Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The n-category Cafe.

In this episode, we dive into John Baez and John Huerta’s paper “The Algebra of Grand Unified Theories” which was awarded the Levi Conant Prize in 2013. The paper gives a crash course in the representation theory underlying the Standard Model of particle physics and its three most well known Grand Unified Theories (GUTs): the SU(5), SO(10) (aka Spin(10)), and Pati-Salam theories. The main result of Baez-Huerta is that the particle representations underlying the three GUTs can in fact be unified via a commutative diagram. We dive deep into the numerology of the standard model to see how the SU(5) theory naturally arises. We then make brief remarks about SO(10) and Pati-Salam theories in order to state the Baez-Huerta theorem about their organization into a commutative square: a unification among grand unifications!Notes:

- While we do provide a crash course on SU(2) and spin, some representation theory jargon is used at times in our discussion. Those unfamiliar should just forge ahead!
- We work in Euclidean signature instead of Lorentzian signature. Other than keeping track of minus signs, no essential details are changed.

- 00:00: Introduction
- 05:50: Climate change
- 09:40: Crackpot index
- 14:50: Eric Weinstein, Brian Keating, Geometric Unity
- 18:13: Overview of “The Algebra of Grand Unified Theories” paper
- 25:40: Overview of Standard Model and GUTs
- 34:25: SU(2), spin, isospin of nucleons 40:22: SO(4), Spin(4), double cover
- 44:24: three kinds of spin

- 49:35: electron and neutrino
- 58:40: quarks
- 1:04:51: the three generations of the Standard Model
- 1:08:25: isospin quantum numbers
- 1:17:11: U(1) representations (“charge”)
- 1:29:01: hypercharge
- 1:34:00: strong force and color
- 1:36:50: SU(3)
- 1:40:45: antiparticles

- 1:41:16: 32 = 2^5 particles
- 1:45:05: Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching
- 2:05:17: Exterior algebra of C^5 and more hypercharge matching
- 2:37:32: SU(5) rep extends Standard Model rep

- 2:41:42: SO(10) rep: brief remarks
- 2:46:28: Pati-Salam rep: brief remarks
- 2:47:17: Commutative diagram: main result
- 2:49:12: What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

If you would like to support this series and future such projects:

Paypal: tim@timothynguyen.org

Bitcoin: 33thftjoPTHFajj8wJFcCB9sFiyQLFVp8S

Ethereum: 0x166a977F411d6f220cF8A56065D16B4FF08a246D

]]>Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The n-category Cafe.

In this episode, we dive into John Baez and John Huerta’s paper “The Algebra of Grand Unified Theories” which was awarded the Levi Conant Prize in 2013. The paper gives a crash course in the representation theory underlying the Standard Model of particle physics and its three most well known Grand Unified Theories (GUTs): the SU(5), SO(10) (aka Spin(10)), and Pati-Salam theories. The main result of Baez-Huerta is that the particle representations underlying the three GUTs can in fact be unified via a commutative diagram. We dive deep into the numerology of the standard model to see how the SU(5) theory naturally arises. We then make brief remarks about SO(10) and Pati-Salam theories in order to state the Baez-Huerta theorem about their organization into a commutative square: a unification among grand unifications!Notes:

- While we do provide a crash course on SU(2) and spin, some representation theory jargon is used at times in our discussion. Those unfamiliar should just forge ahead!
- We work in Euclidean signature instead of Lorentzian signature. Other than keeping track of minus signs, no essential details are changed.

- 00:00: Introduction
- 05:50: Climate change
- 09:40: Crackpot index
- 14:50: Eric Weinstein, Brian Keating, Geometric Unity
- 18:13: Overview of “The Algebra of Grand Unified Theories” paper
- 25:40: Overview of Standard Model and GUTs
- 34:25: SU(2), spin, isospin of nucleons 40:22: SO(4), Spin(4), double cover
- 44:24: three kinds of spin

- 49:35: electron and neutrino
- 58:40: quarks
- 1:04:51: the three generations of the Standard Model
- 1:08:25: isospin quantum numbers
- 1:17:11: U(1) representations (“charge”)
- 1:29:01: hypercharge
- 1:34:00: strong force and color
- 1:36:50: SU(3)
- 1:40:45: antiparticles

- 1:41:16: 32 = 2^5 particles
- 1:45:05: Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching
- 2:05:17: Exterior algebra of C^5 and more hypercharge matching
- 2:37:32: SU(5) rep extends Standard Model rep

- 2:41:42: SO(10) rep: brief remarks
- 2:46:28: Pati-Salam rep: brief remarks
- 2:47:17: Commutative diagram: main result
- 2:49:12: What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

If you would like to support this series and future such projects:

Paypal: tim@timothynguyen.org

Bitcoin: 33thftjoPTHFajj8wJFcCB9sFiyQLFVp8S

Ethereum: 0x166a977F411d6f220cF8A56065D16B4FF08a246D

]]>Additionally, John is a prolific writer and blogger. This first began with John’s column This Week's Finds in Mathematical Physics, which ran 300 issues between 1993 and 2010, which then continued in the form of his ongoing blog Azimuth. Last but not least, John is also a host and contributor of the popular blog The n-category Cafe.

In this episode, we dive into John Baez and John Huerta’s paper “The Algebra of Grand Unified Theories” which was awarded the Levi Conant Prize in 2013. The paper gives a crash course in the representation theory underlying the Standard Model of particle physics and its three most well known Grand Unified Theories (GUTs): the SU(5), SO(10) (aka Spin(10)), and Pati-Salam theories. The main result of Baez-Huerta is that the particle representations underlying the three GUTs can in fact be unified via a commutative diagram. We dive deep into the numerology of the standard model to see how the SU(5) theory naturally arises. We then make brief remarks about SO(10) and Pati-Salam theories in order to state the Baez-Huerta theorem about their organization into a commutative square: a unification among grand unifications!Notes:

- While we do provide a crash course on SU(2) and spin, some representation theory jargon is used at times in our discussion. Those unfamiliar should just forge ahead!
- We work in Euclidean signature instead of Lorentzian signature. Other than keeping track of minus signs, no essential details are changed.

- 00:00: Introduction
- 05:50: Climate change
- 09:40: Crackpot index
- 14:50: Eric Weinstein, Brian Keating, Geometric Unity
- 18:13: Overview of “The Algebra of Grand Unified Theories” paper
- 25:40: Overview of Standard Model and GUTs
- 34:25: SU(2), spin, isospin of nucleons 40:22: SO(4), Spin(4), double cover
- 44:24: three kinds of spin

- 49:35: electron and neutrino
- 58:40: quarks
- 1:04:51: the three generations of the Standard Model
- 1:08:25: isospin quantum numbers
- 1:17:11: U(1) representations (“charge”)
- 1:29:01: hypercharge
- 1:34:00: strong force and color
- 1:36:50: SU(3)
- 1:40:45: antiparticles

- 1:41:16: 32 = 2^5 particles
- 1:45:05: Mapping SU(3) x SU(2) x U(1) to SU(5) and hypercharge matching
- 2:05:17: Exterior algebra of C^5 and more hypercharge matching
- 2:37:32: SU(5) rep extends Standard Model rep

- 2:41:42: SO(10) rep: brief remarks
- 2:46:28: Pati-Salam rep: brief remarks
- 2:47:17: Commutative diagram: main result
- 2:49:12: What about the physics? Spontaneous symmetry breaking and the Higgs mechanism

Twitter: @iamtimnguyen

Webpage: http://www.timothynguyen.org

If you would like to support this series and future such projects:

Paypal: tim@timothynguyen.org

Bitcoin: 33thftjoPTHFajj8wJFcCB9sFiyQLFVp8S

Ethereum: 0x166a977F411d6f220cF8A56065D16B4FF08a246D

]]>In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

- 00:00:00 : Introduction
- 00:03:07 : How did you get into category theory?
- 00:06:29 : Outline of podcast
- 00:09:21 : Motivating category theory
- 00:11:35 : Analogy: Object Oriented Programming
- 00:12:32 : Definition of category
- 00:18:50 : Example: Category of sets
- 00:20:17 : Example: Matrix category
- 00:25:45 : Example: Preordered set (poset) is a category
- 00:33:43 : Example: Category of finite-dimensional vector spaces
- 00:37:46 : Forgetful functor
- 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!
- 00:40:06 : Definition of functor
- 00:42:01 : Example: API change between programming languages is a functor
- 00:44:23 : Example: Groups, group homomorphisms are categories and functors
- 00:47:33 : Resume definition of functor
- 00:49:14 : Example: Functor between poset categories = order-preserving function
- 00:52:28 : Hom Functors. Things are getting meta (no not the tech company)
- 00:57:27 : Category theory is beautiful because of its rigidity
- 01:00:54 : Contravariant functor
- 01:03:23 : Definition: Presheaf
- 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum.
- 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma)
- 01:12:10 : Algebraic topology motivated category theory
- 01:15:44 : Definition: Natural transformation
- 01:19:21 : Example: Indexing category
- 01:21:54 : Example: Change of currency as natural transformation
- 01:25:35 : Isomorphism and natural isomorphism
- 01:27:34 : Notion of isomorphism in different categories
- 01:30:00 : Yoneda Lemma
- 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation
- 01:42:33 : Analogy between Yoneda Lemma and linear algebra
- 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors
- 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.
- 01:55:15 : Language Category
- 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

Further Reading:

- Tai-Danae's Blog: https://www.math3ma.com/categories
- Tai-Danae Bradley. "What is applied category theory?" https://arxiv.org/pdf/1809.05923.pdf
- Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos. "An enriched category theory of language: from syntax to semantics." https://arxiv.org/pdf/2106.07890.pdf

In this episode, we provide a compressed crash course in category theory. We provide definitions and plenty of basic examples for all the basic notions: objects, morphisms, categories, functors, natural transformations. We also discuss the first basic result in category theory which is the Yoneda Lemma. We conclude with a discussion of how Tai-Danae has used category-theoretic methods in her work on language modeling, in particular, in how the passing from syntax to semantics can be realized through category-theoretic notions.

Originally published on July 20, 2022 on YouTube: https://youtu.be/Gz8W1r90olc

Timestamps:

- 00:00:00 : Introduction
- 00:03:07 : How did you get into category theory?
- 00:06:29 : Outline of podcast
- 00:09:21 : Motivating category theory
- 00:11:35 : Analogy: Object Oriented Programming
- 00:12:32 : Definition of category
- 00:18:50 : Example: Category of sets
- 00:20:17 : Example: Matrix category
- 00:25:45 : Example: Preordered set (poset) is a category
- 00:33:43 : Example: Category of finite-dimensional vector spaces
- 00:37:46 : Forgetful functor
- 00:39:15 : Fruity example of forgetful functor: Forget race, gender, we're all part of humanity!
- 00:40:06 : Definition of functor
- 00:42:01 : Example: API change between programming languages is a functor
- 00:44:23 : Example: Groups, group homomorphisms are categories and functors
- 00:47:33 : Resume definition of functor
- 00:49:14 : Example: Functor between poset categories = order-preserving function
- 00:52:28 : Hom Functors. Things are getting meta (no not the tech company)
- 00:57:27 : Category theory is beautiful because of its rigidity
- 01:00:54 : Contravariant functor
- 01:03:23 : Definition: Presheaf
- 01:04:04 : Why are things meta? Arrows, arrows between arrows, categories of categories, ad infinitum.
- 01:07:38 : Probing a space with maps (prelude to Yoneda Lemma)
- 01:12:10 : Algebraic topology motivated category theory
- 01:15:44 : Definition: Natural transformation
- 01:19:21 : Example: Indexing category
- 01:21:54 : Example: Change of currency as natural transformation
- 01:25:35 : Isomorphism and natural isomorphism
- 01:27:34 : Notion of isomorphism in different categories
- 01:30:00 : Yoneda Lemma
- 01:33:46 : Example for Yoneda Lemma: Identity functor and evaluation natural transformation
- 01:42:33 : Analogy between Yoneda Lemma and linear algebra
- 01:46:06 : Corollary of Yoneda Lemma: Isomorphism of objects = Isomorphism of hom functors
- 01:50:40 : Yoneda embedding is fully faithful. Reasoning about this.
- 01:55:15 : Language Category
- 02:03:10 : Tai-Danae's paper: "An enriched category theory of language: from syntax to semantics"

Further Reading:

- Tai-Danae's Blog: https://www.math3ma.com/categories
- Tai-Danae Bradley. "What is applied category theory?" https://arxiv.org/pdf/1809.05923.pdf
- Tai-Danae Bradley, John Terilla, Yiannis Vlassopoulos. "An enriched category theory of language: from syntax to semantics." https://arxiv.org/pdf/2106.07890.pdf

In this episode, John and I discuss a hodgepodge of topics in spectral graph theory. We start off light and discuss the famous Braess's Paradox in which traffic congestion can be increased by opening a road. We then discuss the graph Laplacian to enable us to present Cheeger's Theorem, a beautiful result relating graph bottlenecks to graph eigenvalues. We then discuss various graph embedding and clustering results, and end with a discussion of the PageRank algorithm that powers Google search.

Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg

Timestamps:

I. Introduction

- 00:00: Introduction
- 04:30: Being a professional mathematician and academia vs industry
- 09:41: John's taste in mathematics
- 13:00: Outline
- 17:23: Braess's Paradox: "Opening a highway can increase traffic congestion."
- 25:34: Prisoner's Dilemma. We need social forcing mechanisms to avoid undesirable outcomes (traffic jams).

II. Spectral Graph Theory Basics

- 31:20: What is a graph
- 36:33: Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management.
- 42:44: Quantifying bottlenecks: Cheeger's constant
- 46:43: Cheeger's constant sample computations
- 52:07: NP Hardness
- 55:48: Graph Laplacian
- 1:00:27: Graph Laplacian: 1-dimensional example

III. Cheeger's Inequality and Harmonic Oscillators

- 1:07:35: Cheeger's Inequality: Statement
- 1:09:37: Cheeger's Inequality: A great example of beautiful mathematics
- 1:10:46: Cheeger's Inequality: Computationally tractable approximation of Cheeger's constant
- 1:19:16: Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger's inequality
- 1:22:20: Interlude: Tutte's Spring Embedding Theorem (planar embeddings in terms of springs)
- 1:29:45: Interlude: Graph drawing using eigenfunction

IV. Graph bisection and clustering

- 1:38:26: Summary thus far and graph bisection
- 1:42:44: Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection
- 1:43:40: Graph bisection: 1-dimensional intuition
- 1:47:43: Spectral graph clustering (complementary to graph bisection)

V. Markov chains and PageRank

- 1:52:10: PageRank: Google's algorithm for ranking search results
- 1:53:44: PageRank: Markov chain (Markov matrix)
- 1:57:32: PageRank: Stationary distribution
- 2:00:20: Perron-Frobenius Theorem
- 2:06:10: Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing
- 2:07:56: Conclusion: State of the field, Urschel's recent results
- 2:10:28: Joke: Two kinds of mathematicians

Further Reading:

- A. Ng, M. Jordan, Y. Weiss. "On Spectral Clustering: Analysis and an algorithm"
- D. Spielman. "Spectral and Algebraic Graph Theory"

In this episode, John and I discuss a hodgepodge of topics in spectral graph theory. We start off light and discuss the famous Braess's Paradox in which traffic congestion can be increased by opening a road. We then discuss the graph Laplacian to enable us to present Cheeger's Theorem, a beautiful result relating graph bottlenecks to graph eigenvalues. We then discuss various graph embedding and clustering results, and end with a discussion of the PageRank algorithm that powers Google search.

Originally published on June 9, 2022 on YouTube: https://youtu.be/O6k0JRpA2mg

Timestamps:

I. Introduction

- 00:00: Introduction
- 04:30: Being a professional mathematician and academia vs industry
- 09:41: John's taste in mathematics
- 13:00: Outline
- 17:23: Braess's Paradox: "Opening a highway can increase traffic congestion."
- 25:34: Prisoner's Dilemma. We need social forcing mechanisms to avoid undesirable outcomes (traffic jams).

II. Spectral Graph Theory Basics

- 31:20: What is a graph
- 36:33: Graph bottlenecks. Practical situations: Task assignment, the economy, organizational management.
- 42:44: Quantifying bottlenecks: Cheeger's constant
- 46:43: Cheeger's constant sample computations
- 52:07: NP Hardness
- 55:48: Graph Laplacian
- 1:00:27: Graph Laplacian: 1-dimensional example

III. Cheeger's Inequality and Harmonic Oscillators

- 1:07:35: Cheeger's Inequality: Statement
- 1:09:37: Cheeger's Inequality: A great example of beautiful mathematics
- 1:10:46: Cheeger's Inequality: Computationally tractable approximation of Cheeger's constant
- 1:19:16: Harmonic oscillators: Springs heuristic for lambda_2 and Cheeger's inequality
- 1:22:20: Interlude: Tutte's Spring Embedding Theorem (planar embeddings in terms of springs)
- 1:29:45: Interlude: Graph drawing using eigenfunction

IV. Graph bisection and clustering

- 1:38:26: Summary thus far and graph bisection
- 1:42:44: Graph bisection: Large eigenvalues for PCA vs low eigenvalues for spectral bisection
- 1:43:40: Graph bisection: 1-dimensional intuition
- 1:47:43: Spectral graph clustering (complementary to graph bisection)

V. Markov chains and PageRank

- 1:52:10: PageRank: Google's algorithm for ranking search results
- 1:53:44: PageRank: Markov chain (Markov matrix)
- 1:57:32: PageRank: Stationary distribution
- 2:00:20: Perron-Frobenius Theorem
- 2:06:10: Spectral gap: Analogy between bottlenecks for graphs and bottlenecks for Markov chain mixing
- 2:07:56: Conclusion: State of the field, Urschel's recent results
- 2:10:28: Joke: Two kinds of mathematicians

Further Reading:

- A. Ng, M. Jordan, Y. Weiss. "On Spectral Clustering: Analysis and an algorithm"
- D. Spielman. "Spectral and Algebraic Graph Theory"

In this episode, Richard and I discuss the details of cosmology at large, both technically and historically. We dive into Einstein's equations from general relativity and see what implications they have for an expanding universe alongside a discussion of the cast of characters involved in 20th century cosmology (Einstein, Hubble, Friedmann, Lemaitre, and others). We also discuss inflation, gravitational waves, the story behind Brian Keating's book Losing the Nobel Prize, and the current state of experiments and cosmology as a field.

Originally published on May 3, 2022 on YouTube: https://youtu.be/DiXyZgukRmE

Timestamps:

- 00:00:00 : Introduction
- 00:02:42 : Astronomy must have been one of the earliest sciences
- 00:03:57 : Eric Weinstein and Geometric Unity
- 00:13:47 : Outline of podcast
- 00:15:10 : Brian Keating, Losing the Nobel Prize, Geometric Unity
- 00:16:38 : Big Bang and General Relativity
- 00:21:07 : Einstein's equations
- 00:26:27 : Einstein and Hilbert
- 00:27:47 : Schwarzschild solution (typo in video)
- 00:33:07 : Hubble
- 00:35:54 : One galaxy versus infinitely many
- 00:36:16 : Olbers' paradox
- 00:39:55 : Friedmann and FRLW metric
- 00:41:53 : Friedmann metric was audacious?
- 00:46:05 : Friedmann equation
- 00:48:36 : How to start a fight in physics: West coast vs East coast metric and sign conventions.
- 00:50:05 : Flat vs spherical vs hyperbolic space
- 00:51:40 : Stress energy tensor terms
- 00:54:15 : Conversation laws and stress energy tensor 00:58:28 : Acceleration of the universe
- 01:05:12 : Derivation of a(t) ~ t^2/3 from preceding computations
- 01:05:37 : a = 0 is the Big Bang. How seriously can we take this?
- 01:07:09 : Lemaitre
- 01:11:51 : Was Hubble's observation of an expanding universe in 1929 a fresh observation?
- 01:13:45 : Without Einstein, no General Relativity?
- 01:14:45 : Two questions: General Relativity vs Quantum Mechanics and how to understand time and universe's expansion velocity (which can exceed the speed of light!)
- 01:17:58 : How much of the universe is observable
- 01:24:54 : Planck length
- 01:26:33 : Physics down to the Big Bang singularity
- 01:28:07 : Density of photons vs matter\
- 01:33:41 : Inflation and Alan Guth
- 01:36:49 : No magnetic monopoles?
- 01:38:30 : Constant density requires negative pressure
- 01:42:42 : Is negative pressure contrived?
- 01:49:29 : Marrying General Relativity and Quantum Mechanics
- 01:51:58 : Symmetry breaking
- 01:53:50 : How to corroborate inflation?
- 01:56:21 : Sabine Hossenfelder's criticisms
- 02:00:19 : Gravitational waves
- 02:01:31 : LIGO
- 02:04:13 : CMB (Cosmic Microwave Background)
- 02:11:27 : Relationship between detecting gravitational waves and inflation
- 02:16:37 : BICEP2
- 02:19:06 : Brian Keating's Losing the Nobel Prize and the problem of dust
- 02:24:40 : BICEP3
- 02:26:26 : Wrap up: current state of cosmology

Notes: Easther's blogpost on Eric Weinstein: http://excursionset.com/blog/2013/5/25/trainwrecks-i-have-seen Vice article on Eric Weinstein and Geometric Unity: https://www.vice.com/en/article/z3xbz4/eric-weinstein-says-he-solved-the-universes-mysteries-scientists-disagree

Further learning:

- Matts Roos. "Introduction to Cosmology"
- Barbara Ryden. "Introduction to Cosmology"
- Our Cosmic Mistake About Gravitational Waves: https://www.youtube.com/watch?v=O0D-COVodzY

In this episode, Richard and I discuss the details of cosmology at large, both technically and historically. We dive into Einstein's equations from general relativity and see what implications they have for an expanding universe alongside a discussion of the cast of characters involved in 20th century cosmology (Einstein, Hubble, Friedmann, Lemaitre, and others). We also discuss inflation, gravitational waves, the story behind Brian Keating's book Losing the Nobel Prize, and the current state of experiments and cosmology as a field.

Originally published on May 3, 2022 on YouTube: https://youtu.be/DiXyZgukRmE

Timestamps:

- 00:00:00 : Introduction
- 00:02:42 : Astronomy must have been one of the earliest sciences
- 00:03:57 : Eric Weinstein and Geometric Unity
- 00:13:47 : Outline of podcast
- 00:15:10 : Brian Keating, Losing the Nobel Prize, Geometric Unity
- 00:16:38 : Big Bang and General Relativity
- 00:21:07 : Einstein's equations
- 00:26:27 : Einstein and Hilbert
- 00:27:47 : Schwarzschild solution (typo in video)
- 00:33:07 : Hubble
- 00:35:54 : One galaxy versus infinitely many
- 00:36:16 : Olbers' paradox
- 00:39:55 : Friedmann and FRLW metric
- 00:41:53 : Friedmann metric was audacious?
- 00:46:05 : Friedmann equation
- 00:48:36 : How to start a fight in physics: West coast vs East coast metric and sign conventions.
- 00:50:05 : Flat vs spherical vs hyperbolic space
- 00:51:40 : Stress energy tensor terms
- 00:54:15 : Conversation laws and stress energy tensor 00:58:28 : Acceleration of the universe
- 01:05:12 : Derivation of a(t) ~ t^2/3 from preceding computations
- 01:05:37 : a = 0 is the Big Bang. How seriously can we take this?
- 01:07:09 : Lemaitre
- 01:11:51 : Was Hubble's observation of an expanding universe in 1929 a fresh observation?
- 01:13:45 : Without Einstein, no General Relativity?
- 01:14:45 : Two questions: General Relativity vs Quantum Mechanics and how to understand time and universe's expansion velocity (which can exceed the speed of light!)
- 01:17:58 : How much of the universe is observable
- 01:24:54 : Planck length
- 01:26:33 : Physics down to the Big Bang singularity
- 01:28:07 : Density of photons vs matter\
- 01:33:41 : Inflation and Alan Guth
- 01:36:49 : No magnetic monopoles?
- 01:38:30 : Constant density requires negative pressure
- 01:42:42 : Is negative pressure contrived?
- 01:49:29 : Marrying General Relativity and Quantum Mechanics
- 01:51:58 : Symmetry breaking
- 01:53:50 : How to corroborate inflation?
- 01:56:21 : Sabine Hossenfelder's criticisms
- 02:00:19 : Gravitational waves
- 02:01:31 : LIGO
- 02:04:13 : CMB (Cosmic Microwave Background)
- 02:11:27 : Relationship between detecting gravitational waves and inflation
- 02:16:37 : BICEP2
- 02:19:06 : Brian Keating's Losing the Nobel Prize and the problem of dust
- 02:24:40 : BICEP3
- 02:26:26 : Wrap up: current state of cosmology

Notes: Easther's blogpost on Eric Weinstein: http://excursionset.com/blog/2013/5/25/trainwrecks-i-have-seen Vice article on Eric Weinstein and Geometric Unity: https://www.vice.com/en/article/z3xbz4/eric-weinstein-says-he-solved-the-universes-mysteries-scientists-disagree

Further learning:

- Matts Roos. "Introduction to Cosmology"
- Barbara Ryden. "Introduction to Cosmology"
- Our Cosmic Mistake About Gravitational Waves: https://www.youtube.com/watch?v=O0D-COVodzY

In this episode, we discuss the mathematics behind Loh's novel approach to contact tracing in the fight against COVID, which involves a beautiful blend of graph theory and computer science.

Originally published on March 3, 2022 on Youtube: https://youtu.be/8CLxLBMGxLE

Timestamps:

- 00:00:00 : Introduction
- 00:01:11 : About Po-Shen Loh
- 00:03:49 : NOVID app
- 00:04:47 : Graph theory and quarantining
- 00:08:39 : Graph adjacency definition for contact tracing
- 00:16:01 : Six degrees of separation away from anyone?
- 00:21:13 : Getting the game theory and incentives right
- 00:30:40 : Conventional approach to contact tracing
- 00:34:47 : Comparison with big tech
- 00:39:19 : Neighbor search complexity
- 00:45:15 : Watts-Strogatz small networks phenomenon
- 00:48:37 : Storing neighborhood information
- 00:57:00 : Random hashing to reduce computational burden
- 01:05:24 : Logarithmic probing of sparsity
- 01:09:56 : Two math PhDs struggle to do division
- 01:11:17 : Bitwise-or for union of bounded sets
- 01:16:21 : Step back and recap
- 01:26:15 : Tradeoff between number of hash bins and sparsity
- 01:29:12 : Conclusion

Further reading: Po-Shen Loh. "Flipping the Perspective in Contact Tracing" https://arxiv.org/abs/2010.03806

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In this episode, we discuss the mathematics behind Loh's novel approach to contact tracing in the fight against COVID, which involves a beautiful blend of graph theory and computer science.

Originally published on March 3, 2022 on Youtube: https://youtu.be/8CLxLBMGxLE

Timestamps:

- 00:00:00 : Introduction
- 00:01:11 : About Po-Shen Loh
- 00:03:49 : NOVID app
- 00:04:47 : Graph theory and quarantining
- 00:08:39 : Graph adjacency definition for contact tracing
- 00:16:01 : Six degrees of separation away from anyone?
- 00:21:13 : Getting the game theory and incentives right
- 00:30:40 : Conventional approach to contact tracing
- 00:34:47 : Comparison with big tech
- 00:39:19 : Neighbor search complexity
- 00:45:15 : Watts-Strogatz small networks phenomenon
- 00:48:37 : Storing neighborhood information
- 00:57:00 : Random hashing to reduce computational burden
- 01:05:24 : Logarithmic probing of sparsity
- 01:09:56 : Two math PhDs struggle to do division
- 01:11:17 : Bitwise-or for union of bounded sets
- 01:16:21 : Step back and recap
- 01:26:15 : Tradeoff between number of hash bins and sparsity
- 01:29:12 : Conclusion

Further reading: Po-Shen Loh. "Flipping the Perspective in Contact Tracing" https://arxiv.org/abs/2010.03806

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