2020's Biggest Breakthroughs in Math and Computer Science

Quanta Magazine2 minutes read

Einstein and Turing were intrigued by quantum entanglement and the halting problem, respectively, while recent advancements in mathematics include using quantum entanglement to solve complex problems and digitizing mathematics to create an AI for generating new proofs.

Insights

  • Quantum entanglement allows particles to communicate instantaneously over large distances, a phenomenon that even Albert Einstein found perplexing.
  • Advances in computational complexity theory, such as the halting problem and interactive proofs utilizing quantum entanglement, are reshaping mathematical research and paving the way for AI-powered mathematical proof verification systems like Lean software.

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Recent questions

  • What is quantum entanglement and why did Albert Einstein find it "spooky"?

    Quantum entanglement is a phenomenon where particles can interact instantaneously over vast distances, regardless of the space between them. Albert Einstein found this concept "spooky" because it seemed to violate the principles of classical physics, suggesting a mysterious connection between particles that defied traditional notions of cause and effect.

  • What is the halting problem in computers, as identified by Alan Turing in 1936?

    The halting problem, as identified by Alan Turing in 1936, refers to the issue where a computer program can get stuck in an infinite loop, making it impossible to predict when or if the program will halt or complete its task. This fundamental limitation highlights the inherent complexity and unpredictability of certain computational processes.

  • How did Henry Yuen and co-authors use quantum entanglement to solve complex mathematical questions?

    Henry Yuen and his co-authors developed a landmark proof in computational complexity theory involving interactive proofs, demonstrating how quantum entanglement can be utilized to solve complex mathematical questions. By leveraging the unique properties of quantum entanglement, they were able to tackle intricate mathematical problems in a novel and innovative way.

  • What was the Conway knot problem, and how did Lisa Piccirillo solve it?

    The Conway knot problem revolved around determining whether the Conway knot could be represented as a slice of a higher dimensional knot, a question that had puzzled mathematicians for years. Lisa Piccirillo, a graduate student, successfully proved that the Conway knot was not a slice of a higher dimensional knot, resolving a long-standing mathematical mystery in the field of knot theory.

  • How is Kevin Buzzard at Imperial College London using Lean software to digitize mathematics?

    Kevin Buzzard at Imperial College London is utilizing Lean software to digitize mathematics, enabling the handling of complex proofs and theorems in a digital format. By digitizing mathematical concepts, Buzzard aims to create an artificial intelligence system capable of verifying and generating new mathematical proofs, revolutionizing the field of mathematics through innovative technological advancements.

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Summary

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Revolutionizing Mathematics: From Einstein to AI

  • In 1935, Albert Einstein was intrigued by quantum entanglement, where particles can interact instantly over vast distances, which he found "spooky."
  • Alan Turing, in 1936, identified the halting problem in computers, where they can get stuck in infinite loops, proving it impossible to predict when this will occur.
  • A landmark proof in computational complexity theory, involving interactive proofs, was developed by Henry Yuen and co-authors, showing how quantum entanglement can help solve complex mathematical questions.
  • Lisa Piccirillo, a graduate student, solved the Conway knot problem, proving the Conway knot was not a slice of a higher dimensional knot, which had baffled mathematicians for years.
  • Kevin Buzzard at Imperial College London is digitizing mathematics using Lean software, which can handle complex proofs and theorems, aiming to create an AI capable of verifying and generating new mathematical proofs.
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