Model is featured in figure 5.4 of Visualizing Mathematics with 3D Printing. This is joint work with Keenan Crane.

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A distinguished mathematics professor at Rutgers, has resolved two critical problems in mathematics that have puzzled experts for decades.
He tackled the 1955 Height Zero Conjecture and made significant advancements in the Deligne-Lusztig theory, enhancing theoretical applications in several sciences.
A Rutgers University-New Brunswick professor, dedicated to unraveling the mysteries of higher mathematics, has resolved two separate, fundamental problems that have baffled mathematicians for decades.
The solutions to these long-standing problems could further enhance our understanding of symmetries of structures and objects in nature and science, and of long-term behavior of various random processes arising in fields ranging from chemistry and physics to engineering, computer science and economics.
A Rutgers University-New Brunswick professor who has devoted his career to resolving the mysteries of higher mathematics has solved two separate, fundamental problems that have perplexed mathematicians for decades.
Edward Frenkel is a renowned mathematician, professor of University of California, Berkeley, member of the American Academy of Arts and Sciences, and winner of the Hermann Weyl Prize in Mathematical Physics. In this episode, Edward Frenkel discusses the recent monumental proof in the Langlands program, explaining its significance and how it advances understanding in modern mathematics.
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Edward Frenkel’s previous lecture on TOE [Part 1]: • Revolutionary Math Proof No One Could…
Check out Edward Frenkel’s New York Times Bestselling book “Love and Math” which covers a lot of material in this video: https://amzn.to/4evbBkS
LINKS:
• Edward Frenkel’s Part 1 on TOE for the Langlands Program: • Revolutionary Math Proof No One Could…
• Edward Frenkel’s Twitter: https://twitter.com/edfrenkel.
• Edward Frenkel’s Official Website: https://edwardfrenkel.com.
• Edward Frenkel’s YouTube: / @edfrenkel.
• Edward Frenkel’s Instagram: / edfrenkel.
• Edward Frenkel’s Linkedin / edfrenkel.
• Edward Frenkel’s SoundCloud (DJ Moonstein): / moonstein.
• Edward Frenkel’s 1st TOE Episode: • Edward Frenkel: Infinity, Ai, String…
• Andre Weil’s letter on “Rosetta Stone” of Math: https://www.ams.org/notices/200503/fe…
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A mathematical proof shows that some quantum states can resist nature’s tendency to disorder – but only under very specific conditions.
A black hole analog could tell us a thing or two about an elusive radiation theoretically emitted by the real thing.
Using a chain of atoms in single-file to simulate the event horizon of a black hole, a team of physicists in 2022 observed the equivalent of what we call Hawking radiation – particles born from disturbances in the quantum fluctuations caused by the black hole’s break in spacetime.
This, they say, could help resolve the tension between two currently irreconcilable frameworks for describing the Universe: the general theory of relativity, which describes the behavior of gravity as a continuous field known as spacetime; and quantum mechanics, which describes the behavior of discrete particles using the mathematics of probability.
Two of San Francisco’s leading players in artificial intelligence have challenged the public to come up with questions capable of testing the capabilities of large language models (LLMs) like Google Gemini and OpenAI’s o1. Scale AI, which specializes in preparing the vast tracts of data on which the LLMs are trained, teamed up with the Center for AI Safety (CAIS) to launch the initiative, Humanity’s Last Exam.
Featuring prizes of US$5,000 (£3,800) for those who come up with the top 50 questions selected for the test, Scale and CAIS say the goal is to test how close we are to achieving “expert-level AI systems” using the “largest, broadest coalition of experts in history.”
Why do this? The leading LLMs are already acing many established tests in intelligence, mathematics and law, but it’s hard to be sure how meaningful this is. In many cases, they may have pre-learned the answers due to the gargantuan quantities of data on which they are trained, including a significant percentage of everything on the internet.
The foundation of this simulation, as described by the team, is a well-known cosmological model that describes the universe as expanding uniformly over time. The researchers modeled how a quantum field, initially in a vacuum state (meaning no particles are present), responds to this expansion. As spacetime stretches, the field’s oscillations mix in a process that can create particles where none previously existed. This phenomenon is captured by a transformation that relates the field’s behavior before and after the universe expands, showing how vibrations at different momenta become entangled, leading to particle creation.
To understand how many particles are generated, the researchers used a mathematical tool called the Bogoliubov transformation. This approach describes how the field’s vacuum state evolves into a state where particles can be detected. As the expansion rate increases, more particles are produced, aligning with predictions from quantum field theory. By running this simulation on IBM quantum computers, the team was able to estimate the number of particles created and observe how the quantum field behaves during the universe’s expansion, offering a new way to explore complex cosmological phenomena.
According to the team, the most notable result of the study was the ability to estimate the number of particles created as a function of the expansion rate of the universe. By running their quantum circuit on both simulators and IBM’s 127-qubit Eagle quantum processor, the researchers demonstrated that they could successfully simulate particle creation in a cosmological context. While the results were noisy—particularly for low expansion rates—the error mitigation techniques used helped bring the outcomes closer to theoretical predictions.