charity. She particularly enjoys impressionistic and abstract artworks. Her story and paintings were featured on Artists for Peace, Stone Soup, and Ellen and Cheerio's “One Million Acts of Good.” She is also the grand prize winner of NASA Langley Research Center's Centennial Student Art Contest, as well as a recipient of four Presidential Volunteer Service Awards. Apoorva is also a public speaker who encourages girls, gender minorities, and all youth to pursue STEAM fearlessly. In addition to her art gallery and Gems in STEM, she is a global ambassador for GLAM (Girls Leadership Academy Meetup), where she helps encourage girls aged 8-12 from diverse backgrounds to pursue leadership and careers in tech. Her mission is to encourage others to use their gifts to make a difference in the world. She has helped raise more than $25,000 to support children around the world by using her gifts in math and art to try and give back. In her spare time, Apoorva enjoys playing the violin, practicing kung fu, and reading classical literature. Apoorva aspires to combine pure mathematics, art, and humanities to change the world.

In this episode talk with Ashvni Narayanan, a Graduate Student at the London School of Geometry and Number Theory working under the auspice of Prof. Kevin Buzzard. Ashvni studied Galois representations of elliptic curves during her Master's at the University of British Columbia, Vancouver advised by Prof. Sujatha Ramadorai, and completed a Bachelor's in Mathematics at the Indian Statistical Institute, Bangalore. Ashvni currently works on formalizing algebraic number theory in a program called Lean. We converse about her incredible random walks through life and science; her terrific mentors who've inspired her; automating mathematical theorem proving: nature of research and collaborations in a (post) pandemic world; taking care of one's mental, physical, and emotional health in academia; the fantastic Hmm podcast she started during the lockdown; and many more things! Resources to learn about the Xena Project and Lean: https://xenaproject.wordpress.com/useful-links/ https://leanprover-community.github.io/learn.html

In which years does February have five Sundays? How many right-angled triangles with whole-number sides have a side of length 29? How many shuffles are needed to restore the order of the cards in a pack with two Jokers? Are any of the numbers 11, 111, 1111, 11111, . . . perfect squares? Can one construct a regular polygon with 100 sides if measuring is forbidden? How do prime numbers help to keep our credit cards secure?These are all questions in number theory, the branch of mathematics that's primarily concerned with our counting numbers, 1, 2, 3, etc. Of particular importance are the prime numbers, the 'building blocks' of our number system.The subject is an old one, dating back to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no-one has ever been able to resolve them.This lecture situates the above problems and puzzles in their historical context, drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler and Gauss. Indeed, as Gauss, sometimes described as the 'Prince of Mathematics', has claimed: Mathematics is the Queen of the Sciences, and Number Theory is the Queen of Mathematics.A lecture by Robin Wilson 28 SeptemberThe transcript and downloadable versions of the lecture are available from the Gresham College website: https://www.gresham.ac.uk/lectures-and-events/number-theoryGresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 2,000 lectures free to access or download from the website.Website: http://www.gresham.ac.uk Twitter: http://twitter.com/GreshamCollege Facebook: https://www.facebook.com/greshamcollege Instagram: http://www.instagram.com/greshamcollege

Episode: 3067 The Remarkable Perception of a Mathematical Oddity.  Today, the creativity in all of us.

In this, the second online lecture we are making widely available, Ben Green introduces and delivers a short lecture on Primitive Roots, part of the Number Theory Lecture course for Second Year Undergraduates. We are making these lectures available (there are many more on this YouTube Channel via the Playlist) to give an insight in to the student experience and how we teach Maths in Oxford. All lectures are followed by tutorials where pairs of students spend an hour with their tutor to go through the lectures and accompanying work sheets. An overview of the course and the relevant materials is available here: https://courses.maths.ox.ac.uk/node/44147

In this, the second online lecture we are making widely available, Ben Green introduces and delivers a short lecture on Primitive Roots, part of the Number Theory Lecture course for Second Year Undergraduates. We are making these lectures available (there are many more on this YouTube Channel via the Playlist) to give an insight in to the student experience and how we teach Maths in Oxford. All lectures are followed by tutorials where pairs of students spend an hour with their tutor to go through the lectures and accompanying work sheets. An overview of the course and the relevant materials is available here: https://courses.maths.ox.ac.uk/node/44147

From the softest of interactions of a magnetic field with an electron, to the most violent collisions at the Large Hadron Collider, precision quantum field theory produces numbers and functions with interesting number-theoretic properties. In many examples a co-action principle holds, an invariance under a ”cosmic” Galois group. I will provide several arenas in which this principle can be seen at work, including perhaps the richest set of theoretical data, scattering amplitudes in planar N = 4 super-Yang-Mills theory.

What's it like to be told you need to teach field geology online? This week we talk about distance learning in higher-ed. Fun Paper Friday What's special about 33? Booker, Andrew R. "Cracking the problem with 33." Research in Number Theory 5.3 (2019): 26. Contact us: Show Support us on Patreon! www.dontpanicgeocast.com SWUNG Slack @dontpanicgeo show@dontpanicgeocast.com John Leeman www.johnrleeman.com @geo_leeman Shannon Dulin @ShannonDulin

MAPSOC.ORG // 00:14:00 - Alchemy and John Dee // 01:13:00 - (A Long Transition into) Number Theory of Reality

Episode Notes When was the last time you turned to your number theory text book for career advice? Catherine Dodge, a senior technical program manager at Amazon AWS, made her first big career move when she learned that the NSA hired the most mathematicians from her number theory textbook. She decided to start her career there and talks about her journey from graduating with the wrong degree to working at Amazon. Visit our websiteFind out more at https://modern-working-women.pinecast.co

A musician gives up the rock n' roll dream for number theory, and a glimpse of the infinite.

A musician gives up the rock n' roll dream for number theory, and a glimpse of the infinite.

The boys are back for season two! They try ramping Pascal's Triangle up a couple of dimensions, consider the physics of how we control volume and attempt to help out someone with a fishy problem. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Pascal's Pyramid Sound Quality Fish Tank Siphoning Show Notes Cuboctohedron Stacking (https://robertlovespi.net/2014/03/25/a-space-filling-pair-of-polyhedra-the-cuboctahedron-and-the-octahedron/) Sphere Packing Systems (https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres?fbclid=IwAR3pUsuDc8I0u-EK8WOeh0C5KgRdgH7XWd1_QoDRx3qTssLVqd9AQI7UPDA) Colouring in Pascal's Triangle in mod 3, 4 and 5 (https://www.maa.org/press/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-first-twist-what-is-it) Sierpinski's Pyramid (https://www.researchgate.net/figure/Ray-tracing-Figure-13-The-Sierpinski-Pyramid_fig2_260126327) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Quasi-analytic classes with respect to logarithmically convex sequences by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

Ramin Takloo-Bighash is a Professor and the Director of Undergraduate Studies in the department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. He is the organizer of the Atkin Memorial Lectures & Workshop and an editor of the Journal of the Iranian Mathematical Society. He is a musician, a "wanna-be" painter, and helped translate the collection of poems by Shahram Sheydayi, entitled "Laughing in the Burning House."His research is in Number Theory, Automorphic Forms, Arithmetic Geometry, and Harmonic Analysis on Lie Groups.Ramin has coauthored the books "An Introduction to Mathematical Olympiads", "An Invitation to Modern Number Theory" and, most recently, he is the author of the new book:"A Pythagorean Introduction to Number Theory: Triangles, Sums of Squares & Arithmetic"available here: https://www.amazon.com/Pythagorean-Introduction-Number-Theory-Undergraduate/dp/3030026035His website can be found here:http://homepages.math.uic.edu/~rtakloo/We'd like to thank Ramin for being on our show "Meet a Mathematician" and for sharing his stories and perspective with us!www.sensemakesmath.comPODCAST: http://sensemakesmath.buzzsprout.com/TWITTER: @SenseMakesMathPATREON: https://www.patreon.com/sensemakesmathFACEBOOK: https://www.facebook.com/SenseMakesMathSTORE: https://sensemakesmath.storenvy.comSupport the show (https://www.patreon.com/sensemakesmath)

The boys help organise a rugby training scheme, set robots upon an infinite hotel and discover some lovely patterns in the probability of Snakes and Ladders. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Rugby Training Problem Busy Beavers Gamble Shambles Show Notes Articles written by Alaric on Langton's: * Loops (Introduction) (http://www.alaricstephen.com/main-featured/2017/6/27/langtons-loops) * Loops (Evolving) (http://www.alaricstephen.com/main-featured/2017/6/27/improvements-to-langtons-loops) * Ant (http://www.alaricstephen.com/main-featured/2016/12/13/langtons-ant) Busy Beaver on Wikipedia (https://en.wikipedia.org/wiki/Busy_beaver) Noether's Theorem (https://en.wikipedia.org/wiki/Noether%27s_theorem) The Music Episode (http://www.oddsandevenings.com/21) Maths Jam (that you should totally be attending) (https://mathsjam.com/find-a-jam/) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Appreciation of the phrase Busy Beavers by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys celebrate a year of the podcast, try to get a hold on some of the weirder polygonal numbers and debrief after big maths jam. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Trapezium Numbers Shamir's Shared Secrets Vampire and Werewolf Lies Show Notes Catriona Shearer's Geometry Puzzles on Twitter (https://twitter.com/Cshearer41) Mathigon's Tweet about Trapezium Numbers (https://twitter.com/MathigonOrg/status/1070302146632081409?s=09) An Article on Properties of Trapezium Numbers (https://www.hindawi.com/journals/ijmms/2017/4515249/#B4) Alaric's Lock Gates Article (http://www.alaricstephen.com/main-featured/2017/5/1/lock-gates?rq=key) A Good Quality Shamir's Shared Secrets Article (https://ericrafaloff.com/shamirs-secret-sharing-scheme/) The Vampire and Werewolf MAT Question (https://www.cs.ox.ac.uk/admissions/undergraduate/how_to_apply/MAT2015.pdf) The Answers to the MAT Question (https://www.maths.ox.ac.uk/system/files/attachments/websolutions15_1.pdf) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Silver Service at the Vampire-Werewolf Symposium by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys get to the bottom of a neat pattern that suddenly breaks, contort their limbs into strange shapes in the name of Plato and attempt a very satisfying problem involving some spiders and an ant. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Factorial Terminating Digits Making Polyhedra with your Limbs Spiders and an Ant Pursuit Puzzle Show Notes Nice PDF of the Solution to the Partitions Problem by André Malraux (https://ufile.io/twukd) Alaric's Puzzle from the Conference (https://twitter.com/OddsAndEvenings/status/1064153315280330752) Platonic Solids to Stare at during Sections 2 and 3 of the Show (https://upload.wikimedia.org/wikipedia/commons/d/d3/Platonic_solids.jpg) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Fervent attempts to keep on top of responding to emails for the show by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys get as spooky as they can with cellular automata, have a maths conversation about maths conversations and can't get past the fact that Brunnian and Borromean sound too similar. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Zombies and Vampires The Maths of Conversations Brunnian Pumpkin Carving Show Notes Follow Up Maths Jam Annual Gathering Page (https://www.solipsys.co.uk/cgi-bin/MJ_Wiki.py?FrontPage) The Shade (Alex's New Podcast) (https://hacking.finance/read/?c=the-shade) Maths From the Show SIR Model Introduction by Alaric (http://www.alaricstephen.com/main-featured/2016/9/24/modelling-a-pandemic) SIR Model You Can Play With (https://docs.google.com/spreadsheets/d/1HxS1GttDPT8eDOhTIfOYckueArSJjGb5VfOr1hgqLno/edit?usp=sharing) Borromean Rings (https://en.wikipedia.org/wiki/Borromean_rings) Brunnian Links (https://en.wikipedia.org/wiki/Brunnian_link)(more generalised) Pochhammer's Countour (https://en.wikipedia.org/wiki/Pochhammer_contour) A Classic Video on Brunnian Links (https://www.youtube.com/watch?v=heKK95DAKms&ab_channel=Vihart) (in the form of snakes), by Vi Hart Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Following of the brief: bring halloween problems by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys settle a dicussion about a bike that Alaric has been aching to talk about for a fortnight, Alex finally brings up music and then they both dive down the Gödel rabbit hole with some puzzles. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Bike Pedals Time Signatures Taking an Offer Puzzles Show Notes Follow Up Visualisation of the Codepad Solution from Tom Verduyn (https://bl.ocks.org/TVerduyn/raw/4935615ba55a2c0b73b2550e0eb8764d/) Maths Jam Annual Gathering Page (https://www.solipsys.co.uk/cgi-bin/MJ_Wiki.py?FrontPage) The Shade (Alex's New Podcast) (https://hacking.finance/read/?c=the-shade) Maths From the Show Visuals of the Two Cycloids Mentioned: Prolate (http://mathworld.wolfram.com/ProlateCycloid.html) and Curtate (http://mathworld.wolfram.com/CurtateCycloid.html) Video of the Bike Moving (https://twitter.com/SingMathsJam/status/1042057017148334080) Unusual Time Signatures (https://en.wikipedia.org/wiki/List_of_musical_works_in_unusual_time_signatures) Forever Undecided: A Puzzle Guide to Gödel (https://www.amazon.co.uk/Forever-Undecided-Puzzle-Guide-Godel/dp/0192801414) The book I talk about in the third part of the show. It's great. The Gödel Sentence on Wikipedia (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Truth_of_the_Gödel_sentence) (Related to the puzzle) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Incomplete knowledge of incompleteness theorems by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys move seemlessly between high philosophy and pirate jokes in this eclectic episode. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Keypad Code Guessing Platonism vs Formalism Pirate Plunder Puzzle Show Notes Pirate Plunder Puzzle (http://www.alaricstephen.com/main-featured/2017/4/10/the-pirate-puzzle) Prisoners in Rainbow Hats (http://www.alaricstephen.com/main-featured/2017/6/13/prisoners-in-rainbow-hats) Z-Order Curves (https://en.wikipedia.org/wiki/Z-order_curve) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Dreams of being a philosopher pirate by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys try their hands at problems provided by the listeners in today's special episode. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed Prisoners in Hats Kaprekar's Constant Generalised Egg Dropping Show Notes Prisoners in Rainbow Hats (http://www.alaricstephen.com/main-featured/2017/6/13/prisoners-in-rainbow-hats) Islander Paradox (http://www.alaricstephen.com/main-featured/2017/4/10/the-islander-paradox) Kaprekar's Constant (https://en.wikipedia.org/wiki/6174_(number)) General Egg Problem Solution (https://stackoverflow.com/questions/11603218/three-egg-problem#) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Too many Hat Problem anecdotes by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

The boys both have problems which involve complicating the process of adding. Also snakes. Odds and Evenings Twitter - https://twitter.com/OddsAndEvenings Website - http://www.oddsandevenings.com Topics discussed FRACTRAN Negative Bases Polycube Snakes Show Notes FRACTRAN (https://en.wikipedia.org/wiki/FRACTRAN) Adding in Negative Bases (https://en.wikipedia.org/wiki/Negative_base#Addition) Quarter-Imaginary Base (https://en.wikipedia.org/wiki/Quater-imaginary_base) Fascinating Turing Complete Magic the Gathering (https://www.toothycat.net/~hologram/Turing/HowItWorks.html) Article on Polycube Snakes (http://www.alaricstephen.com/main-featured/2017/5/20/polycube-snakes) Credits Hosted By Alaric - http://alaricstephen.com Alex - http://twitter.com/speakmouthwords Editing by Alex Knuth Pronunciation by Alaric Theme music by David Russell - https://youtube.com/DavidRussell323

A musician gives up the rock n' roll dream for number theory, and a glimpse of the infinite.

Brian Stout, Assistant Professor at the Minerva Project, based in North Stonington, CT, talks about deliberate practice and 10,000 hours of deep work in value investing. He also touches on fascinating aspects of learning math and how mathematics applies to life. Brian is joining Minerva Schools from the U.S. Naval Academy. He earned his Ph.D. and M.A. in Mathematics from the City University of New York (CUNY) Graduate Center in 2013 and 2011, respectively. After finishing his Ph.D., he joined the U.S. Naval Academy as an Assistant Professor from 2013 to 2016. Professor Stout researches in arithmetic dynamics, a confluence of number theory and discrete dynamical systems, where he has had several publications in journals such as Journal of Number Theory and Acta Arithmetica. Professor Stout has taught a broad array of Mathematics and Statistics courses at CUNY and the U.S. Naval Academy. He has also supervised several undergraduate research projects in number theory and dynamics. Professor Stout views mathematics as a quantitative framework that assists in problem solving in wide contexts.

Roger Heath-Brown is one of Oxford's foremost mathematicians. In this interview with fellow Oxford Mathematician Ben Green, Roger reflects on his influences, his achievements and the pleasures that the subject of mathematics has given him. Roger Heath-Brown's work in analytic number theory has been critical to the advances in the subject over the past thirty years and garnered Roger many prizes. On the eve of his retirement Roger spoke to Ben Green, Waynflete Professor of Mathematics in Oxford and himself a leading figure in the field of number theory.

Roger Heath-Brown is one of Oxford's foremost mathematicians. In this interview with fellow Oxford Mathematician Ben Green, Roger reflects on his influences, his achievements and the pleasures that the subject of mathematics has given him. Roger Heath-Brown's work in analytic number theory has been critical to the advances in the subject over the past thirty years and garnered Roger many prizes. On the eve of his retirement Roger spoke to Ben Green, Waynflete Professor of Mathematics in Oxford and himself a leading figure in the field of number theory.

We’re going to play a simple coin-flip game. We take turns flipping a fair coin. The first one to get “heads” wins. You go first. // What’s your chance of winning? // Spiciness: *** out of ****

(Errata alert! This episode was re-uploaded with a correction on Monday March 28 at 10am PST. The corrected version is 6m04s long; the old one is 6m02s. Visit http://www.buzzsprout.com/56982 if your feed contains the old version.) // Here's a game you can play with two fair dice, one red and one green. You throw both dice, and use the throw to generate an infinitely long sequence of numbers, like this: first, you write down the red number. Then you add in the green number, write down the result, add the green again, write down that result, and so on. // You decide to play the game once today. What’s the chance that the sequence made by your dice throw will contain a perfect square? // Spiciness: *** out of ****

Jonathan Zachhuber war zum 12. Weihnachtsworkshop zur Geometrie und Zahlentheorie zurück an seine Alma Mater nach Karlsruhe gekommen und sprach mit Gudrun Thäter über Teichmüllerkurven. Kurven sind zunächst sehr elementare ein-dimensionale mathematische Gebilde, die über den komplexen Zahlen gleich viel reichhaltiger erscheinen, da sie im Sinne der Funktionentheorie als Riemannsche Fläche verstanden werden können und manchmal faszinierende topologische Eigenschaften besitzen. Ein wichtiges Konzept ist dabei das Verkleben von Flächen. Aus einem Rechteck kann man durch Verkleben der gegenüberliegenden Seiten zu einem Torus gelangen (Animation von Kieff zum Verkleben, veröffentlicht als Public Domain): Polynome in mehreren Variablen bieten eine interessante Art Kurven als Nullstellenmengen zu beschreiben: Die Nullstellen-Menge des Polynoms ergibt über den reellen Zahlen den Einheitskreis. Durch Ändern von Koeffizienten kann man die Kurve verformen, und so ist die Nullstellenmenge von eine Ellipse. Über den komplexen Zahlen können diese einfachen Kurven dann aber auch als Mannigfaltigkeiten interpretiert werden, die über Karten und Atlanten beschrieben werden können. Das ist so wie bei einer Straßenkarte, mit der wir uns lokal gut orientieren können. Im Umland oder anderen Städten braucht man weitere Karten, und alle Karten zusammen ergeben bei vollständiger Abdeckung den Straßenatlas. Auch wenn die entstehenden abstrakten Beschreibungen nicht immer anschaulich sind, so erleichtern die komplexen Zahlen den Umgang mit Polynomen in einem ganz wichtigen Punkt: Der Fundamentalsatz der Algebra besagt, dass der Grad des Polynoms gleich der Anzahl der Nullstellen in ihrer Vielfachheit ist. Also hat nun jedes nichtkonstante Polynom mindestens eine Nullstelle, und über den Grad des Polynoms wissen wir, wie viele Punkte sich in der Nullstellenmenge bewegen können, wenn wir an den Koeffizienten Veränderungen vornehmen. Eine gute Methode die entstehenden Flächen zu charakterisieren ist die Bestimmung möglicher geschlossener Kurven, und so gibt es beim Torus beispielsweise zwei unterschiedliche geschlossene Kurven. Die so enstehende Fundamentalgruppe bleibt unter einfachen Deformationen der Flächen erhalten, und ist daher eine Invariante, die hilft die Fläche topologisch zu beschreiben. Eine weitere wichtige topologische Invariante ist das Geschlecht der Fläche. Die Teichmüllerkurven entstehen nun z.B. durch das Verändern von einem Koeffizienten in den Polynomen, die uns durch Nullstellenmengen Kurven beschreiben- sie sind sozusagen Kurven von Kurven. Die entstehenden Strukturen kann man als Modulraum beschreiben, und so diesen Konstruktionen einen Parameterraum mit geometrischer Struktur zuordnen. Speziell entstehen Punkte auf Teichmüllerkurven gerade beim Verkleben von gegenüberliegenden parallelen Kanten eines Polygons; durch Scherung erhält man eine Familie von Kurven, die in seltenen Fällen selbst eine Kurve ist. Ein Beispiel ist das Rechteck, das durch Verkleben zu einem Torus wird, aber durch Scherung um ganz spezielle Faktoren zu einem ganz anderen Ergebnis führen kann. Die durch Verklebung entstandenen Flächen kann man als Translationsflächen in den Griff bekommen. Hier liefert die Translationssymmetrie die Methode um äquivalente Punkte zu identifizieren. Für die weitere Analyse werden dann auch Differentialformen eingesetzt. Translationen sind aber nur ein Beispiel für mögliche Symmetrien, denn auch Rotationen können Symmetrien erzeugen. Da die Multiplikation in den komplexen Zahlen auch als Drehstreckung verstanden werden kann, sind hier Rotationen als komplexe Isomorphismen ganz natürlich, und das findet man auch in den Einheitswurzeln wieder. Literatur und Zusatzinformationen A. Zorich: Flat Surfaces, Frontiers in Number Theory, Physics and Geometry, On Random Matrices, Zeta Functions, and Dynamical Systems, Ed. by P. Cartier, B. Julia, P. Moussa, and P. Vanhove. Vol. 1. Berlin: pp. 439–586, Springer-Verlag, 2006. M. Möller: Teichmüller Curves, Mainly from the Viewpoint of Algebraic Geometry, IAS/Park City Mathematics Series, 2011. J. Zachhuber: Avoidance of by Teichmüller Curves in a Stratum of , Diplomarbeit an der Fakultät für Mathematik am Karlsruher Institut für Technologie (KIT), 2013. C. McMullen: Billiards and Teichmüller curves on Hilbert modular surfaces, Journal of the AMS 16.4, pp. 857–885, 2003. C. McMullen: Prym varieties and Teichmüller curves, Duke Math. J. 133.3, pp. 569–590, 2006. C. McMullen: Dynamics of SL(2,R) over moduli space in genus two, Ann. of Math. (2) 165, no. 2, 397–456, 2007. Weitere Paper von C. McMullen, u.a. The mathematical work of Maryam Mirzakhani Podcast: Modellansatz 040: Topologie mit Prof. Dr. Wolfgang Lück

Bob Garfield and Mike Vuolo ask the question: How did we go from "four and twenty" to "twenty four"? Show notes at www.slate.com/lexiconvalley. Learn more about your ad choices. Visit megaphone.fm/adchoices

Organic DMT (@KnowledgeFountN) joins us on this episode to talk Vibrational Number Theory and also give us a live #DMTtruth speech. It’s an epic podcast that you don’t want to miss.   Hosts of the show: Brando: @Mind_of_Peace Stephen: @Youth_Thinking   Books Mentioned on the Show: Confucian and Taoist Wisdom: http://amzn.to/1hKL28c The Book by Alan Watts: http://amzn.to/1kib5Zg  Day of Wisdom According to Number Vibration: http://amzn.to/1lq3rfx   When you've figured out your Vibrational Number check out OrganicDMT.wordpress.com  for information on that number. 

Album: Dreaming Electronica Genre: PsyTrance, Neo-Trance, Progressive House, Experimental House, Electronica Links: http://radio3.cbc.ca/bands/Contemporary-Sound http://www.facebook.com/pages/Contemporary-Sound/322321701536 http://www.wix.com/contemporarysoundart/contemporarysound?ref=nf http://anastasisk.redbubble.com/ http://soundcloud.com/contemporary-sound http://www.last.fm/music/Contemporary+Sound http://contemporarysound.wordpress.com/ http://www.sonus.ca/ ΙΣ ΗΡ ΝΙ ΚΑ

What is the basis for the common numerical counting systems found in mathematics? And, how do these vary across the globe? On this program, Alex Bellos discussed number theory.

Nu:sounds podcast003 Number Theory in the mix