Category: math

SUMMARY

Professor Terence Tao discusses the transformative impact of AI on mathematics, sharing historical context and modern applications at the IMO.

IDEAS:

• Terence Tao began participating in the IMO at just 11 years, showcasing extraordinary talent early.
• AI tools like AlphaGeometry are revolutionizing how mathematics is approached and solved today.
• Machine assistance in mathematics has historical roots, dating back thousands of years to the abacus.
• Computers have been used in mathematics for about 300-400 years, evolving from mechanical to electronic.
• The term “computer” originally referred to human calculators, particularly during World War II.
• The Online Encyclopedia of Integer Sequences is a valuable resource for identifying mathematical patterns.
• Scientific computation has been used since the 1920s, with early work done by Hendrik Lorentz.
• AI tools now assist in complex mathematical problems that were previously too tedious for humans.
• SAT solvers can analyze logic puzzles and complex statements, but they struggle with scalability.
• AI assistance has enabled the proof of long-standing mathematical conjectures, like the Pythagorean triple problem.
• Formal proof assistants are improving the verification of mathematical arguments and proofs.
• The Four Color Theorem was one of the first major proofs aided by computer assistance.
• Machine learning has recently been applied to discover connections in knot theory and other areas.
• Large language models like GPT-4 can provide solutions to specific mathematical problems, albeit with limitations.
• Formalizing proofs in AI environments can speed up the process of verification and collaboration among mathematicians.
• Collaborative projects using AI have enabled faster and more efficient formalization of complex mathematical proofs.
• The future of mathematics may involve using AI to solve multiple problems simultaneously rather than one at a time.
• Machines can assist in generating conjectures based on large datasets, potentially leading to new discoveries.
• AI’s role in mathematics will remain supportive, enhancing human creativity rather than replacing it.
• Personal interactions and serendipity often lead to new research ideas among mathematicians.
• The integration of AI into mathematics requires mathematicians to retain foundational knowledge to guide AI effectively.

INSIGHTS:

• AI’s integration into mathematics may redefine the boundaries of research and problem-solving methods.
• Historical context reveals that the intersection of machines and mathematics is not a new phenomenon.
• Collaborative mathematical projects can thrive when AI tools assist in the formalization and verification processes.
• Future mathematics could involve large-scale problem exploration facilitated by AI’s computational power.
• Machine learning’s ability to highlight connections in data can lead to innovative mathematical conjectures.
• The evolution of proof assistants has made formal verification more accessible to mathematicians today.
• Humans still play a crucial role in interpreting AI-generated insights and conjectures in mathematics.
• Mathematics is increasingly becoming a collaborative and interdisciplinary field due to technological advancements.
• Serendipity and conversation remain pivotal in shaping research directions in mathematics.
• The potential for AI to automate conjecture generation represents a significant frontier for mathematical exploration.

QUOTES:

• “I hope we all had fun, not just in the competition whether you get a good score or not.”
• “Instead of having three hours to solve a problem, you take months and sometimes you don’t solve it.”
• “We’ve actually been using computers and machines to do mathematics for a long time.”
• “The basic unit of computational power at the time was not the CPU, it was the kilgirl.”
• “In mathematical research, we rely on tables – we call them databases now.”
• “Many promising productive research projects have come up that way.”
• “We’ve been doing scientific computation since the 1920s.”
• “The proof required a few years of computation and it generated a proof certificate.”
• “The future is going to be really exciting.”
• “This may be my most important result to date – better be sure it’s correct.”
• “Every little bubble corresponds to some statement and you don’t need to understand the whole proof.”
• “We’re beginning to sort of prove things that are like 4 or 5 lines long.”
• “AI assistance has enabled the proof of long-standing mathematical conjectures.”
• “I think the future will require more flexibility in research topics.”
• “The hope is that AI will become very good at generating good conjectures.”
• “We still use tables today; we call them databases now, but they’re still the same thing.”

HABITS:

• He reflects fondly on his experiences at the IMO, emphasizing the importance of enjoyment in competition.
• Tao suggests that successful mathematicians often rely on strong mentorship throughout their education.
• Engaging in conversations at conferences can spark new research ideas and collaborations.
• He believes in taking research topics one at a time rather than rushing into multiple areas.
• Tao emphasizes the importance of being flexible in research topics and adapting to new ideas.
• He collaborates with diverse teams, including non-mathematicians, to tackle complex problems.
• Utilizing modern proof assistants has become a regular practice for verifying complex mathematical arguments.
• Tao experiments with AI tools to explore new techniques and approaches in his research.
• He encourages others to learn from mistakes and adapt their strategies when faced with challenges.
• Maintaining foundational knowledge in mathematics is crucial for effectively guiding AI tools.

FACTS:

• The first participant in the IMO to receive a gold medal was Terence Tao at age 13.
• The abacus is one of the earliest machines used for mathematical calculations, dating back thousands of years.
• Computers for mathematical computation have existed in various forms for about 300-400 years.
• The Online Encyclopedia of Integer Sequences contains hundreds of thousands of integer sequences.
• The first major computer-assisted proof was the Four Color Theorem, proven in 1976.
• Scientific computation has been utilized since the 1920s, often involving large human computing teams.
• The proof of the Pythagorean triple problem required a massive computation and was computer-assisted.
• Formal proof assistants are increasingly being used to verify complex mathematical arguments.
• Large language models can provide mathematical solutions, but their accuracy is often limited.
• Machine learning has recently been applied to discover connections between different areas of mathematics.
• The integration of AI in mathematics is projected to enhance collaboration and problem-solving efficiency.
• Collaborative projects in mathematics are becoming more common, often involving interdisciplinary teams.
• The proof of the Kepler conjecture was formalized and completed in 2014 after many years of work.
• Recent advancements in proof assistants have made formal verification processes more efficient and user-friendly.
• AI tools can assist mathematicians by generating conjectures based on large datasets and patterns.
• Mathematics is becoming more collaborative, with mathematicians increasingly sharing ideas and insights.

REFERENCES:

• AlphaGeometry, a tool by DeepMind for answering geometry questions in competitions.
• Online Encyclopedia of Integer Sequences (OEIS), a database of integer sequences.
• Formal proof assistants like Lean and Coq for verifying mathematical arguments.
• The Flyspeck project, which formalized the proof of the Kepler conjecture.
• GitHub Copilot, an AI tool that suggests lines of code for formal proofs.
• The Four Color Theorem, one of the earliest computer-assisted proofs.
• The Birch and Swinnerton-Dyer conjecture, discovered through extensive data tables.
• Condensed mathematics, a field developed by Peter Scholze focusing on functional analysis.
• Various software tools that facilitate collaborative proof formalization projects.
• Notable mathematical events and conferences where ideas and research are shared.

ONE-SENTENCE TAKEAWAY

AI is transforming mathematics by enhancing problem-solving capabilities and facilitating collaborative research among mathematicians.

RECOMMENDATIONS:

• Embrace AI tools to enhance mathematical problem-solving and explore new research avenues.
• Actively participate in mathematical conferences to foster collaboration and share ideas with peers.
• Leverage formal proof assistants to streamline the verification process of complex mathematical proofs.
• Engage with interdisciplinary teams to solve complex mathematical problems effectively.
• Experiment with machine learning to discover unexpected connections in mathematical data.
• Approach mathematical research with flexibility, being open to changing topics and ideas.
• Utilize collaborative project management techniques to break down large proofs into manageable tasks.
• Maintain foundational knowledge in mathematics to effectively guide AI and machine learning tools.
• Seek mentorship throughout educational and research journeys to gain valuable insights and guidance.
• Keep a record of successful problem-solving techniques to reference in future research endeavors.

A wonderful explanation of why $latex q p q^{-1}$ is rotation of $latex p$, where $latex q$ is most easily interpreted with the complex-number like structure of $latex \cos(\theta/2) + \sin(\theta/2)(a i + b j + c k)$ with $latex a i + bj +ck$ normalized (i.e., $latex a^2+b^2+c^2=1$). Note that $latex q^{-1}$ is then just $latex latex \cos(\theta/2) – \sin(\theta/2)(a i + b j + c k)$ (since $latex (a i + b j + c k)^2=-a^2 – b^2 -c^2 = -1$ just like the imaginary $latex i$ for complex number).

Basically quarternion is 4D stereographic projection back to 3D space. When multiply by a quarternion $latex \cos(\alpha) + \sin(\alpha) {\bf v}$ from the left, points parallel to $latex \bf v$ will be translate along $\bf v$ and points penpendicular to $latex \bf v$ will rotate around $latex \bf v$ (following the right hand rule). Similarly, when multiplying from the right by the same quarternion, points parallel to $latex \bf v$ will be translated in the same direction, but points perpendicular to $latex \bf v$ will be rotated in the opposite direction (following left hand rule). So if we multiple $latex q$ and $latex q^{-1}$ from both left and right hand sides, the translation effect will be exactly cancelled, and the the rotation will be doubled. That is why the angle in $latex q$ should be halved.