The Cartesian Cafe Timothy Nguyen

Richard Borcherds is a mathematician and professor at University of California Berkeley known for his work on lattices, group theory, and infinite-dimensional algebras. His numerous accolades include being awarded the Fields Medal in 1998 and being elected a fellow of the American Mathematical Society and the National Academy of Sciences.
Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen
In this episode, Richard and I give an overview of Richard's most famous result: his proof of the Monstrous Moonshine conjecture relating the monster group on the one hand and modular forms on the other. A remarkable feature of the proof is that it involves vertex algebras inspired from elements of string theory. Some familiarity with group theory and representation theory are assumed in our discussion.
I. Introduction
00:25: Biography
02:51 : Success in mathematics
04:04 : Monstrous Moonshine overview and John Conway
09:44 : Technical overview
II. Group Theory
11:31 : Classification of finite-simple groups + history of the monster group
18:03 : Conway groups + Leech lattice
22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions
32:37: Griess algebra
III. Modular Forms
36:42 : Definitions
40:06 : The elliptic modular function
48:58 : Subgroups of SL_2(Z)
IV. Monstrous Moonshine Conjecture Statement
57:17: Representations of the monster
59:22 : Hauptmoduls
1:03:50 : Statement of the conjecture
1:07:06 : Atkin-Fong-Smith's first proof
1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof
V. Sketch of Proof
1:14:47: Vertex algebra and monster Lie algebra
1:21:02 : No ghost theorem from string theory
1:25:24 : What's special about dimension 26?
1:28:33 : Monster Lie algebra details
1:32:30 : Dynkin diagrams and Kac-Moody algebras
1:43:21 : Simple roots and an obscure identity
1:45:13: Weyl denominator formula, Vandermonde identity
1:52:14 : Chasing down where modular forms got smuggled in
1:55:03 : Final calculations
VI. Epilogue
1:57:53 : Your most proud result?
2:00:47 : Monstrous moonshine for other sporadic groups?
2:02:28 : Connections to other fields. Witten and black holes and mock modular forms.
 
Further reading: V Tatitschef. A short introduction to Monstrous Moonshine. https://arxiv.org/pdf/1902.03118.pdf
Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org

Thought I'd share some exciting news about what's happening at The Cartesian Cafe in 2024 and also a personal message to viewers on how they can support the cafe.
Patreon:
https://www.patreon.com/timothynguyen

Tim Maudlin is a philosopher of science specializing in the foundations of physics, metaphysics, and logic. He is a professor at New York University, a member of the Foundational Questions Institute, and the founder and director of the John Bell Institute for the Foundations of Physics.
Patreon (bonus materials + video chat):https://www.patreon.com/timothynguyen
In this very in-depth discussion, Tim and I probe the foundations of science through the avenues of locality and determinism as arising from the Einstein-Poldosky-Rosen (EPR) paradox and Bell's Theorem. These issues are so intricate that even the Nobel Prize committee incorrectly described the significance of Bell's work in their press release for the 2022 prize in physics. Viewers motivated enough to think deeply about these ideas will be rewarded with a conceptually proper understanding of the nonlocal nature of physics and its manifestation in quantum theory.
I. Introduction 00:00 :
00:25: Biography
05:26: Interdisciplinary work
11:54 : Physicists working on the wrong things
16:47 : Bell's Theorem soft overview
24:14: Common misunderstanding of "God does not play dice."
25:59: Technical outline
II. EPR Paradox / Argument
29:14 : EPR is not a paradox
34:57 : Criterion of reality
43:57 : Mathematical formulation
46:32 : Locality: No spooky action at a distance
49:54 : Bertlmann's socks
53:17 : EPR syllogism summarized
54:52 : Determinism is inferred not assumed
1:02:18 : Clarifying analogy: Coin flips
1:06:39 : Einstein's objection to determinism revisited
III. Bohm Segue
1:11:05 : Introduction
1:13:38: Bell and von Neumann's error
1:20:14: Bell's motivation: Can I remove Bohm's nonlocality?
IV. Bell's Theorem and Related Examples
1:25:13 : Setup
1:27:59 : Decoding Bell's words: Locality is the key!
1:34:16 : Bell's inequality (overview)
1:36:46 : Bell's inequality (math)
1:39:15 : Concrete example of violation of Bell's inequality
1:49:42: GHZ Example
V. Miscellany
2:06:23 : Statistical independence assumption
2:13:18: The 2022 Nobel Prize
2:17:43: Misconceptions and hidden variables
2:22:28: The assumption of local realism? Repeat: Determinism is a conclusion not an assumption.
VI. Interpretations of Quantum Mechanics
2:28:44: Interpretation is a misnomer
2:29:48: Three requirements. You can only pick two.
2:34:52: Copenhagen interpretation?
 
Further Reading:
J. Bell. Speakable and Unspeakable in Quantum Mechanics
T. Maudlin. Quantum Non-Locality and Relativity
Wikipedia: Mermin's device, GHZ experiment
 
Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org

Antonio (Tony) Padilla is a theoretical physicist and cosmologist at the University of Nottingham. He serves as the Associate Director of the Nottingham Centre of Gravity, and in 2016, Tony shared the Buchalter Cosmology Prize for his work on the cosmological constant. Tony is also a star of the Numberphile YouTube channel, where his videos have received millions of views and he is also the author of the book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity.
Patreon: https://www.patreon.com/timothynguyen
This episode combines some of the greatest cosmological questions together with mathematical imagination. Tony and I go through the math behind some oft-quoted numbers in cosmology and calculate the age, size, and number of atoms in the universe. We then stretch our brains and consider how likely it would be to find your Doppelganger in a truly large universe, which takes us on a detour through black hole entropy. We end with a discussion of naturalness and the anthropic principle to round out our discussion of fantastic numbers in physics.
Part I. Introduction
00:00 : Introduction
01:06 : Math and or versus physics
12:09 : Backstory behind Tony's book
14:12 : Joke about theoreticians and numbers
16:18 : Technical outline
Part II. Size, Age, and Quantity in the Universe
21:42 : Size of the observable universe
22:32 : Standard candles
27:39 : Hubble rate
29:02 : Measuring distances and time
37:15 : Einstein and Minkowski
40:52 : Definition of Hubble parameter
42:14 : Friedmann equation
47:11 : Calculating the size of the observable universe
51:24 : Age of the universe
56:14 : Number of atoms in the observable universe
1:01:08 : Critical density
1:03:16: 10^80 atoms of hydrogen
1:03:46 : Universe versus observable universe
Part III. Extreme Physics and Doppelgangers
1:07:27 : Long-term fate of the universe
1:08:28 : Black holes and a googol years
1:09:59 : Poincare recurrence
1:13:23 : Doppelgangers in a googolplex meter wide universe
1:16:40 : Finitely many states and black hole entropy
1:25:00 : Black holes have no hair
1:29:30 : Beckenstein, Christodolou, Hawking
1:33:12 : Susskind's thought experiment: Maximum entropy of space
1:42:58 : Estimating the number of doppelgangers
1:54:21 : Poincare recurrence: Tower of four exponents.
Part IV: Naturalness and Anthropics
1:54:34 : What is naturalness? Examples.
2:04:09 : Cosmological constant problem: 10^120 discrepancy
2:07:29 : Interlude: Energy shift clarification. Gravity is key.
2:15:34 : Corrections to the cosmological constant
2:18:47 : String theory landscape: 10^500 possibilities
2:20:41 : Anthropic selection
2:25:59 : Is the anthropic principle unscientific? Weinberg and predictions.
2:29:17 : Vacuum sequestration
Further reading: Antonio Padilla. Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity
Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org

Boaz Barak is a professor of computer science at Harvard University, having previously been a principal researcher at Microsoft Research and a professor at Princeton University. His research interests span many areas of theoretical computer science including cryptography, computational complexity, and the foundations of machine learning. Boaz serves on the scientific advisory boards for Quanta Magazine and the Simons Institute for the Theory of Computing and he was selected for Foreign Policy magazine’s list of 100 leading global thinkers for 2014.
www.patreon.com/timothynguyen
Cryptography is about maintaining the privacy and security of communication. In this episode, Boaz and I go through the fundamentals of cryptography from a foundational mathematical perspective. We start with some historical examples of attempts at encrypting messages and how they failed. After some guesses as to how one might mathematically define security, we arrive at the one due to Shannon. The resulting definition of perfect secrecy turns out to be too rigid, which leads us to the notion of computational secrecy that forms the foundation of modern cryptographic systems. We then show how the existence of pseudorandom generators (which remains a conjecture) ensures that such computational secrecy is achievable, assuming P does not equal NP. Having covered private key cryptography in detail, we then give a brief overview of public key cryptography. We end with a brief discussion of Bitcoin, machine learning, deepfakes, and potential doomsday scenarios.
I. Introduction
00:17 : Biography: Academia vs Industry
10:07 : Military service
12:53 : Technical overview
17:01 : Whiteboard outline
II. Warmup
24:42 : Substitution ciphers
27:33 : Viginere cipher
29:35 : Babbage and Kasiski
31:25 : Enigma and WW2
33:10 : Alan Turing
III. Private Key Cryptography: Perfect Secrecy
34:32 : Valid encryption scheme
40:14 : Kerckhoffs's Principle
42:41 : Cryptography = steelman your adversary
44:40 : Attempt #1 at perfect secrecy
49:58 : Attempt #2 at perfect secrecy
56:02 : Definition of perfect secrecy (Shannon)
1:05:56 : Enigma was not perfectly secure
1:08:51 : Analogy with differential privacy
1:11:10 : Example: One-time pad (OTP)
1:20:07 : Drawbacks of OTP and Soviet KGB misuse
1:21:43 : Important: Keys cannot be reused!
1:27:48 : Shannon's Impossibility Theorem
IV. Computational Secrecy
1:32:52 : Relax perfect secrecy to computational secrecy
1:41:04 : What computational secrecy buys (if P is not NP)
1:44:35 : Pseudorandom generators (PRGs)
1:47:03 : PRG definition
1:52:30 : PRGs and P vs NP
1:55:47: PRGs enable modifying OTP for computational secrecy
V. Public Key Cryptography
2:00:32 : Limitations of private key cryptography
2:09:25 : Overview of public key methods
2:13:28 : Post quantum cryptography
VI. Applications
2:14:39 : Bitcoin
2:18:21 : Digital signatures (authentication)
2:23:56 : Machine learning and deepfakes
2:30:31 : A conceivable doomsday scenario: P = NP
Further reading: Boaz Barak. An Intensive Introduction to Cryptography
Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org

Sean Carroll is a theoretical physicist and philosopher who specializes in quantum mechanics, cosmology, and the philosophy of science. He is the Homewood Professor of Natural Philosophy at Johns Hopkins University and an external professor at the Sante Fe Institute. Sean has contributed prolifically to the public understanding of science through a variety of mediums: as an author of several physics books including Something Deeply Hidden and The Biggest Ideas in the Universe, as a public speaker and debater on a wide variety of scientific and philosophical subjects, and also as a host of his podcast Mindscape which covers topics spanning science, society, philosophy, culture, and the arts.
www.patreon.com/timothynguyen
In this episode, we take a deep dive into The Many Worlds (Everettian) Interpretation of quantum mechanics. While there are many philosophical discussions of the Many Worlds Interpretation available, ours marries philosophy with the technical, mathematical details. As a bonus, the whole gamut of topics from philosophy and physics arise, including the nature of reality, emergence, Bohmian mechanics, Bell's Theorem, and more. We conclude with some analysis of Sean's speculative work on the concept of emergent spacetime, a viewpoint which naturally arises from Many Worlds. This video is most suitable for those with a basic technical understanding of quantum mechanics.
Part I: Introduction
00:00:00 : Introduction
00:05:42 : Philosophy and science: more interdisciplinary work?
00:09:14 : How Sean got interested in Many Worlds (MW)
00:13:04 : Technical outline
Part II: Quantum Mechanics in a Nutshell
00:14:58 : Textbook QM review
00:24:25 : The measurement problem
00:25:28 : Einstein: "God does not play dice"
00:27:49 : The reality problem
Part III: Many Worlds
00:31:53 : How MW comes in
00:34:28 : EPR paradox (original formulation)
00:40:58 : Simpler to work with spin
00:42:03 : Spin entanglement
00:44:46 : Decoherence
00:49:16 : System, observer, environment clarification for decoherence
00:53:54 : Density matrix perspective (sketch)
00:56:21 : Deriving the Born rule
00:59:09 : Everett: right answer, wrong reason. The easy and hard part of Born's rule.
01:03:33 : Self-locating uncertainty: which world am I in?
01:04:59 : Two arguments for Born rule credences
01:11:28 : Observer-system split: pointer-state problem
01:13:11 : Schrodinger's cat and decoherence
01:18:21 : Consciousness and perception
01:21:12 : Emergence and MW
01:28:06 : Sorites Paradox and are there infinitely many worlds
01:32:50 : Bad objection to MW: "It's not falsifiable."
Part IV: Additional Topics
01:35:13 : Bohmian mechanics
01:40:29 : Bell's Theorem. What the Nobel Prize committee got wrong
01:41:56 : David Deutsch on Bohmian mechanics
01:46:39 : Quantum mereology
01:49:09 : Path integral and double slit: virtual and distinct worlds
Part V. Emergent Spacetime
01:55:05 : Setup
02:02:42 : Algebraic geometry / functional analysis perspective
02:04:54 : Relation to MW
Part VI. Conclusion
02:07:16 : Distribution of QM beliefs
02:08:38 : Locality
 
Further reading:
Hugh Everett. The Theory of the Universal Wave Function, 1956.
Sean Carroll. Something Deeply Hidden, 2019.
More Sean Carroll & Timothy Nguyen:
Fragments of the IDW: Joe Rogan, Sam Harris, Eric Weinstein: https://youtu.be/jM2FQrRYyas
Twitter: @iamtimnguyen
Webpage: http://www.timothynguyen.org