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

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00:00:00 - 01:00:00

Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries in this Lex Fridman podcast. He discusses how a two-dimensional creature could comprehend a three-dimensional world, and how mathematics can be used to explore the universe beyond our understanding. If the answer to whether extra dimensions exist is no, it would become a sterile question.

00:00:00 Jordan Ellenberg discusses the beauty and power of mathematics in his book "How Not to Be Wrong." He also discusses the importance of visual proofs in geometry.
00:05:00 Jordan Ellenberg discusses his love of geometry, how it helped him understand the world better, and how teaching it to students can be a transformative experience.
00:10:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, discussing how symmetries can be used to understand similarities and differences between objects. He also discusses how symmetry is fundamental to artificial intelligence and how understanding symmetry is essential to solving problems.
00:15:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, discussing how it is more complicated than previously thought and that it must be formalizable. He talks about a conjecture made by Poincare, and how it impacts the way people think about science and mathematics.
00:20:00 Jordan Ellenberg discusses the role of mathematics in the development of modern science and society, and how it is inspired by the romanticism of the era. He recommends the book Duel at Dawn by Amir Alexander as a fascinating read.
00:25:00 Jordan Ellenberg, a mathematician, is interviewed about his work in high-dimensional shapes and geometry. He discusses how Poincare's conjecture that geometry exists in higher dimensions is now accepted. He also discusses his work in topology and analysis situs.
00:30:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and explains the poincare conjecture. He talks about the need for a rich formalism to enable mathematical thinking about higher dimensions, and how the shape of the universe is unknown but may be flat.
00:35:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, explaining that intrinsic and extrinsic theories exist. He discusses the concept of living on a knot in a knot in three-dimensional space, and how it would be difficult for a cognitive being to internalize the experience. Finally, he discusses the sphere and how it is ubiquitous in the world.
00:40:00 Jordan Ellenberg discusses mathematics of highdimensional shapes and geometries in this Lex Fridman podcast. He discusses how a two-dimensional creature could comprehend a three-dimensional world, and how mathematics can be used to explore the universe beyond our understanding. If the answer to whether extra dimensions exist is no, it would become a sterile question.
00:45:00 The video discusses Jordan Ellenberg's book Mathematics of High-Dimensional Shapes and Geometries, which discusses the Square and the Sphere's differing perspectives on what a cube would be like. The Sphere explains that a cube would be like a three-dimensional version of the Square, while the Square is excited to finally understand what a cube would be like. However, the Sphere is unable to conceptualize that there could be yet another dimension, leading to a religious allegory that I don't really understand theologically. The video finishes by discussing how each person's perspective on a question can be different, and how a group of people can get heated over an answer.
00:50:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries in this Lex Fridman Podcast episode. He discusses how two holes can be considered to be the same hole, and how this relates to the theory of homology. He also discusses how fasting can help clear the mind for geometry discussion.
00:55:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries in his book "How to Think About Numbers." He argues that books are inadequate ways to impart this information, and instead relies on videos and programming to illustrate concepts. Grant Sanderson also discusses visualizing mathematical concepts in his book "How to Visualize Numbers."

01:00:00 - 02:00:00

Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries with Lex Fridman. He talks about the pseudo prime numbers and the twin prime conjecture, and how treating prime numbers as a random process can lead to insights not possible when looking at them as deterministic objects. He also discusses the implications of infinity being a hack in mathematics, and how it can lead to trouble in understanding the real world.

01:00:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, noting that while he sees himself as very different from traditional researchers, his work is creating something new and powerful. He discusses the potential for computers to compete in the field of mathematics as well, pointing out that there is already a market for specialized computer algebra programs.
01:05:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and how difficult it can be to determine whether a knot is slice. Lisa Picarillo solved the problem a few years ago, and the journey to find the proof was difficult but ultimately rewarding. Ellenberg argues that mathematics should be beautiful and simple, and that the human story of perseverance and struggle is the interesting part of mathematics.
01:10:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, explaining that Fermat's last theorem is not as simple as it could be and that it was difficult for mathematicians to prove it. He also discusses the development of number theory over time and the importance of the Fermat problem. Finally, he mentions that Andrew Wiles was the first to prove the theorem.
01:15:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, highlighting the importance of deformation theory in proving the concept of unique factorization. He also discusses the concept of distance and how it is not pre-determined by humans.
01:20:00 In this video, Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries. He talks about the periodic distance, which is a crazy distance, and how it is important for autocomplete and machine translation. He also talks about how close two numbers need to be in order for them to be considered close. He says that two to a large power is a multiplication very small number, and two to a negative power is a very big number. He explains that binary numbers are written backwards, and that this is a good metaphor for how close two numbers need to be in order to be considered close. He says that the goal of mathematics is to help humans understand things, and that proving new theorems is one way to test that. He says that the benchmark for proving something is that it is a new theorem, but that the goal is to understand the problem rather than to know whether it is true or false.
01:25:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries. He argues that in order to understand something, we need to be able to explain it simply. He goes on to say that real things are not always simple, and that in order to understand something, we need to start with a complex idea and simplify it.
01:30:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and how they can be difficult to understand. He also talks about prime numbers and how they are a central part of mathematics, and how they can be difficult to define. He concludes the talk with a discussion of prime gaps and bounded gaps, and how they are important for the future of mathematics.
01:35:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, including the concept of pseudoprimes. He explains that there are various categories of fake prime numbers, and that there is a beautiful geometric proof for determining if a number is prime.
01:40:00 Jordan Ellenberg discusses mathematics of high-dimensional shapes and geometries with Lex Fridman. He talks about the pseudo prime numbers and the twin prime conjecture, which states that on average, prime numbers get farther and farther apart. He explains that it is not random, but deterministic, and that it is phenomenally useful to think of prime numbers as randomly distributed.
01:45:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and how treating prime numbers as a random process can lead to insights not possible when looking at them as deterministic objects. He also discusses the implications of infinity being a hack in mathematics, and how it can lead to trouble in understanding the real world.
01:50:00 Jordan Ellenberg discusses mathematics of high-dimensional shapes and geometries, with a focus on the history of infinity and the implications of computational thinking. He comments on the modern trend of treating infinity as a less-rigorous concept, and cites examples of where infinity is used in computer science and engineering.
01:55:00 Jordan Ellenberg discusses the work of mathematician John Conway and talks about his own experiences as a mathematician growing up in the Soviet Union. Conway was known for his playful nature and for his mathematical analyses of games.

02:00:00 - 02:40:00

Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and how it is useful for physics. He also talks about how topology is a difficult field to understand, and how differential geometry and topology can be beautiful but are hidden underneath dense mathematics.

02:00:00 According to the video, Jordan Ellenberg was a mathematician who worked on high-dimensional shapes and geometries. His research overlapped in some ways with Lex Fridman's, and Fridman would often ask Ellenberg questions in the common room. Ellenberg's responses were so rich and went so many places that they taught him a great deal about mathematics and life in general. Ellenberg's loss is a huge loss to the field of mathematics, and his game of life (a simple algorithm for playing a game with marking squares on a piece of paper) is a wonderful laboratory for exploring complex systems.
02:05:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, pointing out that there are an infinite number of symmetries of a given object. He also mentions group theory, which is the study of the general abstract world that encompasses all these kinds of things.
02:10:00 Jordan Ellenberg discusses the mathematics of high-dimensional shapes and geometries, and how it is useful for physics. He also talks about how topology is a difficult field to understand, and how differential geometry and topology can be beautiful but are hidden underneath dense mathematics.
02:15:00 Jordan Ellenberg discusses how geometry can be used in different ways to help explain real world phenomena. He discusses how geometry can be used to explain poincare's conjecture, and how it is a culmination of many people's work. He also discusses how poetry can be seen as a way to compress information.
02:20:00 Jordan Ellenberg discusses mathematics of high-dimensional shapes and geometries, discussing the reachy flow mechanism and its importance in proving the original shape of the space is the same as the standard space. He also comments on the turning down of a Fields Medal, citing the principle at hand as something against which he must stand.
02:25:00 Jordan Ellenberg discusses the idea of the mathematical Olympiad, how it is not actually a competition where winners receive awards, and how one can integrate mathematical thinking into one's life in a way that is personal to them.
02:30:00 Jordan Ellenberg discusses how he approaches difficult mathematics problems by "falling in love with the process of doing something hard" and enjoying the "enjoying the suffering of it" despite not always understanding the details. He also cites the example of Navy SEAL David Goggins, who has completed many ultra-marathons despite enduring many difficult moments along the way.
02:35:00 Jordan Ellenberg discusses how mathematics underlies the way humans think, and how it can be used to explore the mind. He also discusses how beliefs in God can be based on a person's own experiences, rather than being based on mathematical arguments.
02:40:00 Jordan Ellenberg discusses mathematics in a fascinating way, exploring complex ideas while remaining clear and specific. He also shares an interesting analogy involving x-ray specs and the world.