Podcasts about hausdorff

Nalini AnantharamanGéométrie spectraleCollège de FranceAnnée 2023-2024Colloque - Géométries aléatoires et applications : Sur les graphes aléatoires unimodulairesIntervenant :François BaccelliInria & École normale supérieure ParisRésuméL'exposé introduira d'abord les graphes aléatoires unimodulaires et donnera plusieurs exemples issus de la théorie des processus ponctuels, des processus de branchement, des marches aléatoires et des ensembles aléatoires discrets auto-similaires. Plusieurs types de résultats sur ces graphes seront ensuite passés en revue :Des extensions unimodulaires de théorèmes classiques du calcul de Palm et de la théorie ergodique.Une classification des dynamiques déterministes ou aléatoires sur ces graphes basée sur les propriétés de leurs variétés stables.Deux nouvelles notions de dimension pour de tels graphes, à savoir leurs dimensions unimodulaires de Minkowski et de Hausdorff.Cet exposé est basé sur une série d'articles en collaboration avec M.-O. Haji-Mirsadeghi et A. Khezeli.----Le terme « géométrie aléatoire » désigne tout processus permettant de construire de manière aléatoire un objet géométrique ou des familles d'objets géométriques. Un procédé simple consiste à assembler aléatoirement des éléments de base : sommets et arêtes dans le cas des graphes aléatoires, triangles ou carrés dans certains modèles de surfaces aléatoires, ou encore triangles, « pantalons » ou tétraèdres hyperboliques dans le cadre des géométries hyperboliques. La théorie des graphes aléatoires imprègne toutes les branches des mathématiques actuelles, des plus théoriques (théorie des groupes, algèbres d'opérateurs, etc.) aux plus appliquées (modélisation de réseaux de communication, par exemple). En mathématiques, l'approche probabiliste consiste à évaluer la probabilité qu'une propriété géométrique donnée apparaisse : lorsque l'on ne sait pas si un théorème est vrai, on peut tenter de démontrer qu'il l'est dans 99 % des cas.Une autre méthode classique pour générer des paysages aléatoires consiste à utiliser les séries de Fourier aléatoires, avec de nombreuses applications en théorie du signal ou en imagerie.En physique théorique, les géométries aléatoires sont au cœur de la théorie de la gravité quantique et d'autres théories des champs quantiques. Les différents aspects mathématiques s'y retrouvent curieusement entremêlés, par exemple, la combinatoire des quadrangulations ou des triangulations apparaît dans le calcul de certaines fonctions de partition.Ce colloque offrira un panorama non exhaustif des géométries aléatoires, couvrant des aspects allant des plus abstraits aux applications concrètes en imagerie et télécommunications.

This past Thursday, the International Criminal Court (ICC) issued arrest warrants for Israel's prime minister, Benjamin Netanyahu, and Israel's former defense minister, Yoav Gallant. The warrants were issued on charges of attempting to orchestrate starvation as a method of warfare, and crimes against humanity, of “murder and persecution”, in the ICC's terms. A warrant was also issued for Hamas leader Mohammed Deif, who was killed in an airstrike in Gaza in July.To help us understand the ICC; its role, jurisdiction and credibility; and the wide range of implications of these arrest warrants, our guest is Natasha Hausdorff.Natasha is a British barrister and expert on international law, foreign affairs, and national security policy. She is the Charitable Trust Legal Director of UK Lawyers For Israel (UKLFI). Natasha regularly briefs government leaders and international organizations, and has spoken at parliaments across Europe and at the United Nations. She is a regular commentator on issues of international law, both generally and specifically as they apply to Israel.UK Lawyers For Israel on X: https://x.com/uklfi 

This past Thursday, the International Criminal Court (ICC) issued arrest warrants for Israel's prime minister, Benjamin Netanyahu, and Israel's former defense minister, Yoav Gallant. The warrants were issued on charges of attempting to orchestrate starvation as a method of warfare, and crimes against humanity, of “murder and persecution”, in the ICC’s terms. A warrant was […]

This is the audio from a video we have just published on our YouTube channel – an interview with Natasha Hausdorff. To make sure you never miss great content like this, subscribe to our channel: https://www.youtube.com/@spiked  Israel is easily the most demonised nation on Earth. Despite the Jewish State being a liberal democracy, Western activists spend vastly more time raging against it than they do genuinely despotic regimes. We're told Israel is committing ‘genocide' in Gaza, that it is going out of its way to kill civilians in its war against Hamas, that it is an ‘apartheid state'. But none of this bears any resemblance to reality. Here, barrister Natasha Hausdorff debunks the anti-Israel myths, which she says have become a kind of new blood libel.  Order Brendan O'Neill's new book, After the Pogrom, now from:

Barrister Natasha Hausdorff has teamed up with and debated alongside Douglas Murray to prevent the spread of disinformation around Israel and Hamas. Today, she dispels 5 huge myths. Follow her on Instagram: https://www.instagram.com/natashahausdorff/?hl=en  Follow UK Lawyers for Israel on X: https://x.com/UKLFI  Support my channel: http://andrewgoldheretics.com  Andrew on X: https://twitter.com/andrewgold_ok  Insta: https://www.instagram.com/andrewgold_ok Heretics YouTube channel: https://www.youtube.com/@andrewgoldheretics Learn more about your ad choices. Visit megaphone.fm/adchoices

FDD Senior Vice President Jonathan Schanzer delivers timely situational updates and analysis on headlines of the Middle East, followed by a conversation with Natasha Hausdorff, an international law expert and the director of UK Lawyers for Israel.Learn more at: fdd.org/fddmorningbrief/

Israel is being attacked by the Iranian regime and its proxy armies on SEVEN FRONTS…. But multiple dimensions. There's a media war. There's also a legal war. Lawfare. The way international law is being twisted, legal definitions that only apply to Israel… the weaponization of international institutions. Natasha Hausdorff is a British barrister specializing in public international law, human rights, and national security. Hausdorff has been involved in high-profile legal cases, particularly those concerning national security issues and international humanitarian law. Hausdorff is a fierce advocate for Israel, frequently taking on public debates and speaking engagements to take the fight for Israel to the global stage. Alongside her legal career, she is known for her work in policy circles, often speaking on topics related to international law and Middle Eastern geopolitics. Co-Creator and Host - Eylon LevyCo-Creator and Creative Director - Guy RossExecutive Producer - Asher Westropp-EvansDirector - Lotem SegevGraphics/Assistant Director - Thomas GirschEditor/Assistant Director - Benny GoldmanStay up to date at:https://www.stateofanationpodcast.com/X: https://twitter.com/stateofapodInstagram: https://www.instagram.com/stateofapod/Facebook: https://www.facebook.com/profile.php?... LinkedIn: www.linkedin.com/company/state-of-a-nation

For the anniversary of October 7th I sit down to discuss the latest in the year long war between Israel and Hamas, Hezbollah and Iran with barrister and expert in international law, Natasha Hausdorff.Natasha, fresh off a tour with Douglas Murray, takes me through the failure of the international community, the legality Israel's founding and the legality of the so-called settlements in the West Bank.We also explore geopolitics, specifically Britain and America's approach to Netanyahu and Israel throughout, and what Trump might mean for the region. The capture and failure of institutions in Britain - universities and the mainstream media in particularly.All this and much more in a deep dive you won't get in the corporate media…-----------------------------------------------------------------------------------------------------------------------Linktree: https://linktr.ee/winstonmarshall-----------------------------------------------------------------------------------------------------------------------SUBSCRIBE: If you're liking the show and want to stay updated, don't forget to subscribe to our YouTube channel! Simply hit the 'Subscribe' button below the video, and then click the bell icon to ensure you get all our notifications. Thanks for your support!FOLLOW ME ON SOCIAL MEDIA:Substack: https://www.winstonmarshall.co.uk/X: https://twitter.com/mrwinmarshallInsta: https://www.instagram.com/winstonmarshallLinktree: https://linktr.ee/winstonmarshall----------------------------------------------------------------------------------------------------------------------Chapters 0:00 - October 7th Reflections and Initial Reactions 4:59 - Media Coverage and Public Reactions 8:31 - Hezbollah's Disruption and Israeli Operations23:22- International Law and Media Misrepresentation 42:26 - Historical Context and Legal Frameworks1:17:20- Media Bias and Misinformation1:28:57- Hope for Peace and Cooperation 2:00:01 - Challenges and Future Prospects Hosted on Acast. See acast.com/privacy for more information.

In this episode, I speak with an extraordinary woman, lawyer and legal advocate for Israel - British barrister Natasha Hausdorff - who specializes in international law and human rights. We explore some of the key issues facing Israel, including its “legitimacy” as a state, the legal validity of recent accusations of genocide, and the doctrine of proportionality in military responses under international law. Hausdorff shares insights from her recent appearance with Douglas Murray at the Munk Debate in Toronto -which went viral globally- and discusses the broader implications of anti-Zionism and anti-Semitism worldwide. The episode highlights Hausdorff's role in defending Israel's actions under international law and her efforts to educate and inform on these critical topics. Her deep knowledge, experience and brilliance in explaining complex issues straightforwardly is an opportunity not to be missed.Podcast Notes* JNS “The Quad” episode featuring an interview with Natasha Hausdorff, in studio* Natasha Hausdorff's online biography* Munk Debate, DATE, featuring Natasha Hausdorff, Douglas Murray, Mehdi Hassan and Gideon Levy* UK Lawyers for Israel Charitable Trust* UKLFI upcoming webinar entitled “Unravelling the ICJ Advisory Opinions” with Natasha Hausdorff and Olivia Flasch on Thursday 22 August at 6 pm (UK), 8 pm (IST) and 1 pm (EST)* Article by Iran International reporter Negar Mojtahedi exposing Iran's financing of University of Toronto pro-Hamas encampmentState of Tel Aviv is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber. This is a public episode. If you'd like to discuss this with other subscribers or get access to bonus episodes, visit www.stateoftelaviv.com/subscribe

#ANTISEMITISM: IJC libels. Natasha Hausdorff is an attorney in London and legal director at UK Lawyers for Israel. She clerked for the President of the Supreme Court of Israel in Jerusalem. Natasha was a Fellow at Columbia Law School. She lectures globally on aspects of public international law and national security policy. Malcolm Hoenlein @Conf_of_pres @mhoenlein1 1860 Jerusalem

Natasha Hausdorff is a barrister in London and a Director of UK Lawyers for Israel. She holds law degrees from Oxford and Tel Aviv Universities and was a Fellow in the National Security Law Programme at Columbia Law School. Natasha previously worked for Skadden Arps, in London and Brussels and clerked for the President of the Israeli Supreme Court, Chief Justice Miriam Naor, in Jerusalem. She regularly briefs politicians and international organisations and has spoken at Parliaments across Europe and at the United Nations. Go to https://gdefy.com and use code RELIEF24 for an exclusive $20 off orders of $100 or more Go to https://CozyEarth.com and enter promo code TRIGGERNOMETRY at checkout for up to 35% off Join our Premium Membership for early access, extended and ad-free content: https://triggernometry.supercast.com OR Support TRIGGERnometry Here: Bitcoin: bc1qm6vvhduc6s3rvy8u76sllmrfpynfv94qw8p8d5 Music by: Music by: Xentric | info@xentricapc.com | https://www.xentricapc.com/ YouTube: @xentricapc  Buy Merch Here: https://www.triggerpod.co.uk/shop/ Advertise on TRIGGERnometry: marketing@triggerpod.co.uk Join the Mailing List: https://www.triggerpod.co.uk/#mailinglist Find TRIGGERnometry on Social Media:  https://twitter.com/triggerpod https://www.facebook.com/triggerpod/ https://www.instagram.com/triggerpod/ About TRIGGERnometry:  Stand-up comedians Konstantin Kisin (@konstantinkisin) and Francis Foster (@francisjfoster) make sense of politics, economics, free speech, AI, drug policy and WW3 with the help of presidential advisors, renowned economists, award-winning journalists, controversial writers, leading scientists and notorious comedians. Learn more about your ad choices. Visit megaphone.fm/adchoices

David Harris is joined once again by barrister and specialist in international law, and Director of UK Lawyers for Israel, Natasha Hausdorff, to continue the conversation on Israel advocacy.

On this special edition of Hub Dialogues, publisher Rudyard Grffiths interviews author and journalist Douglas Murray, barrister and expert commentator Natasha Hausdorff, editor-in-chief and founder of Zeteo Mehdi Hasan, and Israeli journalist author and journalist Gideon Levy ahead of the Munk Debate on Anti-Zionism. These four big thinkers will debate the motion Be It Resolved, anti-Zionism is antisemitism at 7:00 PM ET, Monday June 17th at Toronto's Roy Thomson Hall. To find out how you can watch the debate live and archived click here.The Hub Dialogues features The Hub's editor-at-large, Sean Speer, in conversation with leading entrepreneurs, policymakers, scholars, and thinkers on the issues and challenges that will shape Canada's future at home and abroad.If you like what you are hearing on Hub Dialogues consider subscribing to The Hub's free weekly email newsletter featuring our insights and analysis on key public policy issues. Sign up here: https://thehub.ca/free-member-sign-up/. Hosted on Acast. See acast.com/privacy for more information.

On June 17th four debaters will take to the stage at Toronto's Roy Thomson Hall for a sold out debate on Anti-Zionism. The motion up for debate: Be it Resolved, anti-Zionism is antisemitism On this special Munk Dialogue, we speak with each of the debaters to get a sense of their arguments heading into the debate, and what it is about this particular topic that made them want to participate. Arguing for the resolution is award-winning journalist, best-selling author, and former Munk Debater Douglas Murray. His debate partner is Natasha Hausdorff, an international law expert and legal commentator on antisemitism. Opposing the resolution is Mehdi Hasan. Mehdi is a best-selling author, former MSNBC anchor, and the CEO and editor-in-chief of the new media company Zeteo. He will be joined by the award-winning Israeli broadcaster and Haaretz columnist Gideon Levy.   The host of the Munk Debates is Rudyard Griffiths Tweet your comments about this episode to @munkdebate or comment on our Facebook page https://www.facebook.com/munkdebates/ To sign up for a weekly email reminder for this podcast, send an email to podcast@munkdebates.com.   To support civil and substantive debate on the big questions of the day, consider becoming a Munk Member at https://munkdebates.com/membership Members receive access to our 15+ year library of great debates in HD video, a free Munk Debates book, and ticketing privileges at our live events. This podcast is a project of the Munk Debates, a Canadian charitable organization dedicated to fostering civil and substantive public dialogue - https://munkdebates.com/ Executive Producer: Ricki Gurwitz Senior Producer: Daniel Kitts Editor: Kieran Lynch

#ISRAEL Natasha Hausdorff Natasha Hausdorff is an attorney in London and legal director at UK Lawyers for Israel. Malcolm Hoenlein @Conf_of_pres @mhoenlein1 https://www.msn.com/en-us/news/world/what-to-know-about-icj-s-israel-ruling-and-what-happens-next/ar-BB1n0bHj 1594 The Hague

Natasha Hausdorff, specialist in international law and Director of UK Lawyers for Israel, joins David Harris to discuss her advocacy and concisely debunk egregious misconceptions about Israel.

Joining Iain Dale on Cross Question this evening are Conservative MP and chair of the Treasury Committee Harriet Baldwin, Labour MP Charlotte Nichols, Green Party candidate for Mayor of London Zoe Garbett and lawyer Natasha Hausdorff.

What is a proportional response to a terrorist attack? What does genocide involve? And is South Africa's case at the International Court of Justice justified? Presenters: Mark Oppenheimer and Jason Werbeloff Editor and Producer: Jimmy Mullen and Porter Kaufman Brain in a Vat bookshop (Shopify): https://smarturl.it/BrainShop Brain in a Vat bookshop (Amazon): https://smarturl.it/BrainAmazonShop

En el episodio de hoy, hablamos del bonito mundo de la doble tarea, un área que ya de por sí tiene sus características en la sociedad en general y que en el campo de la rehabilitación es una categoría fundamental que tarde o temprano hay que abordar. El tema es extenso a más no poder y ha sido un poco complicado hacer una síntesis, ya que tenemos información de estudios tanto a nivel general de lo que implica la atención a dos tareas, críticas sociales (desde la sociología), estudios de neuropsicología y toda la parte de entrenamiento dual en terapia física y en diferentes patologías. Para esta ocasión, lo que me propongo es dar un marco cultural y sociológico inicial que creo que es importante para entender la globalidad del asunto; después voy a introducir conceptos fundamentales relacionados con la doble tarea y la idea después es transitando hacia estudios de correlatos neurales de la doble tarea y cómo podemos entrenar esa habilidad en neurorrehabilitación, sobre todo en la que concierne a la terapia física. Referencias del episodio: 1. Leone, C., Feys, P., Moumdjian, L., D'Amico, E., Zappia, M., & Patti, F. (2017). Cognitive-motor dual-task interference: A systematic review of neural correlates. Neuroscience and biobehavioral reviews, 75, 348–360. https://doi.org/10.1016/j.neubiorev.2017.01.010 (https://pubmed.ncbi.nlm.nih.gov/28104413/). 2. Kuo, H. T., Yeh, N. C., Yang, Y. R., Hsu, W. C., Liao, Y. Y., & Wang, R. Y. (2022). Effects of different dual task training on dual task walking and responding brain activation in older adults with mild cognitive impairment. Scientific reports, 12(1), 8490. https://doi.org/10.1038/s41598-022-11489-x (https://pubmed.ncbi.nlm.nih.gov/35589771/). 3. Li, K. Z. H., Bherer, L., Mirelman, A., Maidan, I., & Hausdorff, J. M. (2018). Cognitive Involvement in Balance, Gait and Dual-Tasking in Aging: A Focused Review From a Neuroscience of Aging Perspective. Frontiers in neurology, 9, 913. https://doi.org/10.3389/fneur.2018.00913 (https://pubmed.ncbi.nlm.nih.gov/30425679/). 4. Mac-Auliffe D, Chatard B, Petton M, Croizé AC, Sipp F, Bontemps B, Gannerie A, Bertrand O, Rheims S, Kahane P, Lachaux JP. The Dual-Task Cost Is Due to Neural Interferences Disrupting the Optimal Spatio-Temporal Dynamics of the Competing Tasks. Front Behav Neurosci. 2021 Aug 19;15:640178. doi: 10.3389/fnbeh.2021.640178. PMID: 34489652; PMCID: PMC8416616 (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8416616/). 5. McPhee, A. M., Cheung, T. C. K., & Schmuckler, M. A. (2022). Dual-task interference as a function of varying motor and cognitive demands. Frontiers in psychology, 13, 952245. https://doi.org/10.3389/fpsyg.2022.952245 (https://pubmed.ncbi.nlm.nih.gov/36248521/). 6. Piqueres-Juan I, Tirapu-Ustárroz J, García-Sala M. Paradigmas de ejecución dual: aspectos conceptuales. Rev Neurol 2021;72 (10):357-367 (https://neurologia.com/articulo/2020200). 7. Plummer, P., Eskes, G., Wallace, S., Giuffrida, C., Fraas, M., Campbell, G., Clifton, K. L., Skidmore, E. R., & American Congress of Rehabilitation Medicine Stroke Networking Group Cognition Task Force (2013). Cognitive-motor interference during functional mobility after stroke: state of the science and implications for future research. Archives of physical medicine and rehabilitation, 94(12), 2565–2574.e6. https://doi.org/10.1016/j.apmr.2013.08.002 (https://pubmed.ncbi.nlm.nih.gov/23973751/). 8. St George, R. J., Jayakody, O., Healey, R., Breslin, M., Hinder, M. R., & Callisaya, M. L. (2022). Cognitive inhibition tasks interfere with dual-task walking and increase prefrontal cortical activity more than working memory tasks in young and older adults. Gait & posture, 95, 186–191. https://doi.org/10.1016/j.gaitpost.2022.04.021 (https://pubmed.ncbi.nlm.nih.gov/35525151/). 9. Strobach T. (2020). The dual-task practice advantage: Empirical evidence and cognitive mechanisms. Psychonomic bulletin & review, 27(1), 3–14. https://doi.org/10.3758/s13423-019-01619-4 (https://pubmed.ncbi.nlm.nih.gov/31152433/). 10. Watanabe, K., & Funahashi, S. (2014). Neural mechanisms of dual-task interference and cognitive capacity limitation in the prefrontal cortex. Nature neuroscience, 17(4), 601–611. https://doi.org/10.1038/nn.3667 (https://pubmed.ncbi.nlm.nih.gov/24584049/). 11. Ángel L. Martínez Nogueras (2020). Un repaso al paradigma de tarea dual desde la neuropsicología (1ª parte). (https://neurobase.wordpress.com/2020/03/20/un-repaso-al-paradigma-de-tarea-dual-desde-la-neuropsicologia-1a-parte/). 12. Johann Hari (2023). El valor de la atención. Por qué nos la robaron y cómo recuperarla (https://www.planetadelibros.com/libro-el-valor-de-la-atencion/365202). 13. McIsaac, T. L., Lamberg, E. M., & Muratori, L. M. (2015). Building a framework for a dual task taxonomy. BioMed research international, 2015, 591475. https://doi.org/10.1155/2015/591475 (https://pubmed.ncbi.nlm.nih.gov/25961027/). 14. Rémy, F., Wenderoth, N., Lipkens, K., & Swinnen, S. P. (2010). Dual-task interference during initial learning of a new motor task results from competition for the same brain areas. Neuropsychologia, 48(9), 2517–2527. https://doi.org/10.1016/j.neuropsychologia.2010.04.026 (https://pubmed.ncbi.nlm.nih.gov/20434467/). 15. D'Esposito, M., Detre, J. A., Alsop, D. C., Shin, R. K., Atlas, S., & Grossman, M. (1995). The neural basis of the central executive system of working memory. Nature, 378(6554), 279–281. https://doi.org/10.1038/378279a0 (https://pubmed.ncbi.nlm.nih.gov/7477346/9. 16. Just, M. A., Carpenter, P. A., Keller, T. A., Emery, L., Zajac, H., & Thulborn, K. R. (2001). Interdependence of nonoverlapping cortical systems in dual cognitive tasks. NeuroImage, 14(2), 417–426. https://doi.org/10.1006/nimg.2001.0826 (https://pubmed.ncbi.nlm.nih.gov/11467915/).

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Alex Kontorovich is a Professor of Mathematics at Rutgers University and served as the Distinguished Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics in 2020–2021. Alex has received numerous awards for his illustrious mathematical career, including the Levi L. Conant Prize in 2013 for mathematical exposition, a Simons Foundation Fellowship, an NSF career award, and being elected Fellow of the American Mathematical Society in 2017. He currently serves on the Scientific Advisory Board of Quanta Magazine and as Editor-in-Chief of the Journal of Experimental Mathematics. In this episode, Alex takes us from the ancient beginnings to the present day on the subject of circle packings. We start with the Problem of Apollonius on finding tangent circles using straight-edge and compass and continue forward in basic Euclidean geometry up until the time of Leibniz whereupon we encounter the first complete notion of a circle packing. From here, the plot thickens with observations on surprising number theoretic coincidences, which only received full appreciation through the craftsmanship of chemistry Nobel laureate Frederick Soddy. We continue on with more advanced mathematics arising from the confluence of geometry, group theory, and number theory, including fractals and their dimension, hyperbolic dynamics, Coxeter groups, and the local to global principle of advanced number theory. We conclude with a brief discussion on extensions to sphere packings. Patreon: http://www.patreon.com/timothynguyen I. Introduction 00:00: Biography 11:08: Lean and Formal Theorem Proving 13:05: Competitiveness and academia 15:02: Erdos and The Book 19:36: I am richer than Elon Musk 21:43: Overview II. Setup 24:23: Triangles and tangent circles 27:10: The Problem of Apollonius 28:27: Circle inversion (Viette's solution) 36:06: Hartshorne's Euclidean geometry book: Minimal straight-edge & compass constructions III. Circle Packings 41:49: Iterating tangent circles: Apollonian circle packing 43:22: History: Notebooks of Leibniz 45:05: Orientations (inside and outside of packing) 45:47: Asymptotics of circle packings 48:50: Fractals 50:54: Metacomment: Mathematical intuition 51:42: Naive dimension (of Cantor set and Sierpinski Triangle) 1:00:59: Rigorous definition of Hausdorff measure & dimension IV. Simple Geometry and Number Theory 1:04:51: Descartes's Theorem 1:05:58: Definition: bend = 1/radius 1:11:31: Computing the two bends in the Apollonian problem 1:15:00: Why integral bends? 1:15:40: Frederick Soddy: Nobel laureate in chemistry 1:17:12: Soddy's observation: integral packings V. Group Theory, Hyperbolic Dynamics, and Advanced Number Theory 1:22:02: Generating circle packings through repeated inversions (through dual circles) 1:29:09: Coxeter groups: Example 1:30:45: Coxeter groups: Definition 1:37:20: Poincare: Dynamics on hyperbolic space 1:39:18: Video demo: flows in hyperbolic space and circle packings 1:42:30: Integral representation of the Coxeter group 1:46:22: Indefinite quadratic forms and integer points of orthogonal groups 1:50:55: Admissible residue classes of bends 1:56:11: Why these residues? Answer: Strong approximation + Hasse principle 2:04:02: Major conjecture 2:06:02: The conjecture restores the "Local to Global" principle (for thin groups instead of orthogonal groups) 2:09:19: Confession: What a rich subject 2:10:00: Conjecture is asymptotically true 2:12:02: M. C. Escher VI. Dimension Three: Sphere Packings 2:13:03: Setup + what Soddy built 2:15:57: Local to Global theorem holds VII. Conclusion 2:18:20: Wrap up 2:19:02: Russian school vs Bourbaki Image Credits: http://timothynguyen.org/image-credits/

Dr. William P. Hausdorff, Epidemiologist, stops by Supreme Myths to talk all things vaccine and Covid. He answers many of the questions we all have about the pandemic.

In this guest episode, Jeffrey Hausdorff and Nathan Morelli speak about transcranial direct current stimulation (tDCS), its mechanisms of action, current application in research, and where the field is going in the future. In this discussion, we cover many topics which will give you insight into this area of brain stimulation. We begin with the basics of tDCS from its historic origins and therapy fundamentals. Our discussion then progresses to a deep-dive inside some of Prof. Hausdorff's most recent works in collaboration with many world renowned researchers in neurodegenerative disease – notably including recent findings in using tDCS to mitigate freezing of gait in patients with Parkinson's disease. We close with a look into the future of tDCS in research and clinical practice. Given Prof. Hausdorff's expertise there are few people in the world more qualified to speak on Parkinson's disease and non-invasive brain stimulation. As such, it is our immense privilege to present this interview to you.

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Topological Fixed Point Exercises, published by Scott Garrabrant, Sam Eisenstat on the AI Alignment Forum. This is one of three sets of fixed point exercises. The first post in this sequence is here, giving context. 1. (1-D Sperner's lemma) Consider a path built out of n edges as shown. Color each vertex blue or green such that the leftmost vertex is blue and the rightmost vertex is green. Show that an odd number of the edges will be bichromatic. 2. (Intermediate value theorem) The Bolzano-Weierstrass theorem states that any bounded sequence in R n has a convergent subsequence. The intermediate value theorem states that if you have a continuous function f 0 1 → R such that f 0 ≤ 0 and f 1 ≥ 0 , then there exists an x ∈ 0 1 such that f x 0 . Prove the intermediate value theorem. It may be helpful later on if your proof uses 1-D Sperner's lemma and the Bolzano-Weierstrass theorem 3. (1-D Brouwer fixed point theorem) Show that any continuous function f 0 1 → 0 1 has a fixed point (i.e. a point x ∈ 0 1 with f x x ). Why is this not true for the open interval 0 1 4. (2-D Sperner's lemma) Consider a triangle built out of n 2 smaller triangles as shown. Color each vertex red, blue, or green, such that none of the vertices on the large bottom edge are red, none of the vertices on the large left edge are green, and none of the vertices on the large right edge are blue. Show that an odd number of the small triangles will be trichromatic. 5. Color the all the points in the disk as shown. Let f be a continuous function from a closed triangle to the disk, such that the bottom edge is sent to non-red points, the left edge is sent to non-green points, and the right edge is sent to non-blue points. Show that f sends some point in the triangle to the center. 6. Show that any continuous function f from closed triangle to itself has a fixed point. 7. (2-D Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of R 2 to itself has a fixed point. (You may use the fact that given any closed convex subset S of R n , the function from R n to S which projects each point to the nearest point in S is well defined and continuous.) 8. Reflect on how non-constructive all of the above fixed-point findings are. Find a parameterized class of functions where for each t ∈ 0 1 f t 0 1 → 0 1 , and the function t ↦ f t is continuous, but there is no continuous way to pick out a single fixed point from each function (i.e. no continuous function g such that g t is a fixed point of f t for all t 9. (Sperner's lemma) Generalize exercises 1 and 4 to an arbitrary dimension simplex. 10. (Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of R n to itself has a fixed point. 11. Given two nonempty compact subsets A B ⊆ R n , the Hausdorff distance between them is the supremum max sup a ∈ A d a B sup b ∈ B d b A over all points in either subset of the distance from that point to the other subset. We call a set valued function f S → 2 T a continuous Hausdorff limit if there is a sequence f n of continuous functions from S to T whose graphs, x y ∣ y f n x ⊆ S × T , converge to the graph of f x y ∣ f x ∋ y ⊆ S × T , in Hausdorff distance. Show that every continuous Hausdorff limit f T → 2 T from a compact convex subset of R n to itself has a fixed point (a point x such that x ∈ f x 12. Let S and T be nonempty compact convex subsets of R n . We say that a set valued function, f S → 2 T is a Kakutani function if the graph of f x y ∣ f x ∋ y ⊆ S × T , is closed, and f x is convex and nonempty for all x ∈ S . For example, we could take S and T to be the interval 0 1 , and we could have f S → 2 T send each x 1 2 to 0 , map x 1 2 to the whole interval 0 1 , and map x 1 2 to 1 . Show that every Kakutani function is a continuous Hausdorff limit. (H...

Link to bioRxiv paper: http://biorxiv.org/cgi/content/short/2020.06.19.162461v1?rss=1 Authors: van Doorn, J., Xing, M., Cahn, B. R., Delorme, A., Ajilore, O., Leow, A. D. Abstract: Alterations in brain connectivity has been shown for many disease states and groups of people from different levels of cognitive training. To study dynamic functional connectivity, we propose a method for a personalized connectomic state space called Thought Chart. Experienced meditators are an interesting group of healthy subjects for brain connectivity analyses due to their demonstrated differences in resting state dynamics, and altered brain connectivity has been implicated as a potential factor in several psychiatric disorders. Three distinct techniques of meditation are explored: Isha Yoga, Himalayan Yoga, and Vipassana, as well as a meditation-naive group of individuals. All individuals participated in a breath awareness task, an autobiographical thinking task, and one of three different meditation practices according to their expertise, while being recorded by a 64-electrode electroencephalogram (EEG). The functional brain connectivity was estimated using weighted phase lag index (WPLI) and the connectivity dynamics were investigated using a within-individual formulation of Thought Chart, a previously proposed dimensionality reduction method which utilizes manifold learning to map out a state space of functional connectivity. Results showed that the two meditation tasks (breath awareness task and own form of meditation) in all groups were found to have consistently different functional connectivity patterns relative to those of the instructed mind-wandering (IMW) tasks in each individual, as measured using the Hausdorff distance in the state space. The specific meditation state was found to be most similar to the breath awareness state in all groups, as expected in these meditation traditions which incorporate all breath awareness in their practice. The difference in connectivity was found to not be solely driven by specific frequency bands. These results demonstrate that the within-individual form of Thought Chart consistently and reliably separates similar tasks among healthy meditators and non-meditators during resting state-like EEG recordings. We see this dissimilarity between breath awareness/meditation and IMW regardless of meditation experience or tradition, with no significant group differences. Copy rights belong to original authors. Visit the link for more info

Anika used to work as an Engineer in Landscape Architecture for the past 10 years and is now a full-time Yoga Teacher and an Ayurveda Lifestyle Coach! On the podcast she talks about what has worked for her in terms of being healthy while working efficiently and gives a lot of ayurvedic and yogic tips for everyone wanting to do the same while staying creative!

Steffen Winter befasst sich mit fraktaler Geometrie, also mit Mengen, deren Dimension nicht ganzahllig ist. Einen intuitiven Zugang zum Konzept der Dimension bieten Skalierungseigenschaften. Ein einfaches Beispiel, wie das funktioniert, ist das folgende: Wenn man die Seiten eines Würfels halbiert, reduziert sich das Volumen auf ein Achtel (ein Halb hoch 3). Bei einem Quadrat führt die Halbierung der Seitenlänge zu einem Viertel (ein Halb hoch 2) des ursprünglichen Flächeninhalts und die Halbierung einer Strecke führt offenbar auf eine halb so lange Strecke (ein Halb hoch 1). Hier sieht man sehr schnell, dass die uns vertraute Dimension, nämlich 3 für den Würfel (und andere Körper), 2 für das Quadrat (und andere Flächen) und 1 für Strecken (und z.B. Kurven) in die Skalierung des zugehörigen Maßes als Potenz eingeht. Mengen, bei denen diese Potenz nicht ganzzahlig ist, ergeben sich recht ästhetisch und intuitiv, wenn man mit selbstähnlichen Konstruktionen arbeitet. Ein Beispiel ist der Sierpinski-Teppich. Er entsteht in einem iterativen Prozess des fortgesetzten Ausschneidens aus einem Quadrat, hat aber selbst den Flächeninhalt 0. Hier erkennt man durch die Konstruktion, dass die Skalierung ln 8/ln 3 ist, also kein ganzzahliger Wert sondern eine Zahl echt zwischen 1 und 2. Tatsächlich sind das Messen von Längen, Flächen und Volumina schon sehr alte und insofern klassische Probleme und auch die Defizite der beispielsweise in der Schule vermittelten Formeln beim Versuch, sie für Mengen wie den Sierpinski-Teppich anzuwenden, werden schon seit etwa 100 Jahren mit verschiedenen angepassten Maß- und Dimensionskonzepten behoben. Ein Dimensionsbegriff, der ganz ohne die Hilfe der Selbstähnlichkeit auskommt, wurde von Felix Hausdorff vorgeschlagen und heißt deshalb heute Hausdorff-Dimension. Hier werden Überdeckungen der zu untersuchenden Menge mit (volldimensionalen) Kugeln mit nach oben beschränktem (aber ansonsten beliebigem) Durchmesser angeschaut. Die Durchmesser der Kugeln werden zu einer Potenz s erhoben und aufsummiert. Man sucht unter allen Überdeckungen diejenigen, bei denen sich so die kleinste Durchmessersumme ergibt. Nun lässt man den maximal zulässigen Durchmesser immer kleiner werden. Die Hausdorff-Dimension ergibt sich als die kleinstmögliche Potenz s, für die diese minimalen Durchmessersummen gerade noch endlich bleiben. Ein verwandter aber nicht identischer Dimensionsbegriff ist die sogenannte Box-Dimension. Für hinreichend gutartige Mengen stimmen Hausdorff- und Box-Dimension überein, aber man kann zum Beispiel Cantormengen konstruieren, deren Dimensionen verschieden sind. Für die Box-Dimension kann der Fall eintreten, dass die Vereinigung abzählbar vieler Mengen der Dimension 0 zu einer Menge mit Dimension echt größer als 0 führt, was im Kontext von klassischen Dimensionen (und auch für die Hausdorff-Dimension) unmöglich ist und folglich eher als Hinweis zu werten ist, mit der Box-Dimension sehr vorsichtig zu arbeiten. Tatsächlich gibt es weitere Konzepte fraktale Dimensionen zu definieren. Interessant ist der Fakt, dass erst der Physiker und Mathematiker Benoit Mandelbrot seit Ende der 1960er Jahre eine intensivere Beschäftigung mit solchen Konzepten angestoßen hat. Er hatte in vielen physikalischen Phänomenen das Prinzip der Selbstähnlichkeit beobachtet - etwa dass sich Strukturen auf verschiedenen Größenskalen wiederholen. Wenn man z.B. ein Foto von einem Felsen macht und dazu keine Skala weiß, kann man nicht sagen, ob es sich um einen Stein, einen Ausschnitt aus einem mikroskopischen Bild oder um ein Kletterfelsen von 500m Höhe oder mehr handelt. Durch den Einzug von Computern an jedem Arbeitsplatz und später auch in jedem Haushalt (und den Kinderzimmern) wurde die Visualisierung solcher Mengen für jeden und jede sehr einfach möglich und führte zu einem regelrechten populärwissenschaftlichen Boom des Themas Fraktale. Schwierige offene Fragen im Kontext solcher fraktalen Mengen sind z.B., wie man Begriffe wie Oberflächeninhalt oder Krümmung sinnvoll auf fraktale Strukturen überträgt und dort nutzt, oder wie die Wärmeausbreitung und die elektrische Leitfähigkeit in solchen fraktalen Objekten beschrieben werden kann. Literatur und weiterführende Informationen B. Mandelbrot: Die fraktale Geometrie der Natur, Springer-Verlag, 2013. S. Winter: Curvature measures and fractals, Diss. Math. 453, 1-66, 2008. K. Falconer: Fractal geometry, mathematical foundations and applications, John Wiley & Sons, 2004. Podcasts P. Kraft: Julia Sets, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 119, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/julia-sets

With Alex and Nic back in Seattle, Dr. Strand tries to help them solve the mystery of E. Hausdorff. Meanwhile, Alex closes in on some more Strand family secrets. 

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common space-structure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R. The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions, the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact, the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included.