## 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.