title

Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190

description

Jordan Ellenberg is a mathematician and author of Shape and How Not to Be Wrong. Please support this podcast by checking out our sponsors:
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EPISODE LINKS:
Jordan's Website: http://www.jordanellenberg.com
Jordan's Twitter: https://twitter.com/JSEllenberg
PODCAST INFO:
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Apple Podcasts: https://apple.co/2lwqZIr
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Full episodes playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4
Clips playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41
OUTLINE:
0:00 - Introduction
1:01 - Mathematical thinking
4:38 - Geometry
9:15 - Symmetry
19:46 - Math and science in the Soviet Union
27:26 - Topology
42:15 - Do we live in many more than 4 dimensions?
46:45 - How many holes does a straw have
56:11 - 3Blue1Brown
1:01:57 - Will AI ever win a Fields Medal?
1:10:22 - Fermat's last theorem
1:27:41 - Reality cannot be explained simply
1:33:25 - Prime numbers
1:54:54 - John Conway's Game of Life
2:06:46 - Group theory
2:10:03 - Gauge theory
2:18:05 - Grigori Perelman and the Poincare Conjecture
2:28:17 - How to learn math
2:35:26 - Advice for young people
2:37:31 - Meaning of life
SOCIAL:
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- Support on Patreon: https://www.patreon.com/lexfridman

detail

{'title': 'Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190', 'heatmap': [{'end': 487.188, 'start': 387.945, 'weight': 0.707}], 'summary': "Presents a conversation with jordan ellenberg, delving into the power of mathematics, the interplay of algebra and geometry, the intertwining of romanticism and mathematics in the 19th century, mental purification and modern mathematics, simplicity in mathematics and philosophy, prime numbers, john conway's legacy, group theory, challenges in visualizing mathematics, and geometric proofs, emphasizing collaborative nature of mathematical advances and problem-driven learning.", 'chapters': [{'end': 473.561, 'segs': [{'end': 29.431, 'src': 'embed', 'start': 0.089, 'weight': 0, 'content': [{'end': 3.031, 'text': 'The following is a conversation with Jordan Ellenberg,', 'start': 0.089, 'duration': 2.942}, {'end': 13.82, 'text': 'a mathematician at University of Wisconsin and an author who masterfully reveals the beauty and power of mathematics in his 2014 book How Not To Be Wrong,', 'start': 3.031, 'duration': 10.789}, {'end': 22.947, 'text': 'and his new book, just released recently, called Shape The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else.', 'start': 13.82, 'duration': 9.127}, {'end': 29.431, 'text': 'Quick mention of our sponsors, Secret Sauce, ExpressVPN, Blinkist, and Indeed.', 'start': 23.467, 'duration': 5.964}], 'summary': 'Jordan ellenberg, mathematician, discusses his books and sponsors.', 'duration': 29.342, 'max_score': 0.089, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA89.jpg'}, {'end': 168.139, 'src': 'embed', 'start': 137.457, 'weight': 1, 'content': [{'end': 143.501, 'text': "It's hard to imagine doing mathematics without talking about mathematics and sort of thinking in propositions.", 'start': 137.457, 'duration': 6.044}, {'end': 146.723, 'text': "But, you know, maybe it's just because that's the way I do mathematics.", 'start': 143.902, 'duration': 2.821}, {'end': 148.405, 'text': "And maybe I can't imagine it any other way.", 'start': 146.783, 'duration': 1.622}, {'end': 148.885, 'text': "Right It's a.", 'start': 148.465, 'duration': 0.42}, {'end': 160.073, 'text': "Well, what about visualizing shapes, visualizing concepts to which language is not obviously attachable? Ah, that's a really interesting question.", 'start': 149.625, 'duration': 10.448}, {'end': 168.139, 'text': 'And one thing it reminds me of is one thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions.', 'start': 160.293, 'duration': 7.846}], 'summary': 'Mathematics involves language, propositions, and visualization, like dissection proofs.', 'duration': 30.682, 'max_score': 137.457, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA137457.jpg'}, {'end': 257.849, 'src': 'embed', 'start': 228.999, 'weight': 2, 'content': [{'end': 239.183, 'text': 'I think the process of manipulating the visual elements is the same as the process of manipulating the elements of language.', 'start': 228.999, 'duration': 10.184}, {'end': 245.685, 'text': 'And I think probably the manipulating, the aggregation, the stitching stuff together is the important part.', 'start': 239.523, 'duration': 6.162}, {'end': 247.626, 'text': "It's not the actual specific elements.", 'start': 245.845, 'duration': 1.781}, {'end': 251.747, 'text': "It's more like, to me, language is a process and math is a process.", 'start': 247.706, 'duration': 4.041}, {'end': 254.548, 'text': "It's not just specific symbols.", 'start': 251.807, 'duration': 2.741}, {'end': 257.849, 'text': "It's.. It's in action.", 'start': 255.128, 'duration': 2.721}], 'summary': 'Manipulating visual and language elements is a process, not specific symbols, but an action.', 'duration': 28.85, 'max_score': 228.999, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA228999.jpg'}, {'end': 306.694, 'src': 'embed', 'start': 279.177, 'weight': 5, 'content': [{'end': 283.861, 'text': "But I gotta ask you about geometry, and it's a prominent topic in your new book, Shape.", 'start': 279.177, 'duration': 4.684}, {'end': 291.967, 'text': "So for me, geometry is the thing, just like as you're saying, made me fall in love with mathematics when I was young.", 'start': 284.922, 'duration': 7.045}, {'end': 301.051, 'text': 'So being able to prove something visually just did something to my brain that it had this.', 'start': 293.368, 'duration': 7.683}, {'end': 306.694, 'text': 'it planted this hopeful seed that you can understand the world like perfectly.', 'start': 301.051, 'duration': 5.643}], 'summary': 'Geometry in new book, shape, made me fall in love with mathematics when young.', 'duration': 27.517, 'max_score': 279.177, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA279177.jpg'}, {'end': 356.11, 'src': 'embed', 'start': 329.146, 'weight': 4, 'content': [{'end': 333.59, 'text': 'So how do you think about geometry?', 'start': 329.146, 'duration': 4.444}, {'end': 336.012, 'text': 'Why is it a special field in mathematics?', 'start': 334.01, 'duration': 2.002}, {'end': 339.114, 'text': 'And how did you fall in love with it?', 'start': 336.832, 'duration': 2.282}, {'end': 339.555, 'text': 'if you have?', 'start': 339.114, 'duration': 0.441}, {'end': 346.121, 'text': "wow, you've given me like a lot to say, and certainly the experience that you describe is so typical.", 'start': 340.055, 'duration': 6.066}, {'end': 347.842, 'text': "but there's two versions of it.", 'start': 346.121, 'duration': 1.721}, {'end': 351.746, 'text': 'um, you know, one thing i say in the book is that geometry is the cilantro of math.', 'start': 347.842, 'duration': 3.904}, {'end': 352.867, 'text': 'people are not neutral about it.', 'start': 351.746, 'duration': 1.121}, {'end': 356.11, 'text': "there's people who are like, who, like you, are like the rest of it.", 'start': 352.867, 'duration': 3.243}], 'summary': "Geometry is a distinct field in math, evoking strong reactions; it's like the cilantro of math.", 'duration': 26.964, 'max_score': 329.146, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA329146.jpg'}, {'end': 453.436, 'src': 'embed', 'start': 429.994, 'weight': 3, 'content': [{'end': 437.357, 'text': 'And, if you like, just by kind of like focusing in and out, just by kind of looking at this box, looking at this rectangle, I was like well,', 'start': 429.994, 'duration': 7.363}, {'end': 444.88, 'text': "there's six rows of eight holes each, but there's also eight columns of six holes each.", 'start': 437.357, 'duration': 7.523}, {'end': 448.993, 'text': 'Whoa So eight sixes and six eights.', 'start': 445.641, 'duration': 3.352}, {'end': 451.875, 'text': "It's just like the dissection proofs we were just talking about, but it's the same holes.", 'start': 449.033, 'duration': 2.842}, {'end': 453.436, 'text': "It's the same 48 holes.", 'start': 452.335, 'duration': 1.101}], 'summary': 'Analyzing a rectangle with 6 rows of 8 holes and 8 columns of 6 holes reveals 48 holes, similar to dissection proofs.', 'duration': 23.442, 'max_score': 429.994, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA429994.jpg'}], 'start': 0.089, 'title': 'Mathematics and its wonders', 'summary': 'Delves into the power of mathematics through a conversation with jordan ellenberg, explores the connection between mathematical thinking and language, and discusses the allure of geometry in providing visual proofs and a different perspective on understanding the world.', 'chapters': [{'end': 44.839, 'start': 0.089, 'title': 'Unveiling the power of mathematics', 'summary': "Features a conversation with jordan ellenberg, a mathematician at university of wisconsin, discussing his books 'how not to be wrong' and 'shape the hidden geometry of information', while also mentioning sponsors such as secret sauce, expressvpn, blinkist, and indeed.", 'duration': 44.75, 'highlights': ["Jordan Ellenberg, a mathematician at University of Wisconsin, discusses his books 'How Not To Be Wrong' and 'Shape The Hidden Geometry of Information'.", 'The conversation briefly mentions sponsors including Secret Sauce, ExpressVPN, Blinkist, and Indeed.', 'The speaker shares a personal anecdote about how geometry sparked their love for mathematics.']}, {'end': 278.657, 'start': 45.539, 'title': 'Math and language connection', 'summary': 'Explores the connection between mathematical thinking and language, questioning whether mathematical output is akin to linguistic output and delving into the role of visualization in mathematical proofs.', 'duration': 233.118, 'highlights': ["The process of producing mathematical output feels similar to the process of uttering language, and it's hard to imagine doing mathematics in a completely non-linguistic way, implying the strong connection between mathematical thinking and language.", 'The discussion of dissection proofs and the comparison of manipulating visual elements to manipulating elements of language, suggesting that the process of manipulating, aggregating, and stitching elements is more important than the specific elements themselves.', 'The concept that language and math are processes created through action and change, with the idea of a proof being just a certain destination in the evolution of ideas, highlighting the continuous nature of mathematical and linguistic exploration.']}, {'end': 473.561, 'start': 279.177, 'title': 'The magic of geometry', 'summary': 'Explores the allure of geometry in mathematics, discussing its ability to provide visual proofs and a different perspective on understanding the world, with personal anecdotes and contrasting experiences to highlight its divisive nature.', 'duration': 194.384, 'highlights': ['The visual proof of a rectangular array of holes, with six rows of eight holes and eight columns of six holes, providing a unique perspective on geometry and leading to the realization of the same number of holes regardless of how they are counted, showcasing the power of visual proofs in understanding mathematical concepts.', "The contrasting experiences and perspectives on geometry, with some individuals finding it captivating and transformative, while others struggle to comprehend its significance, highlighting the divisive nature of geometry as the 'cilantro of math' where people are not neutral about it.", 'The personal anecdote about the impact of geometry, where the speaker, as a child, had a revelatory moment while observing a rectangular array of holes, leading to a profound realization about the nature of geometry and its ability to provide new perspectives on mathematical concepts.']}], 'duration': 473.472, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA89.jpg', 'highlights': ["Jordan Ellenberg discusses 'How Not To Be Wrong' and 'Shape The Hidden Geometry of Information'.", 'The strong connection between mathematical thinking and language is implied.', 'The process of manipulating, aggregating, and stitching elements is more important than the specific elements themselves.', 'Visual proof of a rectangular array of holes provides a unique perspective on geometry.', "Geometry is described as the 'cilantro of math' with divisive perspectives.", "Geometry's ability to provide new perspectives on mathematical concepts is highlighted."]}, {'end': 1369.968, 'segs': [{'end': 549.507, 'src': 'embed', 'start': 525.453, 'weight': 1, 'content': [{'end': 535.898, 'text': 'in fact, that moment exactly encapsulates the intertwining of algebra and geometry, this algebraic fact that, well, in the instance,', 'start': 525.453, 'duration': 10.445}, {'end': 540.141, 'text': 'eight times six is equal to six times eight, but in general, that whatever two numbers you have,', 'start': 535.898, 'duration': 4.243}, {'end': 543.383, 'text': "you multiply them one way and it's the same as if you multiply them in the other order.", 'start': 540.141, 'duration': 3.242}, {'end': 549.507, 'text': 'It attaches it to this geometric fact about a rectangle, which in some sense makes it true.', 'start': 544.804, 'duration': 4.703}], 'summary': 'Algebra and geometry intertwine in the commutative property: 8x6=6x8.', 'duration': 24.054, 'max_score': 525.453, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA525453.jpg'}, {'end': 642.964, 'src': 'embed', 'start': 613.554, 'weight': 2, 'content': [{'end': 617.575, 'text': 'one of my first loves in mathematics, what i thought about a lot when i was in college.', 'start': 613.554, 'duration': 4.021}, {'end': 629.317, 'text': 'but the notion of symmetry is actually much more general than the things that we would call symmetry if we were looking at like a classical building or a painting or or something like that.', 'start': 617.575, 'duration': 11.742}, {'end': 642.964, 'text': 'um, you know, nowadays in math we could use a symmetry to refer to any kind of transformation of an image or a space or an object.', 'start': 629.317, 'duration': 13.647}], 'summary': 'Symmetry is a broad concept in mathematics, extending beyond traditional examples, with applications in various transformations.', 'duration': 29.41, 'max_score': 613.554, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA613554.jpg'}, {'end': 805.291, 'src': 'embed', 'start': 775.205, 'weight': 0, 'content': [{'end': 782.332, 'text': 'work. So this question of like what are the symmetries and which things that you want to study are invariant under those symmetries is absolutely fundamental.', 'start': 775.205, 'duration': 7.127}, {'end': 785.094, 'text': "Boy, this is getting a little abstract, right? It's not at all abstract.", 'start': 782.352, 'duration': 2.742}, {'end': 791.22, 'text': 'I think this is, this is completely central to everything I think about in terms of artificial intelligence.', 'start': 785.134, 'duration': 6.086}, {'end': 794.903, 'text': "I don't know if you know about the MNIST dataset, what's handwritten digits.", 'start': 791.24, 'duration': 3.663}, {'end': 805.291, 'text': "Yeah, And uh, You know, I don't smoke much weed or any really, but it certainly feels like it when I look at MNIST and think about this stuff,", 'start': 794.983, 'duration': 10.308}], 'summary': 'Understanding symmetries and invariance is crucial for artificial intelligence, like in the case of the mnist dataset.', 'duration': 30.086, 'max_score': 775.205, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA775205.jpg'}, {'end': 1202.966, 'src': 'embed', 'start': 1175.317, 'weight': 4, 'content': [{'end': 1179.86, 'text': "This is happening to France and they're trying to kind of like instantly modernize.", 'start': 1175.317, 'duration': 4.543}, {'end': 1186.164, 'text': "That's fascinating that the humans and mathematics are intricately connected to the history of humans.", 'start': 1180.38, 'duration': 5.784}, {'end': 1194.79, 'text': 'The Cold War is, I think, fundamental to the way people saw science and math in the Soviet Union.', 'start': 1186.805, 'duration': 7.985}, {'end': 1198.193, 'text': "I don't know if that was true in the United States, but certainly was in the Soviet Union.", 'start': 1194.81, 'duration': 3.383}, {'end': 1202.966, 'text': 'It definitely was, and I would love to hear more about how it was in the Soviet Union.', 'start': 1198.661, 'duration': 4.305}], 'summary': 'France aims to modernize; cold war influenced science in soviet union.', 'duration': 27.649, 'max_score': 1175.317, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA1175317.jpg'}, {'end': 1329.266, 'src': 'embed', 'start': 1301.716, 'weight': 5, 'content': [{'end': 1307.098, 'text': "for example, there's like these, beautiful poetic writing about the game of baseball.", 'start': 1301.716, 'duration': 5.382}, {'end': 1312.68, 'text': 'The same was the feeling with mathematics and science in the Soviet Union, and it was in the air.', 'start': 1307.438, 'duration': 5.242}, {'end': 1317.222, 'text': 'Everybody was forced to take high-level mathematics courses.', 'start': 1313.26, 'duration': 3.962}, {'end': 1318.862, 'text': 'You took a lot of math.', 'start': 1317.482, 'duration': 1.38}, {'end': 1322.784, 'text': 'You took a lot of science and a lot of really rigorous literature.', 'start': 1319.163, 'duration': 3.621}, {'end': 1325.405, 'text': 'The level of education.', 'start': 1324.124, 'duration': 1.281}, {'end': 1329.266, 'text': "in Russia, this could be true in China, I'm not sure.", 'start': 1325.405, 'duration': 3.861}], 'summary': 'High-level mathematics and science education was enforced in the soviet union, possibly also in china.', 'duration': 27.55, 'max_score': 1301.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA1301716.jpg'}], 'start': 474.321, 'title': 'The interplay of algebra and geometry', 'summary': 'Explores the transformative understanding of symmetry in algebra and geometry, emphasizing its fundamental role in contemporary mathematics and relevance in artificial intelligence, including its role in the mnist dataset. it also delves into the complexities of symmetry, cognition, and mathematics, and the historical connection between math, science, and geopolitical events.', 'chapters': [{'end': 811.535, 'start': 474.321, 'title': 'The interplay of algebra and geometry', 'summary': 'Delves into the transformative moment of understanding the symmetry in algebra and geometry, the broader concept of symmetry in mathematics, and its fundamental role in contemporary mathematics, emphasizing the importance of symmetries and invariance. it also touches on the relevance of symmetry in artificial intelligence, specifically in the context of the mnist dataset.', 'duration': 337.214, 'highlights': ["The transformative moment of understanding the symmetry in algebra and geometry, particularly encompassing the intertwining of algebraic fact and geometric representation, exemplified through the realization of the commutative property (e.g., 8 times 6 equals 6 times 8) and its connection to a rectangle, highlighting the fundamental nature of this realization in the speaker's career as an algebraic geometer.", "The broader concept of symmetry in mathematics, extending beyond classical symmetries, encompassing the study of various transformations of images, spaces, or objects, and the exploration of the abstract study of possible combinations of symmetries known as group theory, reflecting the speaker's early passion in mathematics during college.", 'The fundamental role of symmetry in contemporary mathematics, emphasizing the question of when two things are considered the same, the concept of invariance under symmetries, and the importance of translation invariance in theories, underlining its significance in mathematical research and its relevance to artificial intelligence, specifically in understanding the MNIST dataset and the similarities between handwritten digits.']}, {'end': 1369.968, 'start': 812.256, 'title': 'Symmetry, cognition, and mathematics', 'summary': 'Delves into the complexities of symmetry, cognition, and mathematics, exploring the role of symmetry in ai, the challenges in formalizing human cognition, and the historical connection between math, science, and geopolitical events.', 'duration': 557.712, 'highlights': ['The role of symmetry in AI and the challenge of formalizing human cognition The chapter explores the significance of symmetry in AI and the challenge of formalizing human cognition, raising questions about the types of symmetries needed to solve problems like handwritten digit recognition.', 'The historical connection between math, science, and geopolitical events The chapter discusses the historical context of math and science, highlighting the connection between geopolitical events like the Franco-Prussian War and the emphasis on math education in France.', 'The impact of mathematics and science in the Soviet Union and the romanticism around it The chapter explores the impact of mathematics and science in the Soviet Union, emphasizing the romanticism and societal significance attributed to excellence in these fields.']}], 'duration': 895.647, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA474321.jpg', 'highlights': ['The fundamental role of symmetry in contemporary mathematics, emphasizing the question of when two things are considered the same, the concept of invariance under symmetries, and the importance of translation invariance in theories, underlining its significance in mathematical research and its relevance to artificial intelligence, specifically in understanding the MNIST dataset and the similarities between handwritten digits.', "The transformative moment of understanding the symmetry in algebra and geometry, particularly encompassing the intertwining of algebraic fact and geometric representation, exemplified through the realization of the commutative property (e.g., 8 times 6 equals 6 times 8) and its connection to a rectangle, highlighting the fundamental nature of this realization in the speaker's career as an algebraic geometer.", "The broader concept of symmetry in mathematics, extending beyond classical symmetries, encompassing the study of various transformations of images, spaces, or objects, and the exploration of the abstract study of possible combinations of symmetries known as group theory, reflecting the speaker's early passion in mathematics during college.", 'The role of symmetry in AI and the challenge of formalizing human cognition The chapter explores the significance of symmetry in AI and the challenge of formalizing human cognition, raising questions about the types of symmetries needed to solve problems like handwritten digit recognition.', 'The historical connection between math, science, and geopolitical events The chapter discusses the historical context of math and science, highlighting the connection between geopolitical events like the Franco-Prussian War and the emphasis on math education in France.', 'The impact of mathematics and science in the Soviet Union and the romanticism around it The chapter explores the impact of mathematics and science in the Soviet Union, emphasizing the romanticism and societal significance attributed to excellence in these fields.']}, {'end': 3215.662, 'segs': [{'end': 1435.438, 'src': 'embed', 'start': 1413.47, 'weight': 0, 'content': [{'end': 1423.753, 'text': 'But Alexander really lays out just how much the way people thought about math in those times in the early 19th century was wound up with, as you say,', 'start': 1413.47, 'duration': 10.283}, {'end': 1424.413, 'text': 'romanticism.', 'start': 1423.753, 'duration': 0.66}, {'end': 1427.094, 'text': "I mean, that's when the romantic movement takes place.", 'start': 1424.473, 'duration': 2.621}, {'end': 1432.836, 'text': 'And he really outlines how people were predisposed to think about mathematics in that way,', 'start': 1427.134, 'duration': 5.702}, {'end': 1435.438, 'text': 'because they thought about poetry that way and they thought about music that way.', 'start': 1432.836, 'duration': 2.602}], 'summary': 'In the early 19th century, the romantic movement influenced the way people thought about math, as described by alexander.', 'duration': 21.968, 'max_score': 1413.47, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA1413470.jpg'}, {'end': 1823.938, 'src': 'embed', 'start': 1800.302, 'weight': 1, 'content': [{'end': 1807.387, 'text': "he's always motivated by physics, but the physics drove him to need to think about spaces of higher dimension,", 'start': 1800.302, 'duration': 7.085}, {'end': 1810.569, 'text': 'and so he needed a formalism that was rich enough to enable him to do that.', 'start': 1807.387, 'duration': 3.182}, {'end': 1814.692, 'text': 'and once you do that, that formalism is also going to include things that are not physical.', 'start': 1810.569, 'duration': 4.123}, {'end': 1815.572, 'text': 'and then you have two choices.', 'start': 1814.692, 'duration': 0.88}, {'end': 1823.938, 'text': "you can be like oh well, that stuff's trash, or but and this is more the mathematicians frame of mind if you have a formalistic framework that like,", 'start': 1815.572, 'duration': 8.366}], 'summary': 'Physics led to consideration of higher dimensions and non-physical elements.', 'duration': 23.636, 'max_score': 1800.302, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA1800302.jpg'}, {'end': 2131.728, 'src': 'embed', 'start': 2105.397, 'weight': 2, 'content': [{'end': 2113.14, 'text': "I mean, how are you even supposed to think about the shape of a thing that doesn't have anything outside of it?", 'start': 2105.397, 'duration': 7.743}, {'end': 2116.645, 'text': "I mean, ah, but that's exactly what topology does.", 'start': 2114.144, 'duration': 2.501}, {'end': 2118.785, 'text': "Topology is what's called an intrinsic theory.", 'start': 2116.705, 'duration': 2.08}, {'end': 2120.285, 'text': "That's what's so great about it.", 'start': 2119.425, 'duration': 0.86}, {'end': 2125.927, 'text': 'This question about the mug, you could answer it without ever leaving the mug right?', 'start': 2120.305, 'duration': 5.622}, {'end': 2131.728, 'text': "Because it's a question about a loop drawn on the surface of the mug and what happens if it never leaves that surface.", 'start': 2125.947, 'duration': 5.781}], 'summary': 'Topology is an intrinsic theory that deals with shapes, allowing questions about objects to be answered without leaving the object.', 'duration': 26.331, 'max_score': 2105.397, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA2105397.jpg'}, {'end': 2552.169, 'src': 'embed', 'start': 2516.59, 'weight': 3, 'content': [{'end': 2521.493, 'text': 'So we live in the three-dimensional world, maybe with the time component four-dimensional,', 'start': 2516.59, 'duration': 4.903}, {'end': 2529.938, 'text': 'and then math allows us to go into high dimensions comfortably and explore the world from those perspectives.', 'start': 2521.493, 'duration': 8.445}, {'end': 2544.765, 'text': 'Is it possible? that the universe is many more dimensions than the ones we experience as human beings.', 'start': 2535.381, 'duration': 9.384}, {'end': 2552.169, 'text': 'So if you look at the you know, especially in physics, theories of everything,', 'start': 2545.205, 'duration': 6.964}], 'summary': 'Math allows exploration of high dimensions beyond human experience.', 'duration': 35.579, 'max_score': 2516.59, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA2516590.jpg'}, {'end': 2795.164, 'src': 'embed', 'start': 2770.687, 'weight': 4, 'content': [{'end': 2777.412, 'text': 'It may be beyond our cognitive capabilities to visualize a four-dimensional cube a tesseract,', 'start': 2770.687, 'duration': 6.725}, {'end': 2780.854, 'text': 'as some like to call it or a five-dimensional cube or a six-dimensional cube,', 'start': 2777.412, 'duration': 3.442}, {'end': 2788.059, 'text': 'but it is not beyond our cognitive capabilities to figure out how many corners a six-dimensional cube would have.', 'start': 2780.854, 'duration': 7.205}, {'end': 2789.22, 'text': "That's what's so cool about us.", 'start': 2788.099, 'duration': 1.121}, {'end': 2792.202, 'text': 'Whether we can visualize it or not, we can still talk about it.', 'start': 2789.44, 'duration': 2.762}, {'end': 2793.182, 'text': 'We can still reason about it.', 'start': 2792.222, 'duration': 0.96}, {'end': 2795.164, 'text': 'We can still figure things out about it.', 'start': 2793.463, 'duration': 1.701}], 'summary': 'We can reason about six-dimensional cubes despite visualization challenges.', 'duration': 24.477, 'max_score': 2770.687, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA2770687.jpg'}], 'start': 1370.208, 'title': '19th century math and romanticism', 'summary': "Explores the intertwining of romanticism and mathematics in the 19th century, focusing on mathematicians everest galois and poincaré, the three-body problem, and chaotic dynamics. it also discusses poincaré's contributions to geometry and topology, the concept of topology as an intrinsic theory, exploring higher dimensions, and the concept of dimensions and holes.", 'chapters': [{'end': 1629.288, 'start': 1370.208, 'title': 'Math and romanticism in 19th century', 'summary': 'Explores the intertwining of romanticism and mathematics in the 19th century, focusing on the stories of mathematicians such as everest galois and poincaré, and delves into the complexity of the three-body problem and chaotic dynamics.', 'duration': 259.08, 'highlights': ['The intertwining of romanticism and mathematics in the 19th century, as exemplified by the stories of mathematicians like Everest Galois and Poincaré, is outlined, shedding light on the influence of romantic movement and the transcendent aspirations on mathematical thinking.', "Amir Alexander's book 'Duel at Dawn' provides a detailed account of the romanticized nature of math history and its connection to the romantic era's mood, emphasizing the influence of romanticism on the way people thought about mathematics during the early 19th century.", 'The chapter discusses the story of mathematician Everest Galois, known for his role in developing the theory of groups and symmetries, and his untimely death in a duel in his early twenties, highlighting the romantic and largely unpublished nature of his life.', "The influence of the romantic era's mood on the perception of mathematics is highlighted, with a focus on how people of that time were predisposed to think about mathematics in a romantic way due to their similar approach to poetry and music.", 'The intertwining of human history and math history is emphasized, underscoring that mathematics is inseparable from human experiences, societal influences, and the individuals involved in its development.', "The significance of Poincaré's contribution to creating the geometric world and his notable success in addressing the three-body problem, a complex issue in understanding the stability of the motion of three astronomical objects under gravity, is discussed in detail, shedding light on the challenges and complexities involved in this area of study."]}, {'end': 2105.397, 'start': 1629.949, 'title': "Poincaré's contributions to geometry and topology", 'summary': "Discusses poincaré's pioneering work in understanding higher dimensional spaces, development of topology, and the poincaré conjecture about three-dimensional spaces, highlighting his impact on geometry and physics as well as the ongoing debate on the shape of the universe.", 'duration': 475.448, 'highlights': ["Poincaré's pioneering work in understanding higher dimensional spaces and development of topology Poincaré developed the subject now known as topology, emphasizing the need to study higher dimensional spaces and formalism rich enough to enable it.", "The Poincaré conjecture about three-dimensional spaces The conjecture states that there's a simple way to determine if a three-dimensional space is the standard one, based on its fundamental group having nothing interesting in it.", 'Ongoing debate on the shape of the universe The chapter delves into the ongoing debate about the shape of the universe, discussing ideas such as the universe being flat-ish or having dodecahedral symmetry, emphasizing the open question and the legitimate cosmological debate surrounding it.']}, {'end': 2430.312, 'start': 2105.397, 'title': 'Topology and intrinsic theory', 'summary': 'Discusses the concept of topology as an intrinsic theory, explaining the difference between intrinsic and extrinsic properties, and how one can determine their world without leaving it, with a focus on living on a circle or a knot, and the peculiar features of the real projective plane.', 'duration': 324.915, 'highlights': ['The chapter discusses the concept of topology as an intrinsic theory, explaining the difference between intrinsic and extrinsic properties, and how one can determine their world without leaving it. Topology is presented as an intrinsic theory, enabling one to understand the properties of their world without leaving it, contrasting with the visualization challenges of extrinsic properties.', "Explanation of living on a circle or a knot, and the difficulties in determining intrinsic features of one's world. Living on a circle or a knot is explored, highlighting the challenge of determining intrinsic features of one's world without relying on extrinsic factors.", "Discussion on the peculiar features of the real projective plane, such as the loop of string that can't be pulled closed but can be closed by looping around it twice. The real projective plane is described, emphasizing its peculiar features, including the loop of string that exhibits a unique behavior, providing insight into the nature of one's world."]}, {'end': 2711.368, 'start': 2430.352, 'title': 'Exploring higher dimensions', 'summary': 'Discusses the concept of higher dimensions using the example of flatland, exploring the possibility of a universe with more dimensions than the three we perceive, and considering the implications in physics and theology.', 'duration': 281.016, 'highlights': ['The concept of higher dimensions and their potential existence is discussed, considering the possibility of a universe with more dimensions than the three we perceive.', 'The example of Flatland is used to illustrate the challenges of perceiving higher dimensions, drawing parallels to the difficulty of understanding dimensions beyond our direct experience.', 'The potential implications of higher dimensions in physics and theories of everything are explored, raising questions about their impact on observable phenomena and the tools of mathematics.', 'The Christian subtext in the book Flatland is highlighted, showcasing the idea of transcending earthly perception to see the truth from a higher perspective, as perceived by the author, Edwin Abbott, a minister.']}, {'end': 3215.662, 'start': 2711.368, 'title': 'Dimensions and holes', 'summary': 'Delves into the concept of dimensions, discussing the inability of the sphere to conceptualize a fourth dimension, the cognitive capability of humans to reason about high-dimensional objects, and the intriguing debate about the number of holes in a straw, with diverse perspectives and the analogy to the theory of homology.', 'duration': 504.294, 'highlights': ["The sphere's inability to conceptualize a fourth dimension and the discussion on human cognitive capability to reason about high-dimensional objects. The sphere is unable to conceptualize a fourth dimension, highlighting the cognitive limitations in understanding higher dimensions, while emphasizing the human capability to reason about high-dimensional objects.", 'The intriguing debate about the number of holes in a straw, showcasing diverse perspectives and the analogy to the theory of homology. The debate about the number of holes in a straw presents diverse perspectives, leading to an intriguing analogy to the theory of homology, which emphasizes the arithmetic nature of holes and their geometric significance.', 'The discussion on the relationship between holes in a straw and a pair of pants, providing an interesting analogy and highlighting the concept of homology in modern topology. The relationship between holes in a straw and a pair of pants is discussed, offering an interesting analogy and emphasizing the concept of homology in modern topology, showcasing the arithmetic nature of holes and their geometric significance.']}], 'duration': 1845.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA1370208.jpg', 'highlights': ['The intertwining of romanticism and mathematics in the 19th century, as exemplified by mathematicians like Everest Galois and Poincaré, is outlined, shedding light on the influence of romantic movement on mathematical thinking.', "Poincaré's pioneering work in understanding higher dimensional spaces and development of topology is discussed, emphasizing the need to study higher dimensional spaces and formalism rich enough to enable it.", 'The chapter discusses the concept of topology as an intrinsic theory, enabling one to understand the properties of their world without leaving it, contrasting with the visualization challenges of extrinsic properties.', 'The concept of higher dimensions and their potential existence is considered, raising questions about their impact on observable phenomena and the tools of mathematics.', "The sphere's inability to conceptualize a fourth dimension and the discussion on human cognitive capability to reason about high-dimensional objects is highlighted, emphasizing the human capability to reason about high-dimensional objects."]}, {'end': 4002.541, 'segs': [{'end': 3296.422, 'src': 'embed', 'start': 3215.782, 'weight': 0, 'content': [{'end': 3220.986, 'text': "Is that your weekly routine or just in preparation for talking about geometry for three hours? Exactly, it's just for this.", 'start': 3215.782, 'duration': 5.204}, {'end': 3224.108, 'text': "It's hardship to purify the mind.", 'start': 3221.927, 'duration': 2.181}, {'end': 3225.169, 'text': "No, it's for the first time.", 'start': 3224.149, 'duration': 1.02}, {'end': 3226.65, 'text': 'I just wanted to try the experience.', 'start': 3225.189, 'duration': 1.461}, {'end': 3227.051, 'text': 'Oh, wow.', 'start': 3226.731, 'duration': 0.32}, {'end': 3232.073, 'text': 'And just to to pause, to do things that are out of the ordinary,', 'start': 3227.211, 'duration': 4.862}, {'end': 3239.954, 'text': 'to pause and to reflect on how grateful I am to be just alive and be able to do all the cool shit that I get to do so.', 'start': 3232.073, 'duration': 7.881}, {'end': 3242.755, 'text': 'Did you drink water? Yes, yes, yes, yes, yes.', 'start': 3239.974, 'duration': 2.781}, {'end': 3244.215, 'text': 'Water and salt.', 'start': 3243.175, 'duration': 1.04}, {'end': 3246.595, 'text': 'So like electrolytes and all those kinds of things.', 'start': 3244.615, 'duration': 1.98}, {'end': 3253.636, 'text': 'But anyway, so the inflow on the top of the pants equals to the outflow on the bottom of the pants.', 'start': 3247.195, 'duration': 6.441}, {'end': 3258.249, 'text': 'Exactly. So this idea that I mean.', 'start': 3254.737, 'duration': 3.512}, {'end': 3265.015, 'text': 'I think Poincaré really had this idea, this sort of modern idea I mean building on stuff other people did.', 'start': 3258.249, 'duration': 6.766}, {'end': 3269.839, 'text': 'Betti is an important one of this kind of modern notion of relations between wholes.', 'start': 3265.015, 'duration': 4.824}, {'end': 3275.504, 'text': "But the idea that wholes really had an arithmetic, the really modern view was really Emmy Noether's idea.", 'start': 3269.859, 'duration': 5.645}, {'end': 3280.648, 'text': 'So she kind of comes in and sort of truly puts the subject on the table.', 'start': 3275.604, 'duration': 5.044}, {'end': 3283.33, 'text': 'on its modern footing that we have, that we have now.', 'start': 3280.908, 'duration': 2.422}, {'end': 3285.212, 'text': "so you know, it's always a challenge.", 'start': 3283.33, 'duration': 1.882}, {'end': 3291.177, 'text': "you know, in the book i'm not going to say i give like a course so that you read this chapter and then you're like oh, it's just like i took,", 'start': 3285.212, 'duration': 5.965}, {'end': 3293.179, 'text': 'like a semester of algebraic anthropology.', 'start': 3291.177, 'duration': 2.002}, {'end': 3294.36, 'text': "it's not like this and it's always a.", 'start': 3293.179, 'duration': 1.181}, {'end': 3296.422, 'text': "you know, it's always a challenge writing about math,", 'start': 3294.36, 'duration': 2.062}], 'summary': 'Discussion on preparation for geometry, experience, and gratitude.', 'duration': 80.64, 'max_score': 3215.782, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA3215782.jpg'}, {'end': 3669.219, 'src': 'embed', 'start': 3647.185, 'weight': 4, 'content': [{'end': 3655.332, 'text': "And to be honest, it's something that I think we haven't quite figured out how to value inside academic mathematics in the same way.", 'start': 3647.185, 'duration': 8.147}, {'end': 3660.797, 'text': "And this is a bit older that I think we haven't quite figured out how to value the development of computational infrastructure.", 'start': 3655.372, 'duration': 5.425}, {'end': 3669.219, 'text': 'You know, we all have computers as our partners now and people build computers that sort of assist and participate in our mathematics.', 'start': 3660.837, 'duration': 8.382}], 'summary': 'Academic math needs to value computational infrastructure development.', 'duration': 22.034, 'max_score': 3647.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA3647185.jpg'}], 'start': 3215.782, 'title': 'Mental purification and modern mathematics', 'summary': "Discusses mental purification through reflection and its unique experience while preparing to talk about geometry. it also explores modern mathematics, its outreach, the impact of emmy noether's idea, and the importance of various teaching modes including books, youtube, and computational infrastructure.", 'chapters': [{'end': 3258.249, 'start': 3215.782, 'title': 'Mental purification through reflection', 'summary': 'Discusses a unique experience of mental purification and reflection while preparing to talk about geometry, emphasizing the importance of gratitude and the balance of inflow and outflow.', 'duration': 42.467, 'highlights': ['The speaker engages in a unique experience of mental purification and reflection, emphasizing the importance of gratitude and the balance of inflow and outflow.', 'The speaker reflects on the importance of gratitude and the ability to do extraordinary things in life.', 'The speaker prepares for a three-hour discussion on geometry by engaging in a weekly routine of mental purification.']}, {'end': 4002.541, 'start': 3258.249, 'title': 'Modern mathematics and its outreach', 'summary': "Discusses the modern notion of relations between wholes, the impact of emmy noether's idea on modern mathematics, the challenges of writing about math, and the importance and impact of different modes of teaching mathematics including books, youtube, and computational infrastructure.", 'duration': 744.292, 'highlights': ["The impact of Emmy Noether's idea on modern mathematics is emphasized. Emmy Noether's idea is highlighted as truly putting the subject of modern mathematics on the table and on its modern footing.", "The challenges of writing about math and the importance of conveying the beauty of the subject are discussed. The chapter explores the challenges of writing about math and the author's efforts to convey the beauty of the subject through meaningful and interesting discussions.", 'The significance and impact of different modes of teaching mathematics are emphasized, including books, YouTube, and computational infrastructure. The chapter highlights the importance of different modes of teaching mathematics, such as books, YouTube videos by creators like 3Blue1Brown and Numberphile, and the development of computational infrastructure as integral parts of modern mathematical outreach.']}], 'duration': 786.759, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA3215782.jpg', 'highlights': ['The speaker prepares for a three-hour discussion on geometry by engaging in a weekly routine of mental purification.', 'The speaker engages in a unique experience of mental purification and reflection, emphasizing the importance of gratitude and the balance of inflow and outflow.', 'The speaker reflects on the importance of gratitude and the ability to do extraordinary things in life.', "The impact of Emmy Noether's idea on modern mathematics is emphasized. Emmy Noether's idea is highlighted as truly putting the subject of modern mathematics on the table and on its modern footing.", 'The significance and impact of different modes of teaching mathematics are emphasized, including books, YouTube, and computational infrastructure. The chapter highlights the importance of different modes of teaching mathematics, such as books, YouTube videos by creators like 3Blue1Brown and Numberphile, and the development of computational infrastructure as integral parts of modern mathematical outreach.', "The challenges of writing about math and the importance of conveying the beauty of the subject are discussed. The chapter explores the challenges of writing about math and the author's efforts to convey the beauty of the subject through meaningful and interesting discussions."]}, {'end': 5583.082, 'segs': [{'end': 4034.51, 'src': 'embed', 'start': 4002.841, 'weight': 1, 'content': [{'end': 4006.483, 'text': 'One thing about it is her paper was rather, was very short.', 'start': 4002.841, 'duration': 3.642}, {'end': 4008.083, 'text': 'It was very short and simple.', 'start': 4006.843, 'duration': 1.24}, {'end': 4009.644, 'text': 'Nine pages of which two were pictures.', 'start': 4008.163, 'duration': 1.481}, {'end': 4017.245, 'text': 'very short, for, like a paper solving a major conjecture, and it really makes you think about what we mean by difficulty in mathematics.', 'start': 4011.724, 'duration': 5.521}, {'end': 4020.846, 'text': "like, do you say oh, actually the problem wasn't difficult because you could solve it so simply?", 'start': 4017.245, 'duration': 3.601}, {'end': 4025.127, 'text': "or do you say, like, well, no, evidently it was difficult because, like the world's top topologists,", 'start': 4020.846, 'duration': 4.281}, {'end': 4027.328, 'text': 'many you know worked on it for 20 years and nobody could solve it.', 'start': 4025.127, 'duration': 2.201}, {'end': 4028.628, 'text': 'so therefore it is difficult.', 'start': 4027.328, 'duration': 1.3}, {'end': 4034.51, 'text': "or is it that we need sort of some new category of things, that about which it's difficult to figure out that they're not difficult?", 'start': 4028.628, 'duration': 5.882}], 'summary': 'A nine-page paper, with two pictures, solved a major conjecture, raising questions about the concept of difficulty in mathematics.', 'duration': 31.669, 'max_score': 4002.841, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4002841.jpg'}, {'end': 4093.73, 'src': 'embed', 'start': 4066.17, 'weight': 2, 'content': [{'end': 4069.613, 'text': "or at least it feels like, I hope there's just simple answers to everything.", 'start': 4066.17, 'duration': 3.443}, {'end': 4080.242, 'text': "That we'll look and it'll be simple laws that govern the universe, simple explanation of what is consciousness, of what is love.", 'start': 4070.153, 'duration': 10.089}, {'end': 4082.184, 'text': 'is mortality fundamental to life??', 'start': 4080.242, 'duration': 1.942}, {'end': 4083.225, 'text': "What's the meaning of life??", 'start': 4082.364, 'duration': 0.861}, {'end': 4093.73, 'text': "Are humans special or we're just another sort of reflection of all that is beautiful in the universe?", 'start': 4084.746, 'duration': 8.984}], 'summary': 'Seeking simple answers to complex questions about life, consciousness, and human significance.', 'duration': 27.56, 'max_score': 4066.17, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4066170.jpg'}, {'end': 4144.689, 'src': 'embed', 'start': 4112.779, 'weight': 3, 'content': [{'end': 4120.805, 'text': "I think, just like symmetry and the breaking of symmetry is beautiful, somehow there's something beautiful about simplicity.", 'start': 4112.779, 'duration': 8.026}, {'end': 4125.27, 'text': "I think it, what is that? It's aesthetic, yeah.", 'start': 4121.466, 'duration': 3.804}, {'end': 4129.313, 'text': "But it's aesthetic in the way that happiness is an aesthetic.", 'start': 4125.689, 'duration': 3.624}, {'end': 4133.636, 'text': 'Like. why is that so joyful?', 'start': 4130.935, 'duration': 2.701}, {'end': 4140.483, 'text': 'that a simple explanation that governs a large number of cases is really appealing?', 'start': 4133.636, 'duration': 6.847}, {'end': 4144.689, 'text': "Even when it's not like.", 'start': 4142.287, 'duration': 2.402}], 'summary': 'Beauty of simplicity: joy in a simple explanation.', 'duration': 31.91, 'max_score': 4112.779, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4112779.jpg'}, {'end': 4202.536, 'src': 'embed', 'start': 4168.772, 'weight': 0, 'content': [{'end': 4170.053, 'text': 'but why is that so appealing?', 'start': 4168.772, 'duration': 1.281}, {'end': 4175.417, 'text': 'And in this nice forms in mathematics, like you, look at the Einstein papers.', 'start': 4170.473, 'duration': 4.944}, {'end': 4177.881, 'text': 'Why are those so beautiful?', 'start': 4176.6, 'duration': 1.281}, {'end': 4183.024, 'text': "And why is the Andrew Wiles proof of the Fermat's last theorem not quite so beautiful?", 'start': 4178.062, 'duration': 4.962}, {'end': 4190.349, 'text': "Like, what's beautiful about that story is the human struggle of like the human story of perseverance,", 'start': 4183.245, 'duration': 7.104}, {'end': 4196.032, 'text': 'of the drama of not knowing if the proof is correct, and ups and downs and all of those kinds of things.', 'start': 4190.349, 'duration': 5.683}, {'end': 4196.973, 'text': "That's the interesting part.", 'start': 4196.053, 'duration': 0.92}, {'end': 4202.536, 'text': "But the fact that the proof is huge and nobody understands, well, from my outsider's perspective, nobody understands what the heck it is.", 'start': 4197.233, 'duration': 5.303}], 'summary': 'The beauty of mathematics lies in human struggle and perseverance, not just in the complexity of proofs.', 'duration': 33.764, 'max_score': 4168.772, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4168772.jpg'}, {'end': 4259.31, 'src': 'embed', 'start': 4231.66, 'weight': 4, 'content': [{'end': 4237.661, 'text': "there's a few problems that are deemed by the world throughout its history to be exceptionally difficult.", 'start': 4231.66, 'duration': 6.001}, {'end': 4245.583, 'text': 'And that one in particular is really simple to formulate and really hard to come up with a proof for.', 'start': 4237.821, 'duration': 7.762}, {'end': 4250.926, 'text': 'And it was like taunted as simple by Fermat himself.', 'start': 4246.383, 'duration': 4.543}, {'end': 4259.31, 'text': "There's something interesting to be said about that X to the N plus Y to the N equals Z to the N for N of three or greater.", 'start': 4251.386, 'duration': 7.924}], 'summary': "Fermat's last theorem is simple to state but hard to prove.", 'duration': 27.65, 'max_score': 4231.66, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4231660.jpg'}, {'end': 4416.653, 'src': 'embed', 'start': 4388.143, 'weight': 13, 'content': [{'end': 4395.265, 'text': "I understand the appeal of saying like wouldn't it be cool if this very simple equation there was like a very simple, clever,", 'start': 4388.143, 'duration': 7.122}, {'end': 4397.346, 'text': 'wonderful proof that you could do in a page or two?', 'start': 4395.265, 'duration': 2.081}, {'end': 4398.967, 'text': 'And that would be great, but you know what?', 'start': 4397.466, 'duration': 1.501}, {'end': 4404.369, 'text': "There's lots of equations like that that are solved by very clever methods like that, including the special cases that Fermat wrote about,", 'start': 4398.987, 'duration': 5.382}, {'end': 4406.849, 'text': 'the method of descent, which is like very wonderful and important.', 'start': 4404.369, 'duration': 2.48}, {'end': 4416.653, 'text': "But in the end, those are nice things that like, you know, you teach an undergraduate class and it is what it is, but they're not big.", 'start': 4406.95, 'duration': 9.703}], 'summary': "Many equations are solved by clever methods, including those fermat wrote about, but they're not significant.", 'duration': 28.51, 'max_score': 4388.143, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4388143.jpg'}, {'end': 4456.325, 'src': 'embed', 'start': 4426.043, 'weight': 5, 'content': [{'end': 4435.006, 'text': 'I mean, work on the Fermat problem developed this incredible richness of number theory that we now live in today.', 'start': 4426.043, 'duration': 8.963}, {'end': 4440.528, 'text': 'And not, by the way, just Wiles, Andrew Wiles being the person who together with Richard Taylor finally proved this theorem.', 'start': 4435.066, 'duration': 5.462}, {'end': 4446.154, 'text': 'But you have this whole moment that people try to prove this theorem and they fail.', 'start': 4441.749, 'duration': 4.405}, {'end': 4456.325, 'text': "And there's a famous false proof by LeMay from the 19th century, where Kummer, in understanding what mistake LeMay had made in this incorrect proof,", 'start': 4446.534, 'duration': 9.791}], 'summary': 'Work on the fermat problem led to rich number theory. wiles and taylor proved the theorem after failed attempts by others.', 'duration': 30.282, 'max_score': 4426.043, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4426043.jpg'}, {'end': 4567.897, 'src': 'embed', 'start': 4538.893, 'weight': 6, 'content': [{'end': 4543.195, 'text': 'Or however you sort of chop the number up, you end up with the same set of prime factors.', 'start': 4538.893, 'duration': 4.302}, {'end': 4552.961, 'text': 'And indeed what people finally understood at the end of the 19th century is that if you work in number,', 'start': 4544.896, 'duration': 8.065}, {'end': 4562.786, 'text': "systems slightly more general than the ones we're used to, which it turns out are relevant to Fermat, all of a sudden this stops being true.", 'start': 4552.961, 'duration': 9.825}, {'end': 4564.767, 'text': 'Things get..', 'start': 4564.307, 'duration': 0.46}, {'end': 4567.897, 'text': 'I mean, things get more complicated.', 'start': 4565.816, 'duration': 2.081}], 'summary': 'In number systems beyond the norm, prime factor patterns vary.', 'duration': 29.004, 'max_score': 4538.893, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4538893.jpg'}, {'end': 4831.813, 'src': 'embed', 'start': 4798.508, 'weight': 8, 'content': [{'end': 4806.317, 'text': 'You know, in the natural language processing community in AI, there might be some notion of semantic distance or lexical distance between two words.', 'start': 4798.508, 'duration': 7.809}, {'end': 4808.68, 'text': 'How much do they tend to arise in the same context?', 'start': 4806.357, 'duration': 2.323}, {'end': 4816.386, 'text': "That's incredibly important, for you know, doing auto-complete and machine translation and stuff like that, and it doesn't have anything to do with.", 'start': 4808.7, 'duration': 7.686}, {'end': 4817.886, 'text': 'are they next to each other in the dictionary right?', 'start': 4816.386, 'duration': 1.5}, {'end': 4819.007, 'text': "It's a different kind of distance.", 'start': 4817.906, 'duration': 1.101}, {'end': 4825.09, 'text': 'Okay, ready? In this kind of number theory, there is a crazy distance called the p-adic distance.', 'start': 4819.327, 'duration': 5.763}, {'end': 4828.651, 'text': "I didn't write about this that much in the book because even though I love it and it's a big part of my research life,", 'start': 4825.13, 'duration': 3.521}, {'end': 4829.852, 'text': 'it gets a little bit into the weeds, but', 'start': 4828.651, 'duration': 1.201}, {'end': 4831.813, 'text': 'Your listeners are going to hear about it now.', 'start': 4830.532, 'duration': 1.281}], 'summary': 'Natural language processing involves semantic distance for auto-complete and machine translation, distinct from lexical distance. p-adic distance in number theory is complex.', 'duration': 33.305, 'max_score': 4798.508, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4798508.jpg'}, {'end': 5087.747, 'src': 'embed', 'start': 5004.291, 'weight': 7, 'content': [{'end': 5006.251, 'text': 'Why is that interesting? Yeah, the NEP.', 'start': 5004.291, 'duration': 1.96}, {'end': 5008.192, 'text': "Because that's the kind of deformation that comes up.", 'start': 5006.291, 'duration': 1.901}, {'end': 5017.226, 'text': "in Wiles' proof that defamation, where moving something a little bit means a little bit in this, to add a sense.", 'start': 5009.42, 'duration': 7.806}, {'end': 5020.088, 'text': "No, I mean, it's such a, I mean, I just get excited talking about it.", 'start': 5018.167, 'duration': 1.921}, {'end': 5022.79, 'text': 'And I just taught this like in the fall semester that.', 'start': 5020.108, 'duration': 2.682}, {'end': 5025.973, 'text': 'But it like reformulating, why is..', 'start': 5024.071, 'duration': 1.902}, {'end': 5041.239, 'text': 'So you pick a different measure of distance over which you can talk about very tiny changes and then use that to then prove things about the entire thing.', 'start': 5029.505, 'duration': 11.734}, {'end': 5048.337, 'text': "Yes, Although you know honestly what I would say mean it's true that we use it to prove things.", 'start': 5042.48, 'duration': 5.857}, {'end': 5053.642, 'text': 'but i would say we use it to understand things and then, because we understand things better, then we can prove things.', 'start': 5048.337, 'duration': 5.305}, {'end': 5055.383, 'text': 'but you know the goal is always the understanding.', 'start': 5053.642, 'duration': 1.741}, {'end': 5057.925, 'text': 'the goal is not so much to prove things.', 'start': 5055.383, 'duration': 2.542}, {'end': 5059.847, 'text': "the goal is not to know what's true or false.", 'start': 5057.925, 'duration': 1.922}, {'end': 5065.852, 'text': "i mean this is something i read about in the book near the end, and it's something that it's a wonderful, wonderful essay by, uh, by bill thurston,", 'start': 5059.847, 'duration': 6.005}, {'end': 5070.595, 'text': 'kind of one of the great geometers of our time, who unfortunately passed away a few years ago.', 'start': 5065.852, 'duration': 4.743}, {'end': 5076.101, 'text': 'um called on proof and progress mathematics, and he writes very wonderfully about how.', 'start': 5070.595, 'duration': 5.506}, {'end': 5080.504, 'text': "you know we're not, it's not a theorem factory where we have a production quota.", 'start': 5076.101, 'duration': 4.403}, {'end': 5087.747, 'text': "i mean the point of mathematics is to help humans understand things and the way we test that is that we're proving new theorems along the way.", 'start': 5080.504, 'duration': 7.243}], 'summary': 'Discussion about using nep to understand and prove mathematical theorems, emphasizing the goal of understanding over proving, as highlighted by bill thurston.', 'duration': 83.456, 'max_score': 5004.291, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA5004291.jpg'}], 'start': 4002.841, 'title': 'Mathematics and philosophy', 'summary': "Delves into simplicity in mathematics and philosophy, beauty of mathematical proofs, the drama behind fermat's last theorem's proof, deformation theory, and ai's use of distance functions in tasks like natural language processing.", 'chapters': [{'end': 4168.772, 'start': 4002.841, 'title': 'Simplicity in mathematics and philosophy', 'summary': 'Discusses the concept of simplicity in mathematics and philosophy, questioning the definition of difficulty and expressing a hope for simple answers to complex questions, including the meaning of life and consciousness.', 'duration': 165.931, 'highlights': ['The concept of simplicity in solving difficult problems is explored, such as the case of a short and simple paper solving a major conjecture, raising questions about the definition of difficulty in mathematics.', 'The desire for simple answers is expressed, hoping for simple laws to govern the universe and provide explanations for consciousness, love, and the meaning of life.', 'The appeal of simplicity and aesthetic beauty in symmetry is discussed, acknowledging the potential for simple explanations to govern a large number of cases, despite the risks associated with disconnectedness from reality and potential harmful effects.']}, {'end': 4625.466, 'start': 4168.772, 'title': "Beauty of mathematics and fermat's last theorem", 'summary': "Discusses the appeal of beautiful mathematical proofs, the human struggle and drama behind andrew wiles' proof of fermat's last theorem, the history and significance of fermat's last theorem, and the profound insights gained from the proof.", 'duration': 456.694, 'highlights': ["Andrew Wiles' proof of Fermat's Last Theorem and its human struggle and drama The human struggle and drama behind Andrew Wiles' proof of Fermat's Last Theorem is highlighted, emphasizing the appeal of beautiful mathematical proofs.", "Significance and history of Fermat's Last Theorem The history and significance of Fermat's Last Theorem, including its simple formulation and difficult proof, are discussed, emphasizing the exceptional difficulty of proving the theorem.", "Fermat's insights and the development of number theory The development of number theory and the insights gained from Fermat's Last Theorem, including Fermat's contributions and the subsequent rich development of number theory, are highlighted, emphasizing the profound impact of the theorem on mathematics.", 'Unique factorization and the complexity of number systems The concept of unique factorization and the complexity of number systems, including the implications of non-unique factorization in more general number systems, are discussed, emphasizing the richness and opportunities for new study arising from complexity.']}, {'end': 5130.893, 'start': 4625.486, 'title': 'Deformation theory and distance metrics', 'summary': "Discusses deformation theory, emphasizing the significance of understanding distance metrics, including the p-adic distance and its role in wiles' proof, and highlighting the goal of mathematics to aid in human understanding rather than solely producing theorems.", 'duration': 505.407, 'highlights': ["Deformation theory involves understanding how objects can be deformed and moved around, which was crucial in Andrew Wiles' proof of Fermat's Last Theorem. Understanding deformation theory, its importance in Wiles' proof", 'The concept of distance is crucial in mathematics and can be varied, as seen in the example of the p-adic distance, which has applications in number theory. Significance of varied notions of distance, application of p-adic distance in number theory', 'The goal of mathematics is to aid human understanding rather than simply producing theorems, as highlighted by the essay by Bill Thurston. Emphasis on the goal of mathematics, as explained by Bill Thurston']}, {'end': 5583.082, 'start': 5130.893, 'title': 'Ai and distance functions', 'summary': 'Explores the concept of distance functions in ai, emphasizing the importance of finding accurate and complex distance measures for tasks like natural language processing and word embeddings.', 'duration': 452.189, 'highlights': ['The importance of accurate and complex distance measures in AI for tasks like natural language processing and word embeddings.', 'The discussion about the complexity of explaining real things and the notion that learning to explain something helps understand it.', 'The different perspectives on mathematical truth and its relation to art and religion.', "The beliefs of mathematicians and physicists about explaining things simply and the idea of 'God's book' in mathematics."]}], 'duration': 1580.241, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA4002841.jpg', 'highlights': ["The human struggle and drama behind Andrew Wiles' proof of Fermat's Last Theorem is highlighted, emphasizing the appeal of beautiful mathematical proofs.", 'The concept of simplicity in solving difficult problems is explored, such as the case of a short and simple paper solving a major conjecture, raising questions about the definition of difficulty in mathematics.', 'The desire for simple answers is expressed, hoping for simple laws to govern the universe and provide explanations for consciousness, love, and the meaning of life.', 'The appeal of simplicity and aesthetic beauty in symmetry is discussed, acknowledging the potential for simple explanations to govern a large number of cases, despite the risks associated with disconnectedness from reality and potential harmful effects.', "The history and significance of Fermat's Last Theorem, including its simple formulation and difficult proof, are discussed, emphasizing the exceptional difficulty of proving the theorem.", "The development of number theory and the insights gained from Fermat's Last Theorem, including Fermat's contributions and the subsequent rich development of number theory, are highlighted, emphasizing the profound impact of the theorem on mathematics.", 'The concept of unique factorization and the complexity of number systems, including the implications of non-unique factorization in more general number systems, are discussed, emphasizing the richness and opportunities for new study arising from complexity.', "Understanding deformation theory, its importance in Wiles' proof", 'Significance of varied notions of distance, application of p-adic distance in number theory', 'Emphasis on the goal of mathematics, as explained by Bill Thurston', 'The importance of accurate and complex distance measures in AI for tasks like natural language processing and word embeddings.', 'The discussion about the complexity of explaining real things and the notion that learning to explain something helps understand it.', 'The different perspectives on mathematical truth and its relation to art and religion.', "The beliefs of mathematicians and physicists about explaining things simply and the idea of 'God's book' in mathematics."]}, {'end': 6193.042, 'segs': [{'end': 5640.831, 'src': 'embed', 'start': 5603.726, 'weight': 0, 'content': [{'end': 5604.587, 'text': 'That are not like that.', 'start': 5603.726, 'duration': 0.861}, {'end': 5608.17, 'text': 'All this talk of primes got me hungry for primes.', 'start': 5605.628, 'duration': 2.542}, {'end': 5616.939, 'text': 'You wrote a blog post, The Beauty of Bounding Gaps, a huge discovery about prime numbers and what it means for the future of math.', 'start': 5609.431, 'duration': 7.508}, {'end': 5620.464, 'text': 'Can you tell me about prime numbers?', 'start': 5619.164, 'duration': 1.3}, {'end': 5621.625, 'text': 'What the heck are those?', 'start': 5620.785, 'duration': 0.84}, {'end': 5622.725, 'text': 'What are twin primes?', 'start': 5621.825, 'duration': 0.9}, {'end': 5623.626, 'text': 'What are prime gaps?', 'start': 5622.765, 'duration': 0.861}, {'end': 5625.986, 'text': 'What are bounding gaps in primes?', 'start': 5623.666, 'duration': 2.32}, {'end': 5627.527, 'text': 'What are all these things?', 'start': 5626.707, 'duration': 0.82}, {'end': 5631.408, 'text': 'And what, if anything or what exactly is beautiful about them??', 'start': 5627.867, 'duration': 3.541}, {'end': 5640.831, 'text': 'Yeah, so you know, prime numbers are one of the things that number theorists study the most and have for millennia.', 'start': 5632.269, 'duration': 8.562}], 'summary': 'Discussion about prime numbers, twin primes, prime gaps, and bounding gaps in primes, revealing their significance in number theory.', 'duration': 37.105, 'max_score': 5603.726, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA5603726.jpg'}, {'end': 5814.923, 'src': 'embed', 'start': 5783.937, 'weight': 4, 'content': [{'end': 5786.814, 'text': 'Would you be able to go over 100? I think so.', 'start': 5783.937, 'duration': 2.877}, {'end': 5788.895, 'text': "There's always those ones that trip people up.", 'start': 5787.074, 'duration': 1.821}, {'end': 5793.338, 'text': "There's a famous one, the Grotendieck prime 57, like sort of Alexander Grotendieck,", 'start': 5789.055, 'duration': 4.283}, {'end': 5798.641, 'text': 'the great algebraic geometer was sort of giving some lecture involving a choice of a prime in general.', 'start': 5793.338, 'duration': 5.303}, {'end': 5806.567, 'text': "And somebody said like, can't you just choose a prime? And he said, okay, 57, which is in fact not prime, it's three times 19.", 'start': 5798.701, 'duration': 7.866}, {'end': 5809.869, 'text': "But it was like, I promise you in some circles, that's a funny story.", 'start': 5806.567, 'duration': 3.302}, {'end': 5814.923, 'text': "Okay But, There's a humor in it.", 'start': 5809.909, 'duration': 5.014}], 'summary': 'Famous math joke about 57 not being prime, told by grotendieck during lecture.', 'duration': 30.986, 'max_score': 5783.937, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA5783937.jpg'}, {'end': 5931.712, 'src': 'embed', 'start': 5896.88, 'weight': 1, 'content': [{'end': 5898.262, 'text': 'Ready, let me give you the simplest version of it.', 'start': 5896.88, 'duration': 1.382}, {'end': 5900.324, 'text': "You can dress it up a little bit, but here's the basic idea.", 'start': 5898.282, 'duration': 2.042}, {'end': 5912.039, 'text': "I take the number, the mystery number, I raise two to that power, So let's say your mystery number is six.", 'start': 5902.646, 'duration': 9.393}, {'end': 5917.302, 'text': "Are you sorry you asked me? Are you ready? It's not- No, you're breaking my brain again, but yes.", 'start': 5912.719, 'duration': 4.583}, {'end': 5918.123, 'text': "Okay, let's do it.", 'start': 5917.362, 'duration': 0.761}, {'end': 5919.524, 'text': "We're gonna do a live demonstration.", 'start': 5918.163, 'duration': 1.361}, {'end': 5922.586, 'text': "Let's say your number is six.", 'start': 5921.225, 'duration': 1.361}, {'end': 5925.428, 'text': "So I'm gonna raise two to the sixth power.", 'start': 5923.326, 'duration': 2.102}, {'end': 5931.712, 'text': "Okay, so if I were working on it, I'd be like, that's two cubed squared, so that's eight times eight, so that's 64.", 'start': 5926.028, 'duration': 5.684}], 'summary': 'Raising a mystery number to the power of 2, e.g., 2^6 = 64.', 'duration': 34.832, 'max_score': 5896.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA5896880.jpg'}], 'start': 5583.082, 'title': 'Prime numbers', 'summary': 'Explores prime numbers, their definition as non-factorable numbers, the concept of pseudo primes, limitations of primality tests, and the existence of infinitely many twin primes, with examples of easily mistaken prime numbers like 57 and 51.', 'chapters': [{'end': 5895.539, 'start': 5583.082, 'title': 'Understanding prime numbers and their significance', 'summary': 'Discusses prime numbers and their significance, including their definition as numbers that cannot be factored, the concept of pseudo primes, and the limitations of primality tests, along with examples such as 57 and 51 being easily mistaken as prime numbers.', 'duration': 312.457, 'highlights': ['Prime numbers are numbers that cannot be factored and serve as the building blocks of all numbers. Prime numbers are the atoms, the building blocks, of all numbers, and they cannot be factored into smaller numbers.', 'The concept of pseudo primes is based on a primality test devised by Fermat, which has limitations as some numbers, like 341, pass the test but are not prime. There is a concept of pseudo primes, which are numbers passing a primality test devised by Fermat, but it is not a definitive test for primality as some numbers such as 341 pass the test but are not prime.', 'Numbers like 57 and 51 are easily mistaken to be prime due to their appearance, but they are not prime numbers. Numbers like 57 and 51 are often mistakenly thought to be prime, but they are not prime numbers as they can be factored into smaller numbers.']}, {'end': 6193.042, 'start': 5896.88, 'title': 'The mystery of prime numbers', 'summary': 'Delves into the concept of prime numbers, explaining a method to determine if a number is prime, discussing the infinite nature of prime numbers and pondering about the existence of infinitely many twin primes.', 'duration': 296.162, 'highlights': ['The concept of prime numbers and a method to determine if a number is prime is explained using the example of raising 2 to a mystery number and checking the remainder when divided by the original number. Explanation of the method to determine if a number is prime using examples.', 'The confirmation of the infinite nature of prime numbers and the increasing gap between primes as numbers grow larger is discussed. Confirmation of the infinite nature of prime numbers and the increasing gap between primes as numbers grow larger.', 'The question of the existence of infinitely many twin primes is pondered, although it is confirmed that the average gap between primes becomes larger as the numbers increase. Pondering about the existence of infinitely many twin primes and the confirmation that the average gap between primes becomes larger as the numbers increase.']}], 'duration': 609.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA5583082.jpg', 'highlights': ['Prime numbers are the atoms, the building blocks, of all numbers, and they cannot be factored into smaller numbers.', 'The confirmation of the infinite nature of prime numbers and the increasing gap between primes as numbers grow larger is discussed.', 'The concept of pseudo primes is based on a primality test devised by Fermat, which has limitations as some numbers, like 341, pass the test but are not prime.', 'Confirmation of the infinite nature of prime numbers and the increasing gap between primes as numbers grow larger.', 'Numbers like 57 and 51 are often mistakenly thought to be prime, but they are not prime numbers as they can be factored into smaller numbers.', 'Pondering about the existence of infinitely many twin primes and the confirmation that the average gap between primes becomes larger as the numbers increase.']}, {'end': 7600.785, 'segs': [{'end': 6763.032, 'src': 'embed', 'start': 6737.935, 'weight': 7, 'content': [{'end': 6745.24, 'text': "Well, he'd say like well, it's like you sort of divide the length of this line segment by the length of this other line segment,", 'start': 6737.935, 'duration': 7.305}, {'end': 6748.723, 'text': 'and then you make them a little shorter and you divide again, and then you make them a little shorter and you divide again,', 'start': 6745.24, 'duration': 3.483}, {'end': 6751.605, 'text': "and then you just keep on doing that until they're like infinitely short and then you divide them again.", 'start': 6748.723, 'duration': 2.882}, {'end': 6763.032, 'text': "These quantities that are like they're not zero, but they're also smaller than any actual number, these infinitesimals.", 'start': 6752.505, 'duration': 10.527}], 'summary': 'Mathematical concept of infinitesimals involves dividing line segments infinitely to create quantities smaller than any actual number.', 'duration': 25.097, 'max_score': 6737.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA6737935.jpg'}, {'end': 6939.922, 'src': 'embed', 'start': 6914.051, 'weight': 1, 'content': [{'end': 6922.415, 'text': "with John Conway looking at the game of life, Stephen Wolfram's work that I've been a big fan of for a while of cellular automata.", 'start': 6914.051, 'duration': 8.364}, {'end': 6929.518, 'text': 'I was wondering if you have ever encountered these kinds of objects you ever looked at them as a mathematician,', 'start': 6922.595, 'duration': 6.923}, {'end': 6939.922, 'text': 'where you have very simple rules of tiny little objects that, when taken as a whole, create incredible complexities but are very difficult to analyze,', 'start': 6929.518, 'duration': 10.404}], 'summary': 'Transcript discusses cellular automata and their complex behavior.', 'duration': 25.871, 'max_score': 6914.051, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA6914051.jpg'}, {'end': 7056.65, 'src': 'embed', 'start': 7026.059, 'weight': 4, 'content': [{'end': 7033.141, 'text': "Just, I don't know, from my outsider's perspective, there's not many mathematicians that stand out throughout the history of the 20th century.", 'start': 7026.059, 'duration': 7.082}, {'end': 7034.501, 'text': "And he's one of them.", 'start': 7033.161, 'duration': 1.34}, {'end': 7037.542, 'text': "I feel like he's not sufficiently recognized.", 'start': 7035.041, 'duration': 2.501}, {'end': 7039.32, 'text': "I think he's pretty recognized.", 'start': 7038.157, 'duration': 1.163}, {'end': 7040.903, 'text': 'Okay, well.', 'start': 7040.141, 'duration': 0.762}, {'end': 7044.951, 'text': 'I mean, he was a full professor at Princeton for most of his life.', 'start': 7041.083, 'duration': 3.868}, {'end': 7048.623, 'text': 'He was sort of certainly at the pinnacle of yeah, but i found myself.', 'start': 7044.991, 'duration': 3.632}, {'end': 7056.65, 'text': 'every time i talk about conway and how excited i am about him, i have to constantly explain to people who he is,', 'start': 7048.623, 'duration': 8.027}], 'summary': 'Conway, a prominent mathematician, deserves more recognition, being a full professor at princeton.', 'duration': 30.591, 'max_score': 7026.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7026059.jpg'}, {'end': 7300.473, 'src': 'embed', 'start': 7274.761, 'weight': 3, 'content': [{'end': 7280.309, 'text': "Essentially, it's a game that you play with marking little squares on a sheet of graph paper.", 'start': 7274.761, 'duration': 5.548}, {'end': 7283.994, 'text': 'And in the 70s, I think he was like literally doing it with like a pen on graph paper.', 'start': 7280.329, 'duration': 3.665}, {'end': 7285.957, 'text': 'You have some configuration of squares.', 'start': 7284.235, 'duration': 1.722}, {'end': 7288.741, 'text': 'Some of the squares in the graph paper are filled in, some are not.', 'start': 7286.017, 'duration': 2.724}, {'end': 7297.83, 'text': "And then there's a rule, a single rule that tells you at the next stage which squares are filled in and which squares are not.", 'start': 7289.002, 'duration': 8.828}, {'end': 7300.473, 'text': "Sometimes an empty square gets filled in, that's called birth.", 'start': 7298.151, 'duration': 2.322}], 'summary': 'Game involves marking squares on graph paper, with birth rule.', 'duration': 25.712, 'max_score': 7274.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7274761.jpg'}, {'end': 7581.215, 'src': 'embed', 'start': 7530.13, 'weight': 0, 'content': [{'end': 7533.652, 'text': 'the gods will not allow us to predict the cellular automata.', 'start': 7530.13, 'duration': 3.522}, {'end': 7538.994, 'text': "But, uh, that's fascinating that we can't, I'm not sure what to make of it.", 'start': 7534.412, 'duration': 4.582}, {'end': 7547.897, 'text': "And there's power to calling this particular set of rules game of life, as Conway did, because not exactly sure,", 'start': 7539.194, 'duration': 8.703}, {'end': 7557.221, 'text': "but I think he had a sense that there's some core ideas here that are fundamental to life, to complex systems, to the way life emerge on earth.", 'start': 7547.897, 'duration': 9.324}, {'end': 7561.176, 'text': "I'm not sure I think Conway thought that.", 'start': 7559.332, 'duration': 1.844}, {'end': 7562.538, 'text': "It's something that I mean.", 'start': 7561.777, 'duration': 0.761}, {'end': 7565.925, 'text': 'Conway always had a rather ambivalent relationship with the game of life,', 'start': 7562.538, 'duration': 3.387}, {'end': 7573.793, 'text': 'because I think he saw it as It was certainly the thing he was most famous for in the outside world.', 'start': 7565.925, 'duration': 7.868}, {'end': 7581.215, 'text': 'And I think that he, his view, which is correct, is that he had done things that were much deeper mathematically than that.', 'start': 7574.713, 'duration': 6.502}], 'summary': "Cellular automata are unpredictable; conway's game of life has fundamental ideas for complex systems and life emergence.", 'duration': 51.085, 'max_score': 7530.13, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7530130.jpg'}], 'start': 6193.222, 'title': "Primes, infinity, and john conway's legacy", 'summary': "Discusses treating primes as random numbers, the nature of infinity in mathematics, and the legacy of mathematician john conway, exploring his impact on mathematics and computer science, including conway's game of life.", 'chapters': [{'end': 6489.031, 'start': 6193.222, 'title': 'Primes as random numbers', 'summary': 'Discusses the productive approach of treating primes as random numbers, although they are deterministic, and the usefulness of this perspective in understanding their behavior and making predictions, with a parallel drawn to the concept of free will. insights are generated by pretending primes are random, resulting in a greater understanding, despite the fact that they are deterministic, and the parallel is drawn to understanding people and machines through a similar lens.', 'duration': 295.809, 'highlights': ['The productive approach of treating primes as random numbers despite being deterministic and the usefulness of this perspective in understanding their behavior and making predictions. Treating primes as random numbers, although they are deterministic, and the usefulness of this perspective in understanding their behavior and making predictions.', 'The concept that insights are generated by pretending primes are random, resulting in a greater understanding, despite the fact that they are deterministic. Insights are generated by pretending primes are random, resulting in a greater understanding, despite the fact that they are deterministic.', 'The parallel drawn to understanding people and machines through a similar lens, treating them as beings with wants and needs, even though it may not be true. The parallel drawn to understanding people and machines through a similar lens, treating them as beings with wants and needs, even though it may not be true.']}, {'end': 6751.605, 'start': 6490.811, 'title': 'The nature of infinity in mathematics', 'summary': 'Discusses the concept of finitism, ultra-finitism, and intuitionism in mathematics, questioning the validity of infinity and its application in the real world, while highlighting the historical perspective and discomfort associated with the concept of infinity.', 'duration': 260.794, 'highlights': ['The notion of finitism, ultra-finitism, and intuitionism in mathematics is explored, challenging the concept of infinity and its practical application in the real world.', 'The discomfort with infinity is highlighted, mentioning psychological unease and suspicion regarding its potential oversimplification of reality.', 'The historical perspective of mathematics before Cantor and the modern theory of infinity is discussed, emphasizing the widespread discomfort with the notion of infinity and the agnostic approach in mathematical practices.']}, {'end': 7250.018, 'start': 6752.505, 'title': 'The legacy of john conway', 'summary': 'Explores the legacy of mathematician john conway, his playful approach to mathematics, creation of new number systems, and exploration of infinitesimals, while discussing the impact of his work on the field of mathematics and computer science, including the connection to machine learning and cellular automata.', 'duration': 497.513, 'highlights': ["John Conway's playful approach to mathematics and creation of new number systems, including a system where each number is a game, demonstrates his unique perspective and contribution to the field of mathematics.", "Conway's exploration of infinitesimals, reimagining them as certain kinds of two-player games, showcases his innovative thinking and impact on mathematical concepts.", "The discussion of Conway's influence on the field of mathematics and computer science, as well as his connection to machine learning and cellular automata, highlights the breadth and depth of his impact on various areas of study.", 'The chapter also touches on the significance of elevating mathematical and scientific thinking in society, emphasizing the potential benefits in both understanding human complexity and driving technological advancements.']}, {'end': 7600.785, 'start': 7251.747, 'title': "Conway's game of life", 'summary': "Discusses conway's game of life, a simple algorithmic game that generates complex structures, with variants and challenges, and the mystery of predicting its evolution, revealing the brilliance and frustration of its creator.", 'duration': 349.038, 'highlights': ["Conway's Game of Life is a simple algorithmic game that starts from a small set of boxes and generates rich and complex structures, with variants available on the Golly app for iOS devices.", "The challenge of predicting the evolution of the Game of Life's rules, like the rule 30 challenge, remains unsolved, reflecting the mystery and frustration surrounding it.", "Conway's work on the Game of Life reflects his Darwin-like exploration of simple ideas, resulting in the creation of a thousand-page book and challenging predictions about the game's evolution.", 'Conway saw the Game of Life as a core idea fundamental to complex systems, despite his ambivalent relationship with it and his deeper mathematical contributions.']}], 'duration': 1407.563, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA6193222.jpg', 'highlights': ['Treating primes as random numbers, despite being deterministic, provides useful perspectives for understanding their behavior and making predictions.', 'Insights are generated by pretending primes are random, resulting in a greater understanding, despite their deterministic nature.', 'The parallel drawn to understanding people and machines through a similar lens, treating them as beings with wants and needs, even though it may not be true, offers valuable insights.', 'The discomfort with infinity is highlighted, mentioning psychological unease and suspicion regarding its potential oversimplification of reality.', "John Conway's playful approach to mathematics and creation of new number systems, including a system where each number is a game, demonstrates his unique perspective and contribution to the field of mathematics.", "Conway's exploration of infinitesimals, reimagining them as certain kinds of two-player games, showcases his innovative thinking and impact on mathematical concepts.", "Conway's Game of Life is a simple algorithmic game that starts from a small set of boxes and generates rich and complex structures, with variants available on the Golly app for iOS devices.", "The challenge of predicting the evolution of the Game of Life's rules, like the rule 30 challenge, remains unsolved, reflecting the mystery and frustration surrounding it."]}, {'end': 8283.299, 'segs': [{'end': 7718.808, 'src': 'embed', 'start': 7688.898, 'weight': 4, 'content': [{'end': 7697.202, 'text': 'So a person has two symmetries, a rectangle four, a square eight, different kinds of shapes have different numbers of symmetries.', 'start': 7688.898, 'duration': 8.304}, {'end': 7701.564, 'text': "And the real observation is that that's just not like a set of things.", 'start': 7698.863, 'duration': 2.701}, {'end': 7705.06, 'text': 'um, they can be combined.', 'start': 7702.799, 'duration': 2.261}, {'end': 7707.702, 'text': 'you do one symmetry, then you do another.', 'start': 7705.06, 'duration': 2.642}, {'end': 7711.044, 'text': 'the result of that is some third symmetry.', 'start': 7707.702, 'duration': 3.342}, {'end': 7718.808, 'text': 'so a group really abstracts away this notion of saying um,', 'start': 7711.044, 'duration': 7.764}], 'summary': 'Different shapes have different numbers of symmetries, which can be combined to create new symmetries.', 'duration': 29.91, 'max_score': 7688.898, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7688898.jpg'}, {'end': 7781.481, 'src': 'embed', 'start': 7750.689, 'weight': 2, 'content': [{'end': 7754.111, 'text': 'If I shuffle and then you shuffle, the result is some symmetry.', 'start': 7750.689, 'duration': 3.422}, {'end': 7759.954, 'text': 'other kind of thing you might call a double shuffle, which is a more complicated symmetry.', 'start': 7755.052, 'duration': 4.902}, {'end': 7766.956, 'text': 'so group theory is kind of the study of the general abstract world that encompasses all these kinds of things.', 'start': 7759.954, 'duration': 7.002}, {'end': 7773.538, 'text': 'but then, of course, like lots of things that are way more complicated than that, like infinite groups of symmetries, for instance,', 'start': 7766.956, 'duration': 6.582}, {'end': 7781.481, 'text': "so they can be infinite, huh, oh yeah, okay, well, okay, ready, think about the symmetries of the line you're like.", 'start': 7773.538, 'duration': 7.943}], 'summary': 'Group theory studies symmetries in abstract worlds, including infinite groups.', 'duration': 30.792, 'max_score': 7750.689, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7750689.jpg'}, {'end': 7939.841, 'src': 'embed', 'start': 7910.588, 'weight': 0, 'content': [{'end': 7911.669, 'text': "no, it wasn't changing its shape.", 'start': 7910.588, 'duration': 1.081}, {'end': 7914.49, 'text': 'you were just wrong about what counted as a symmetry.', 'start': 7911.669, 'duration': 2.821}, {'end': 7919.211, 'text': 'now that we have this new group, the so-called lorenz group, now that we understand what the symmetries really are,', 'start': 7914.49, 'duration': 4.721}, {'end': 7922.932, 'text': 'we see it was just an illusion that the the thing was changing its shape.', 'start': 7919.211, 'duration': 3.721}, {'end': 7929.294, 'text': 'yeah, so you can then describe the sameness of things under this weirdness that is, that is general relativity, for example.', 'start': 7922.932, 'duration': 6.362}, {'end': 7939.841, 'text': 'yeah, yeah, still, um, i wish there was a simpler explanation of like exact, i mean get you know, gauge symmetries,', 'start': 7930.976, 'duration': 8.865}], 'summary': 'The symmetries in general relativity were misunderstood, clarified with the new lorenz group, revealing the illusion of shape change.', 'duration': 29.253, 'max_score': 7910.588, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7910588.jpg'}, {'end': 8042.72, 'src': 'embed', 'start': 8017.149, 'weight': 3, 'content': [{'end': 8023.21, 'text': "That's my hater formulation But at the same time, I'm sure that's very necessary to do sort of rigorous discussion.", 'start': 8017.149, 'duration': 6.061}, {'end': 8026.271, 'text': "But I feel like- But don't you think that's what gauge symmetry is like?", 'start': 8023.23, 'duration': 3.041}, {'end': 8029.532, 'text': "I mean it's not a field I know well, but it certainly seems like- Yes, it is like that.", 'start': 8026.311, 'duration': 3.221}, {'end': 8040.277, 'text': "Okay But my problem with topology, okay, and even like differential geometry is like you're talking about beautiful things.", 'start': 8029.892, 'duration': 10.385}, {'end': 8042.72, 'text': 'Like if they could be visualized.', 'start': 8040.998, 'duration': 1.722}], 'summary': 'Discussing gauge symmetry and challenges of visualizing topology and differential geometry.', 'duration': 25.571, 'max_score': 8017.149, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA8017149.jpg'}], 'start': 7600.805, 'title': 'Group theory and challenges in visualizing mathematics', 'summary': 'Introduces group theory, emphasizing its application in physics, and discusses the challenges in visualizing mathematical concepts, emphasizing the need for simplification and visual representation in disciplines like topology and differential geometry.', 'chapters': [{'end': 7944.003, 'start': 7600.805, 'title': 'Group theory and symmetry', 'summary': "Introduces group theory as the study of symmetries, illustrating how different shapes have varying numbers of symmetries and how group theory applies to physics, particularly in understanding the symmetries of physical laws and the shift in the fundamental group of symmetries before and after einstein's theory of relativity.", 'duration': 343.198, 'highlights': ['Group theory abstracts the notion of transformations and their combination as symmetries, applicable to various shapes and permutations, such as shuffling a deck of cards, where the result of applying a symmetry to the original deck is a new order of the cards. Group theory abstracts the notion of transformations and their combination as symmetries, applicable to various shapes and permutations, such as shuffling a deck of cards, where the result of applying a symmetry to the original deck is a new order of the cards.', "The concept of gauge theory and gauge symmetry is discussed as a useful notion in physics to understand the symmetries under which physical laws don't change, exemplifying the shift in the fundamental group of symmetries before and after Einstein's theory of relativity. The concept of gauge theory and gauge symmetry is discussed as a useful notion in physics to understand the symmetries under which physical laws don't change, exemplifying the shift in the fundamental group of symmetries before and after Einstein's theory of relativity.", "The symmetries of the physical world and its laws are explored, highlighting the significant shift in the fundamental group of symmetries before and after Einstein's theory of relativity, leading to a new understanding of symmetries and the sameness of things under the theory of general relativity. The symmetries of the physical world and its laws are explored, highlighting the significant shift in the fundamental group of symmetries before and after Einstein's theory of relativity, leading to a new understanding of symmetries and the sameness of things under the theory of general relativity."]}, {'end': 8283.299, 'start': 7946.744, 'title': 'Challenges in visualizing mathematics', 'summary': 'Discusses the challenges in visualizing mathematical concepts, highlighting the need for simplification and visual representation in disciplines like topology and differential geometry, and the potential for stimulating creativity through distillation and compression of ideas.', 'duration': 336.555, 'highlights': ['The need for simplification and visual representation in disciplines like topology and differential geometry The speaker emphasizes the challenges in visualizing mathematical concepts, particularly in disciplines like topology and differential geometry, and the importance of simplifying and visually representing complex ideas.', 'The potential for stimulating creativity through distillation and compression of ideas The chapter explores the potential for stimulating creativity in the mathematics community through the distillation and compression of ideas, drawing a parallel to the relation between prose and poetry as a form of compression.', 'The importance of varied approaches in explaining complex ideas The speaker underscores the importance of varied approaches in explaining complex ideas, acknowledging that different individuals respond to different modes of explanation and advocating against monoculture in teaching.']}], 'duration': 682.494, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA7600805.jpg', 'highlights': ['Group theory abstracts the notion of transformations and their combination as symmetries, applicable to various shapes and permutations, such as shuffling a deck of cards.', "The concept of gauge theory and gauge symmetry is discussed as a useful notion in physics to understand the symmetries under which physical laws don't change.", "The symmetries of the physical world and its laws are explored, highlighting the significant shift in the fundamental group of symmetries before and after Einstein's theory of relativity.", 'The need for simplification and visual representation in disciplines like topology and differential geometry.', 'The potential for stimulating creativity through distillation and compression of ideas.', 'The importance of varied approaches in explaining complex ideas.']}, {'end': 8896.381, 'segs': [{'end': 8693.949, 'src': 'embed', 'start': 8655.756, 'weight': 0, 'content': [{'end': 8662.962, 'text': "Second of all, there's like these weird rules to it that it's only three people and some projects have a huge number of people, and it's like this,", 'start': 8655.756, 'duration': 7.206}, {'end': 8666.764, 'text': "I don't know.", 'start': 8664.143, 'duration': 2.621}, {'end': 8674.926, 'text': "it doesn't kind of highlight the way science has done on some of these projects in the best possible way, but in general the prizes are great.", 'start': 8666.764, 'duration': 8.162}, {'end': 8693.949, 'text': "But what this kind of teaches me and reminds me is sometimes in your life there'll be moments when the thing that you would really like to do society would really like you to do is the thing that goes against something you believe in,", 'start': 8674.966, 'duration': 18.983}], 'summary': 'Challenges with nobel prize rules, but prizes are great.', 'duration': 38.193, 'max_score': 8655.756, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA8655756.jpg'}], 'start': 8283.299, 'title': 'Mathematics and geometric proofs', 'summary': "Explores grigori perelman's proof of the poincaré conjecture, the concept of ricci flow in deforming three-dimensional spaces, and the significance of declining the fields medal, emphasizing collaborative nature of mathematical advances and the impact of personal principles on professional decisions.", 'chapters': [{'end': 8396.094, 'start': 8283.299, 'title': "Poincaré conjecture and perelman's proof", 'summary': "Discusses the inspiring story of grigori perelman's proof of the poincaré conjecture, highlighting the need to study higher levels of geometry and the collaborative nature of mathematical advances.", 'duration': 112.795, 'highlights': ["Grigori Perelman's proof of the Poincaré conjecture showcases the recurring need to study higher levels of geometry, as seen in real-world problems.", "Perelman's understanding of the question about a certain kind of three-dimensional object demonstrates the necessity to go beyond the usual three-dimensional space, emphasizing the collaborative nature of mathematical advances involving many people like Richard Hamilton.", 'The collaborative nature of mathematical advances is exemplified by the involvement of several individuals, such as Gerhard Frey, Maser, and Ken Ribbit, in building the other pieces of the arch before the culmination of the program.', "The chapter emphasizes the need to study a higher level of geometry, which is a recurring theme in real-world problems and is exemplified by Perelman's understanding of Poincaré's question about a certain kind of three-dimensional object."]}, {'end': 8520.142, 'start': 8396.094, 'title': 'Ricci flow and geometry of three-dimensional spaces', 'summary': "Discusses the ricci flow, a process that continuously deforms three-dimensional spaces, eventually leading to the standard three-dimensional space, with the aim of proving that the original shape must have been the same as the standard shape, alluding to the idea that it's not about the geometry you think it's about, but the geometry of all geometries.", 'duration': 124.048, 'highlights': ['The Ricci flow continuously deforms three-dimensional spaces until it reaches the standard three-dimensional space, proving that the original shape must have been the same as the standard shape. The Ricci flow process continuously deforms three-dimensional spaces until it reaches the standard three-dimensional space, without any sharp transitions, providing a mechanism to understand the geometry of all three-dimensional geometries.', "The process aims to prove that it's not about the geometry you think it's about, but the geometry of all geometries. The discussion emphasizes that the nature of the proof lies in the understanding that it's not about the geometry you think it's about, but the geometry of all geometries, which requires being 'jerked out of Flatland' to comprehend.", "The hard part is proving that the process doesn't acquire any singularities or sharp kinks along the way. The difficulty lies in proving that the process doesn't acquire any singularities or sharp kinks, ensuring a smooth trajectory through the space of all three-dimensional geometries."]}, {'end': 8896.381, 'start': 8520.182, 'title': 'Fields medal and nobel prize: a mathematical perspective', 'summary': "Discusses the significance of turning down the fields medal, the complexities of mathematical awards, and the impact of personal principles on professional decisions, with a focus on a mathematician's decision to decline the prize.", 'duration': 376.199, 'highlights': ["The chapter discusses the significance of turning down the Fields Medal and the impact of personal principles on professional decisions. The conversation revolves around the mathematician's decision to decline the Fields Medal and the potential reasons behind it, highlighting the clash between personal principles and societal expectations in professional pursuits.", "The complexities of mathematical awards, including the Nobel Prize and Fields Medal, are explored, with emphasis on their limitations and the underlying hierarchies and individualism inherent in the process. The discussion delves into the complexities of mathematical awards, such as the Nobel Prize and Fields Medal, shedding light on their limitations, hierarchical structures, individualism, and the potential impact of personal experiences and ideologies on one's perception of such accolades.", 'The impact of personal principles on professional decisions is emphasized, illustrating the significance of integrity and the potential conflicts between personal beliefs and societal expectations. The conversation highlights the importance of upholding personal principles in professional decisions, emphasizing the value of integrity and the potential conflicts that may arise between individual beliefs and the expectations of society.']}], 'duration': 613.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA8283299.jpg', 'highlights': ["Grigori Perelman's proof of the Poincaré conjecture showcases the recurring need to study higher levels of geometry, as seen in real-world problems.", 'The Ricci flow continuously deforms three-dimensional spaces until it reaches the standard three-dimensional space, proving that the original shape must have been the same as the standard shape.', 'The chapter discusses the significance of turning down the Fields Medal and the impact of personal principles on professional decisions.']}, {'end': 9698.273, 'segs': [{'end': 9079.531, 'src': 'embed', 'start': 9055.939, 'weight': 5, 'content': [{'end': 9063.743, 'text': "i mean he knew me well enough to say like you're gonna learn because you're gonna be working on a problem and then there's gonna be a fact from ega you need in order to solve your problem that you want to solve,", 'start': 9055.939, 'duration': 7.804}, {'end': 9065.304, 'text': "and that's how you're going to learn it.", 'start': 9063.743, 'duration': 1.561}, {'end': 9068.065, 'text': "you're not going to learn it without a problem to bring you into it.", 'start': 9065.304, 'duration': 2.761}, {'end': 9069.726, 'text': 'and so for a lot of people, i think,', 'start': 9068.065, 'duration': 1.661}, {'end': 9079.531, 'text': "if you're like i'm trying to understand machine learning and i'm like i can see that there's sort of some mathematical technology that i don't have,", 'start': 9069.726, 'duration': 9.805}], 'summary': 'Learning through problem-solving is essential in understanding machine learning and related mathematical technology.', 'duration': 23.592, 'max_score': 9055.939, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA9055939.jpg'}, {'end': 9374.039, 'src': 'embed', 'start': 9345.669, 'weight': 1, 'content': [{'end': 9348.431, 'text': 'I actually do have to give advice to two young people all the time.', 'start': 9345.669, 'duration': 2.762}, {'end': 9350.112, 'text': "They don't listen, but I still give it.", 'start': 9348.451, 'duration': 1.661}, {'end': 9355.315, 'text': 'You know one thing I often say to students.', 'start': 9353.154, 'duration': 2.161}, {'end': 9360.078, 'text': "I don't think I've actually said this to my kids yet, but I say it to students a lot, is you know?", 'start': 9355.315, 'duration': 4.763}, {'end': 9366.512, 'text': 'you come to these decision points And everybody is beset by self-doubt, right?', 'start': 9360.078, 'duration': 6.434}, {'end': 9374.039, 'text': "It's like not sure, like what they're capable of like, not sure what they really wanna do.", 'start': 9366.632, 'duration': 7.407}], 'summary': 'Advising young people on decision-making and self-doubt.', 'duration': 28.37, 'max_score': 9345.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA9345669.jpg'}, {'end': 9596.637, 'src': 'embed', 'start': 9570.302, 'weight': 0, 'content': [{'end': 9576.746, 'text': 'you know, the mathematical argument was like um, about certain reasons for behaving in a certain way.', 'start': 9570.302, 'duration': 6.444}, {'end': 9581.068, 'text': "but he basically said like look, like math doesn't tell you that god's there or not.", 'start': 9576.746, 'duration': 4.322}, {'end': 9584.67, 'text': "like if god's there, he'll tell you you know, you don't?", 'start': 9581.068, 'duration': 3.602}, {'end': 9585.731, 'text': 'i love this.', 'start': 9584.67, 'duration': 1.061}, {'end': 9586.771, 'text': 'so you have.', 'start': 9585.731, 'duration': 1.04}, {'end': 9589.132, 'text': 'you have mathematics, you have.', 'start': 9586.771, 'duration': 2.361}, {'end': 9589.733, 'text': 'uh, what do you?', 'start': 9589.132, 'duration': 0.601}, {'end': 9596.637, 'text': "what do you have like ways to explore the mind, let's say psychedelics you have, like incredible technology.", 'start': 9589.733, 'duration': 6.904}], 'summary': "Math doesn't prove god's existence; various tools like psychedelics and technology exist for mind exploration.", 'duration': 26.335, 'max_score': 9570.302, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA9570302.jpg'}, {'end': 9666.39, 'src': 'embed', 'start': 9638.034, 'weight': 4, 'content': [{'end': 9641.236, 'text': 'And I probably should have drawn this picture earlier in the process.', 'start': 9638.034, 'duration': 3.202}, {'end': 9643.057, 'text': 'Maybe it would have made my organization easier.', 'start': 9641.276, 'duration': 1.781}, {'end': 9644.578, 'text': 'I actually drew it only at the end.', 'start': 9643.097, 'duration': 1.481}, {'end': 9648.26, 'text': 'And many of the things we talked about are on this map.', 'start': 9645.498, 'duration': 2.762}, {'end': 9652.563, 'text': 'The connections are yet to be fully dissected and investigated.', 'start': 9648.7, 'duration': 3.863}, {'end': 9655.945, 'text': 'And yes, God is in the picture.', 'start': 9652.703, 'duration': 3.242}, {'end': 9658.743, 'text': 'Right on the edge, right on the edge, not in the center.', 'start': 9656.761, 'duration': 1.982}, {'end': 9661.906, 'text': 'Thank you so much for talking to me.', 'start': 9660.825, 'duration': 1.081}, {'end': 9664.588, 'text': 'It is a huge honor that you would waste your valuable time with me.', 'start': 9661.966, 'duration': 2.622}, {'end': 9666.39, 'text': 'Thank you, Lex.', 'start': 9665.849, 'duration': 0.541}], 'summary': 'Late visualization hindered organization. god on edge. grateful for time.', 'duration': 28.356, 'max_score': 9638.034, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA9638034.jpg'}], 'start': 8896.401, 'title': 'Mathematical thinking and problem-driven learning', 'summary': "Explores integrating mathematical thinking into daily life, learning from books like martin gardner's, and solving contest problems. it emphasizes problem-driven learning, connecting math to real problems, and enjoying the process of overcoming difficulties.", 'chapters': [{'end': 8997.797, 'start': 8896.401, 'title': 'Incorporating mathematical thinking into your life', 'summary': "Explores various ways for people of different ages to learn and integrate mathematical thinking into their lives, including reading math textbooks, exploring martin gardner's books, and working on contest style problems.", 'duration': 101.396, 'highlights': ['Reading math textbooks can be a viable way for some people to learn math, while not being effective for others.', "Martin Gardner's books, which embody the spirit of play and joy, offer a fresh and enjoyable way into the subject of math, providing a different approach from formal textbooks.", 'Working on contest style problems, such as Russian and Bulgarian problems, can be a motivating way to delve into math for certain individuals.', "There's no one-size-fits-all approach to learning math; it's a journey of self-knowledge, and different methods work for different people."]}, {'end': 9698.273, 'start': 8997.797, 'title': 'Importance of problem-driven learning in math', 'summary': 'Emphasizes the importance of problem-driven learning, advising learners to connect their mathematical learning to problems they care about, as learning math from a textbook or video requires accepting and enjoying the process of overcoming difficulties.', 'duration': 700.476, 'highlights': ['The importance of problem-driven learning The chapter emphasizes the significance of letting problems that individuals care about drive their mathematical learning, connecting mathematical learning to real-world problems.', 'The value of enjoying the process of overcoming difficulties in learning math The chapter discusses the importance of falling in love with the process of doing something hard, emphasizing the value of enjoying the journey of overcoming challenges and becoming a better person.', 'Advice on making high self-esteem choices The chapter advises making high self-esteem choices, encouraging individuals to act in the way that a more confident version of themselves would act when faced with decisions.', "The limitations of using mathematics to answer transcendent questions The chapter explores the limitations of using mathematics to answer transcendent questions, as evidenced by Blaise Pascal's belief in God being based on a mystical experience rather than a mathematical argument."]}], 'duration': 801.872, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/tueAcSiiqYA/pics/tueAcSiiqYA8896401.jpg', 'highlights': ['The chapter emphasizes problem-driven learning and connecting math to real problems.', "Martin Gardner's books offer a fresh and enjoyable way into the subject of math.", 'Working on contest style problems can be a motivating way to delve into math.', "There's no one-size-fits-all approach to learning math; different methods work for different people.", 'The chapter discusses the importance of falling in love with the process of doing something hard.', 'The chapter advises making high self-esteem choices.', 'The chapter explores the limitations of using mathematics to answer transcendent questions.']}], 'highlights': ['The intertwining of algebra and geometry, emphasizing the commutative property and its connection to a rectangle.', 'The fundamental role of symmetry in contemporary mathematics and its relevance to artificial intelligence.', 'The intertwining of romanticism and mathematics in the 19th century, exemplified by mathematicians like Everest Galois and Poincaré.', "The impact of Emmy Noether's idea on modern mathematics and the significance of different modes of teaching mathematics.", "The human struggle and drama behind Andrew Wiles' proof of Fermat's Last Theorem and the concept of simplicity in solving difficult problems.", 'The significance of varied notions of distance, application of p-adic distance in number theory, and the importance of accurate and complex distance measures in AI.', 'The concept of prime numbers as the building blocks of all numbers and the confirmation of their infinite nature.', 'The treatment of primes as random numbers and the insights generated by pretending primes are random.', "John Conway's playful approach to mathematics, creation of new number systems, and exploration of infinitesimals.", 'The concept of group theory abstracting the notion of transformations and their combination as symmetries, applicable to various shapes and permutations.', "Grigori Perelman's proof of the Poincaré conjecture and the significance of turning down the Fields Medal.", 'The emphasis on problem-driven learning, connecting math to real problems, and the importance of falling in love with the process of doing something hard.']}