Poincare Conjecture and Ricci Flow | A Million Dollar Problem in Topology Aleph 0 2020-06-24 | How do we use Riemannian Geometry and Surgery Theory to crack a million-dollar problem in topology? Ricci flow, that's how. In this video, we tackle the only Millennium Prize Problem that's been solved so far, and find the deep mathematics uncovered in the process.---Official Problem Statement:claymath.org/millennium-problems/poincar%C3%A9-conjecture---Follow me!Twitter: twitter.com/00aleph00Instagram: instagram.com/00aleph00__Music Info: Documentary - AShamaluevMusic.Music Link: ashamaluevmusic.comIntro: (0:00)Poincare Conjecture: (0:45)Riemannian Geometry: (2:31) Ricci Flow: (4:17)Surgery Theory: (7:10) Proof of Poincare Conjecture: (7:26)
Something weird happens in dimension 8 Aleph 0 2025-05-01 | What do oranges, 8-dimensional space, and a Fields Medal have in common? This is the story of one of the most beautiful proofs of the 21st century.Help fund future projects: patreon.com/aleph0An equally valuable form of support is to simply share the videos.CORRECTIONSAt 0:01 and 0:22, the sphere packings presented are not the densest packings in 3D. The densest packing in 3D is achieved by the face-centered cubic, which is different from the packing I showed here. (Note that the hexagonal close packing in 3D also achieves the densest packing.) At 3:30, the triangular lattice should be a rhomboidal lattice. NEWSLETTERI have a weekly math newsletter where I collate resources to self-study a specific topic and deliver it to your inbox. If this sounds interesting, fill out the form below to sign up: https://forms.gle/Rt1f5StAj3yZtakE6SOURCES and REFERENCES for Further Reading:This video is a quick-and-dirty introduction to Viazovska’s work and the broader story of sphere packing. There are many technical details that I couldn't cover in full, so if you’d like to dive deeper, here are some excellent references:(a) Henry Cohn’s expository article explaining Viazovska’s result:ams.org/publications/journals/notices/201702/rnoti-p102.pdf(b) Maryna Viazovska’s original proof for the E8 lattice packing:https://annals.math.princeton.edu/2017/185-3/p07(c) The follow-up paper on sphere packing in 24 dimensions:https://annals.math.princeton.edu/2017/185-3/p08Proving Cohn and Elkies' theorem for general sphere packings (that aren't lattice packings):This is explained in detail in Henry Cohn's article, Theorem 3 (see above). The idea is to use the fact that lattice packings come arbitrarily close to the optimal packing density and then use a "shifted" version of Poisson summation. Follow me!Twitter: @00aleph00MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.com
Math isnt ready to solve this problem Aleph 0 2025-03-09 | An introduction to the rank conjecture, an unsolved problem about elliptic curves. ONLINE COURSE If you're interested in learning more about this subject, I'm teaching a live online course about elliptic curves and cryptography! It will have:a) a series of weekly live zoom calls,b) a curated list of problems and exercises to practice the content, andc) a supportive online community to discuss problems with.If this is interesting to you, fill out the form below and I’ll let you know when the course is ready. LINK TO THE FORM: https://forms.gle/FgNSSbR29sEvP3vg7Follow me!Twitter: @00aleph00___MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.com00:00-00:40 Intro00:40-2:25 The Circle2:25-3:52 Elliptic Curves3:52-4:40 Announcement!4:40-7:30 Group Law7:30-9:20 What is the rank?9:20-11:20 Rank Conjecture11:20-13:36 Known results
Manifolds, explained intuitively Aleph 0 2025-02-18 | A high-level explanation of what a manifold is.
What is algebraic topology? Aleph 0 2025-02-10 | An introduction to homology, a key concept in algebraic topology. Take your personal data back with Incogni! Use code ALEPH at the link below and get 60% off an annual plan: http://incogni.com/aleph.Help fund future projects: patreon.com/c/aleph0An equally valuable form of support is to simply share the videos.This video is a collaboration with Trevor from the channel Mathemaniac. Check out his video here: youtube.com/watch?v=H6YofGjvZIQ&ab_channel=Mathemaniac.A HUGE thank you to Brendan Shuttleworth for working with me to make the script and storyboard for this video. You rock Brendan! NEWSLETTERI have a weekly math newsletter where I collate resources to self-study a specific topic and deliver it to your inbox. if this is interesting to you, fill out the form below to signup: https://forms.gle/Rt1f5StAj3yZtakE6SOURCES and REFERENCES for Further Reading:This video is a quick-and-dirty introduction to homology. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.(a) Video Lectures by Pierre Albin: youtube.com/playlist?list=PLpRLWqLFLVTCL15U6N3o35g4uhMSBVA2b (b) “Algebraic Topology” by Allen Hatcher: https://pi.math.cornell.edu/~hatcher/AT/ATpage.htmlCORRECTIONS:12:59 You take n-cycles/(n+1)-boundaries not n-chains.Follow me!Twitter: @00aleph00___MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.com00:00-1:11 Intro1:11-2:15 The Overall Idea2:25-4:25 Cells4:25-6:15 The Circle6:15-7:00 Sponsored Message7:00-8:55 The Torus8:55-11:10 Homology, in general11:10-14:37 Singular Homology
How to learn machine learning as a complete beginner: a self-study guide Aleph 0 2024-05-17 | A step-by-step roadmap of how to learn machine learning as a beginner.If you'd like to sign up for the Aleph 0 math / machine learning newsletter, fill out the form here: https://forms.gle/Rt1f5StAj3yZtakE6------------BOOK RECOMMENDATIONSGrokking Deep Learning by Andrew TraskThe 100-page Machine Learning Handbook by Andriy BurkovDeep Learning with PyTorch by Laura Mitchell, Sri Yogesh K, and Vishnu Subramanian1a. FEED-FORWARD NEURAL NETWORKSChapter 1 of Book by Michael Nielsen: neuralnetworksanddeeplearning.com/chap1.htmlGrokking Deep Learning (Chapters 2,3)The 100-page Machine Learning Handbook (Chapter 3.1, 3.2, Chapter 6)1b. GRADIENT DESCENT / BACKPROPAGATIONGrokking Deep Learning (Chapters 4,6)Chapter 2 of Book by Michael Nielsen: neuralnetworksanddeeplearning.com/chap2.html 2. CONVOLUTIONAL NEURAL NETWORKSTwo videos by Computerphile:Blurs and filters: youtube.com/watch?v=C_zFhWdM4ic&ab_channel=ComputerphileEdge detection: youtube.com/watch?v=uihBwtPIBxM&ab_channel=ComputerphileIntro to CNNs: youtube.com/watch?v=py5byOOHZM8&t=240s&ab_channel=ComputerphileDeep Learning with PyTorch (Chapter 5)3. RECURRENT NEURAL NETWORKSGrokking Deep Learning (Chapters 11 and 12)Video by Serrano Academy: youtube.com/watch?v=UNmqTiOnRfg&ab_channel=Serrano.AcademyStat Quest: youtube.com/watch?app=desktop&v=AsNTP8Kwu80&ab_channel=StatQuestwithJoshStarmer4. AUTOENCODERSDeep Learning with Pytorch (Chapter 6).Video playlist by Digital Sreeni:youtube.com/playlist?list=PLZsOBAyNTZwb-uK_a6ywrU3t0hy80G5QP 5. REINFORCEMENT LEARNINGDeep Learning with Pytorch (Chapter 9).6. ATTENTIONBlog post by Jay Alammar: jalammar.github.io/illustrated-transformer Lecture by Stanford Online: youtube.com/watch?v=XfpMkf4rD6E&t=252s&ab_channel=StanfordOnlineIntro: (0:00)Three book recommendations: (0:53)Feed-Forward Neural Networks: (2:06)Convolutional Neural Networks: (4:12)Recurrent Neural Networks: (5:21)Autoencoders: (6:36)Reinforcement Learning: (7:20)Attention: (7:54)General Tips: (9:06)
The shocking connection between complex numbers and geometry. Aleph 0 2024-04-26 | A peek into the world of Riemann surfaces, and how complex analysis is algebra in disguise. Secure your privacy with Surfshark! Enter coupon code ALEPH for an extra 3 months free at https://surfshark.deals/ALEPH.Help fund future projects: patreon.com/aleph0An equally valuable form of support is to simply share the videos.SOURCES and REFERENCES for Further Reading:This video is a quick-and-dirty introduction to Riemann Surfaces. But as with any quick introduction, there are many details that I gloss over. To learn these details rigorously, I've listed a few resources down below.(a) Complex Analysis To learn complex analysis, I really like the book "Visual Complex Functions: An Introduction with Phase Portraits" by Elias Wegert. It explains the whole subject using domain coloring front and center. Another one of my favorite books is "A Friendly Approach To Complex Analysis" by Amol Sasane and Sara Maad Sasane. I think it motivates all the concepts really well and is very thoroughly explained. (b) Riemann Surfaces and Algebraic Curves A beginner-friendly resource to learn this is "A Guide to Plane Algebraic Curves" by Keith Kendig. It starts off elementary with lots of pictures and visual intuition. Later on in the book, it talks about Riemann surfaces. A more advanced graduate book is "Algebraic Curves and Riemann Surfaces" by Rick Miranda. SOCIALSFollow me!Twitter: @00aleph00___MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.com00:00-00:54 Intro00:55-04:30 Complex Functions4:31-5:53 Riemann Sphere5:54-6:50 Sponsored Message6:51-11:06 Complex Torus11:07-11:50 Riemann Surfaces12:11-13:53 Riemann's Existence Theorem
What is a hole? Aleph 0 2023-12-29 | An introduction to the fundamental group, a key concept in algebraic topology. This video is sponsored by Brilliant. To try it out for free for 30 days, head to brilliant.org/Aleph0/. The first 200 people to sign up will get 20% off a yearly subscription. Help fund future projects: patreon.com/aleph0.An equally valuable form of support is to simply share the videos.A HUGE thank you to Waleed Qaisar for working with me to make the script and storyboard for this video. You rock Waleed! And thank you to Davide Radaelli for patiently listening to the script and offering helpful feedback!CORRECTIONS:At the start of the video, I said that Poincare’s paper Analysis Situs was published in 1985. This is a typo - the paper was published in 1895. Thanks to those who spotted the error.SOURCES and REFERENCES for Further Reading:This video is a quick-and-dirty introduction to the fundamental group. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.(a) Video Lectures by Pierre Albin: youtube.com/playlist?list=PLpRLWqLFLVTCL15U6N3o35g4uhMSBVA2b(b) “Algebraic Topology” by Allen Hatcher: https://pi.math.cornell.edu/~hatcher/AT/ATpage.htmlFollow me!Twitter: @00aleph00___MUSIC CREDITS:Music: purple-planet.comSong: Thinking Ahead
What is algebraic geometry? Aleph 0 2023-10-17 | Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry. Help fund future projects here: patreon.com/aleph0An equally valuable form of support is to simply share the videos.----A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video! And another thank you to Davide Radaelli for helpful conversations when making this video. ----CORRECTIONS:At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2. SOURCES and REFERENCES for Further Reading!(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.----MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.comFollow me!Twitter: @00aleph00Instagram: @00aleph00What is algebraic geometry?: (0:00)Coordinate Ring: (3:04)How algebra detects reducibility: (3:54)How algebra detects a node: (5:15)Schemes!: (8:00)
The unsolvable problem that launched a revolution in set theory Aleph 0 2023-02-27 | An introduction to the Continuum Hypothesis - a problem in set theory that cannot be proved correct or incorrect. _______Help fund future projects: patreon.com/aleph0An equally valuable form of support is to simply share the videos.A HUGE thank you to Luciano Salvetti, a graduate student at the University of Toronto in set theory, for helping me make this video!_____MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.comFollow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)Continuum Hypothesis: (0:59)What is Independence?: (2:04)ZFC Axioms: (3:15)Model of ZFC: (4:03)Godel's Strategy: (5:23)Cohen's Strategy: (5:35)
The bridge between number theory and complex analysis Aleph 0 2022-04-14 | How the discoveries of Ramanujan in 1916, combined with the insights of Eichler and Shimura in the 50's, led to the proof of Fermat's Last Theorem. Help fund future projects: patreon.com/aleph0An equally valuable form of support is to simply share the videos.SOURCES and REFERENCES for Further Reading!This video is a quick-and-dirty introduction to modular forms and elliptic curves. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.(a) ELLIPTIC CURVESThe book "Elliptic Curves: Number theory and cryptography" by Lawrence Washington is really good for self-study. It also has tons of numerical examples, making it good for self-study. The subject only really clicked for me after I read this book, so I'd highly recommend reading it.Professor Alvaro Lozano-Robledo also has a wonderful YouTube playlist which explains all the details of the subject rigorously: youtube.com/playlist?list=PLYpVTXjEi1oe1OeAllJpNhFoI4B7Ws8Yl(b) MODULAR FORMSFor modular forms, a great book is "Modular Forms: A Classical And Computational Introduction" by Lloyd Kilford. It also has plenty of numerical examples and you can also code up a bunch of the sections as well, which makes it nice to work through. (c) EICHLER SHIMURA THEORY (how to go from modular forms to elliptic curves)The book "Elliptic Curves" by Anthony Knapp (see Chapter 11: "Eichler Shimura Theory") contains the main content of this video with the integrating and lattices and all that. This is a dense book, but it is really beautifully written. The first chapter contains an extended numerical example that illustrates how to go from modular forms to elliptic curves. I wouldn't read this book as an introduction, because it's very comprehensive and can be a little overwhelming. But rather, it's great as a second pass after reading the intro books I mentioned at the start. (d) STRATEGY OF WILES' PROOF (how to go from elliptic curves to modular forms)The book "Elliptic Curves, Modular Forms, and the Proof of Fermat's Last Theorem" edited by John Coates and ST Yau has a full rigorous explanation of Wiles' proof in the first chapter. It is very dense, and it requires a solid grounding in algebraic number theory (see the last video in this channel for resources to learn this). The chapter describes all the new techniques that Wiles invented essentially from scratch to tackle Taniyama-Shimura. The key to Wiles' approach was a technique called a "modularity lifting theorem". This is not easy reading: it is aimed at graduate students and researchers in number theory. But it is beatifully written and by far the clearest rigorous exposition of FLT I've seen so far.If you really want to know: what are the 'curved arcs' from 2:40? The rigorous definition is: the "curved arc" is really a geodesic connecting two cusps that are equivalent under the action of Gamma_0(11). Equivalently, it is a homology class (with integral coefficients) in the modular curve X_0(11). These are the details you would find in "Elliptic Curves" by Knapp, see part (c) above.-----MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.comTHANK YOUs:Extra special thanks to Davide Radaelli and Grant Sanderson for feedback and helpful conversations while making this video.Follow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)Eichler-Shimura: (2:04)From Lattices to Number Theory: (3:21)Counting Solutions: (5:00)Taniyama-Shimura: (7:21)
Algebraic number theory - an illustrated guide | Is 5 a prime number? Aleph 0 2022-03-10 | This video is an introduction to Algebraic Number Theory, and a subfield of it called Iwasawa Theory. It describes how prime numbers factor in infinite towers of number rings. Help fund future projects: patreon.com/aleph0An equally valuable form of support is to simply share the videos.Minor corrections:at 4:58: I should have said "closed under addition and *subtraction*" instead of "closed under addition and multiplication." The text on the screen is correct.at 16:32: Instead of the "power of p dividing the class number", this should read the "p-part of the class number". SOURCES and REFERENCES for Further Reading!This video is a quick-and-dirty introduction to Algebraic Number Theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.(a) ALGEBRAIC NUMBER THEORYAlgebraic Number Theory notes by Professor Robert Ash: https://faculty.math.illinois.edu/~r-ash/ANT.html. These notes are quite thorough and they have a lot of core material needed for algebraic number theory. Algebraic Number Theory videos by Billy Woods: youtube.com/playlist?list=PLSibAQEfLnTwq2-zCB-t9v2WvnnVKd0wn. These videos are *very* well made and they have all the core intuitions in them. I personally loved how visual they are; they make the concepts feel a lot more visceral.PREREQUISITES: The prerequisites for learning Algebraic Number Theory are: group theory, ring theory, and Galois theory. It's possible to get a basic non-rigorous feel for the subject without these prerequisites, which is what I tried to do in this video. But if you want to know the details (for example: you might have asked: what *exactly* is a number ring?), then these prerequisites are essential. To learn these prereqs, check out the previous video on this channel, "How to self study math", where there are a bunch of resources to learn these prereqs. (b) IWASAWA THEORY Introduction to Cyclotomic Fields by Lawrence Washington: This book is AMAZING! To see Iwasawa theory in action, skip directly to chapter 13, Iwasawa's theory of Zp extensions. (You don't need to read the book in sequential order because the chapters are largely independent.) The proof of this theorem is just miraculous. PREREQUISITES: Algebraic Number Theory (that is, the previous section). -----WORKS CITEDThe data of class numbers for the cyclotomic number rings was from here: oeis.org/A055513This list is only for cyclotomic number rings (Z adjoin a p-th root of unity) where p is a *prime* number.The two examples of class numbers (class numbers 100 and 2000) was from the L-functions and Modular Forms database:lmfdb.org/NumberField-----MUSIC CREDITS:The song is “Taking Flight”, by Vince Rubinetti.vincentrubinetti.comTHANK YOUs:Extra special thanks to Davide Radaelli and Vivek Verma (@vcubingx) for helpful conversations while making this video. you guys rock!!Follow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)Number Rings: (1:41)Ideals: (4:46)Unique Factorization: (8:55)Class Numbers: (11:41)Iwasawa Theory: (14:53)Thank you!: (18:37)Learning Resources: (18:49)Patreon: (19:45)
How to self study pure math - a step-by-step guide Aleph 0 2021-12-24 | This video has a list of books, videos, and exercises that goes through the undergrad pure mathematics curriculum from start to finish. ---REAL ANALYSISBook: “Understanding Analysis” by Stephen Abbott.Videos: Lectures by Francis Su (youtube.com/playlist?list=PL0E754696F72137EC)LINEAR ALGEBRABook: “Linear Algebra Done Right” by Sheldon AxlerVideos: Sheldon Axler’s Playlist(youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmSinFV0wEir2Vw)POINT SET TOPOLOGYOnline Notes with Problems: MAT327 Course Notes (http://www.math.toronto.edu/ivan/mat327/?resources)COMPLEX ANALYSISIntro Book: “Visual Complex Functions: an Introduction with Phase Portraits” by Elias WegertMore Technical Book: “Complex Analysis” by Serge LangVideos: Wesleyan University Playlist (youtube.com/playlist?list=PL_onPhFCkVQjdQTbG0eQk42eH0RaBoYJf)GROUP THEORY Book: “Topics in Algebra” by Herstein (Chapter 2)Videos: Lectures by Benedict Gross (youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5)GALOIS THEORYNotes by Tom Leinster: maths.ed.ac.uk/~tl/gt/gt.pdfDIFFERENTIAL GEOMETRYBook: Introduction to Differentiable Manifolds and Riemannian Geometry by BoothbyALGEBRAIC TOPOLOGYBook: Algebraic Topology by Allen Hatcher (available for free on his website: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)Videos: Lectures by Pierre Albin (youtube.com/playlist?list=PL41FDABC6AA085E78)Intro: (0:00)Linear Algebra: (0:36)Real Analysis: (2:20)Point Set Topology: (3:19)Complex Analysis: (4:09)Group Theory: (5:46)Galois Theory: (6:54)Differential Geometry: (7:23)Algebraic Topology: (8:44)
What is the square root of two? | The Fundamental Theorem of Galois Theory Aleph 0 2021-11-25 | This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups. __SOURCES and REFERENCES for Further Reading!This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below. (a) Galois Theory Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: maths.ed.ac.uk/~tl/gt/gt.pdf)This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. (youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt) (b) Group TheoryGroup Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here: youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5)MUSIC CREDITS:The song is called “Taking Flight”, by Vince Rubinetti.vincentrubinetti.comTHANK YOUs:Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video!SOFTWARE USED:Adobe Premiere Pro for EditingFollow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)What is the square root of 2?: (1:08)Fields and Automorphisms: (6:04)Examples: (8:55)Group Theory: (16:34)The Fundamental Theorem: (18:25)
The Insolvability of the Quintic Aleph 0 2021-02-20 | This video is an introduction to Galois Theory, which spells out a beautiful connection between fields and their Galois Groups. Using this, we'll prove that the quintic has no general formula in radicals.__SOURCES and REFERENCES for further reading!As with any quick introduction, there are details that I gloss over for the sake of brevity. If you’d like to learn these details more rigorously, I've listed a few resources down below.(a) Group TheoryThis playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5(b) Galois Theory“Galois Theory” by David Cox is a skinny little book that goes through the main theorems of Galois Theory. The first few chapters give historical background, and the remaining chapters lay out the key theorems and applications.“Galois Theory for Beginners” by Jorg Bewersdorff explains the insolvability of the quintic in intuitive terms. It doesn’t assume any prior background knowledge, and all chapters but the last can be understood without group theory. The last chapter formulates the theorem using the language of groups and field extensions, but it explains all the definitions as it goes along.This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt__SOME NOTES ABOUT THE VIDEO CONTENT:Galois Theory is normally introduced at the end of a course in abstract algebra, and for good reason. There’s a lot of technical machinery involved, and I’ve deliberately omitted certain parts that I felt were not immediately relevant towards the insolvability of the quintic. If you’re interested in seeing how the ideas in this video differ from the standard treatment, read on.1) (The Galois Group.) In this video, we define the Galois Group of a *polynomial*. In the modern treatment, however, we normally talk about the Galois Group of a *field extension* (not a polynomial), and we define it as the set of all automorphisms of the top field that fix the bottom field pointwise. When I refer to the Galois Group of a polynomial, I am referring to the Galois Group of its *splitting field*, viewed as a field extension of the rational numbers. But obviously, that’s quite a mouthful. That’s why I took the route I did; I felt that introducing all this machinery – automorphisms, splitting fields, etc. – would have obscured the main point of the video. 2) (Normal Subgroups.) We observed in the final example that the subgroup partitions the group table into squares, and many of the squares had the same elements, just in a different order. A subgroup that splits the group table into squares so that any two squares are either equal or mutually disjoint is called a “normal subgroup”. This is a non-standard definition of a normal subgroup – although, it is equivalent to the standard definition. I felt that this definition of a normal subgroup was a lot more intuitive than the standard definition (which, for the record, I still find quite mysterious, even after having taken a course in group theory!) ___MUSIC CREDITS:Music: purple-planet.comSong: Thinking AheadSOFTWARE USED:Adobe Premiere Pro for EditingFollow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)Field Extensions: (0:48)Galois Groups: (3:22)The Insolvability of the Quintic: (8:20)
The derivative isnt what you think it is. Aleph 0 2020-11-03 | The derivative's true nature lies in its connection with topology. In this video, we'll explore what this connection is through two fields of algebraic topology: homology and cohomology. __SOURCES and REFERENCES for Further Reading!In this video, I give a quick-and-dirty introduction to differential forms and cohomology. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below that I found helpful. Differential Forms: The book “A Geometric Approach to Differential Forms” by David Bachman is a treasure. Instead of leading with the formalism, it gives a nice intuitive picture of what forms do, and then provides the precise definitions.Homology: This lecture series by Pierre Albin is a beauty. There are a few lectures that cover homology in a slow, accessible way with lots of computational examples. (youtube.com/watch?v=I2GbdKDN9zg&t=3s)Cohomology and De Rham’s Theorem: The amazing Fredrich Schuller (who I have raved about in previous videos) has a crystal-clear lecture on Cohomology. (youtube.com/watch?v=QLnzIOGIvfo)More on Cohomology: I also came across the book “Differential Forms in Algebraic Topology” by Bott and Tu, which starts off with De Rham’s Theorem and goes into much more depth about the relationship between the boundary and the exterior derivative. This is quite advanced (read: I only got through the first few chapters before I stopped understanding what all the words meant ...), but if you’re up for it, read along!_____LINKSThe Work of Emmy Noether, one of the great pioneers of Homology: britannica.com/biography/Emmy-NoetherThurston’s Paper about the Interpretations of the Derivative: arxiv.org/abs/math/9404236___MUSIC CREDITS:Music: purple-planet.comSong: Thinking AheadSOFTWARE USED:Adobe Premiere Elements for EditingBlender 2.8 for AnimationsFollow me!Twitter: @00aleph00Instagram: @00aleph00Intro: (0:00)Homology: (1:08)Cohomology: (3:41)De Rham's Theorem: (7:45)The Punch Line: (9:02)
Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem Aleph 0 2020-07-26 | Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!QUESTIONS:1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)? 5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this? 6. Does anyone understand Andrew Wiles’ paper? LOLSOURCES I USED TO STUDY:Keith Conrad’s Lectures on Modular Forms (8 part video series): youtube.com/watch?v=LolxzYwN1TQKeith Conrad’s Notes on Modular Forms: https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo. (The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)“The Arithmetic of Elliptic Curves” by Joseph SilvermanHOMEWORK IDEA CREDIT goes to Looking Glass Universe! SAGE RESOURCES:“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html Download SAGE: sagemath.org/download.htmlProof of the Hasse-Weil Bound on Terry Tao’s Blog: terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-boundOTHER VIDEOS ON THESE TOPICS:Numberphile Playlist: youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlWElliptic Curves and Modular Forms: youtube.com/watch?v=A8fsU97g3tgSOFTWARE USED TO MAKE THIS VIDEO:SAGE for the code and the graphsgithub.com/hernanat/dcolor for domain coloringAdobe Premiere Elements For Video EditingMUSIC:Music Info: Documentary - AShamaluevMusic.Music Link: ashamaluevmusic.comFollow me!Twitter: twitter.com/00aleph00Instagram: instagram.com/00aleph00Intro: (0:00)Elliptic Curves: (0:58)Modular Forms: (3:26) Taniyama Shimura Conjecture: (7:26)Fermat's Last Theorem: (8:02) Questions for you!: (8:51)
Navier Stokes Equation | A Million-Dollar Question in Fluid Mechanics Aleph 0 2020-06-03 | The Navier-Stokes Equations describe everything that flows in the universe. If you can prove that they have smooth solutions, you'll win a million dollars.---Official Problem Statement:claymath.org/millennium-problems/navier%E2%80%93stokes-equationA very informative talk by Dr. Edriss Titi:youtube.com/watch?v=VH4oawCiHPU---Follow me!Twitter: twitter.com/00aleph00Instagram: instagram.com/00aleph00Facebook: facebook.com/00aleph00__Music Info: Documentary - AShamaluevMusic.Music Link: ashamaluevmusic.com
Stokes Theorem on Manifolds Aleph 0 2020-05-03 | Stokes' Theorem is the crown jewel of differential geometry. It extends the fundamental theorem of Calculus to manifolds in n-dimensional space.---This video aims to give an intuitive discussion of Stokes' Theorem, without the complicated equations and formalism. For those interested in the details, here's a thorough treatment of the topic: arxiv.org/abs/1604.07862(Be warned, this rabbit hole goes very deep!)---Follow me!Twitter: twitter.com/00aleph00Instagram: instagram.com/00aleph00Facebook: facebook.com/00aleph00__Pictures:Boundary and Green's Theorem - "Calculus: Early Transcendentals - 8th Edition" by James Stewart Exterior Derivative - "A Geometric Approach to Differential Forms" by David BachmanMusic:Relaxing Guitar Music - Acoustic - Calming Music for Stress Relief, Studying
