From Ronald Ross to ChatGPT: the birth and strange life of the random walk - Jordan Ellenberg

Introduction

Thank you all for coming. It's great to see such a big crowd here today. I hope you're all here because you want to spend your early evening talking about math, and not just because we're in one of the few air-conditioned rooms in greater London!

Let's start with a puzzle. It may not look like a math puzzle at first, but bear with me. As you contemplate this strange list of things, let me introduce myself. I write books about math, teach math, and also write research math papers. One common aspect I've found in all these areas is that they all involve telling a story about mathematics. So tonight, I want to tell you an interesting story that I discovered while researching my last book. It begins with this gentleman.

This is Ronald Ross, who in 1904 was a major celebrity in the medical field. He had recently won the second-ever Nobel Prize for Medicine for discovering that mosquitoes were the vector for malaria.

Ross was somewhat bitter because he never felt he got the recognition he deserved. Despite his achievements in medicine, what he truly esteemed was math. He even developed a model of epidemic growth, the Ross-Hudson model. But in 1904, he was stuck on a particular problem involving mosquito eradication. If mosquitoes are removed from an area, how long before they reinhabit it? Ross couldn't solve this problem, so he turned to the mathematical community for help.

Ross's main model was to use random walks to understand mosquito movement. His model assumed that mosquitoes moved a fixed distance in random directions each day. This, he believed, would have significant implications for malaria control.

Carl Pearson, a British statistician, helped make Ross's problem more famous by posting it in the journal "Nature," calling it the "Problem of the Random Walk." Pearson omitted any mention of malaria to make it more appealing to mathematicians.

Interestingly, several other notable figures around the world were simultaneously pondering similar random walk problems. Albert Einstein and Louis Bachelier were among them, the former using it to support molecular theory and the latter applying it to finance.

Andrey Markov, a Russian mathematician, devised the Markov chain, a critical development in random walks. Markov used his model to challenge theological arguments for free will, demonstrating that even dependent events could settle into statistical averages.

Move forward to the 1940s, and Claude Shannon advanced the field by developing information theory. Shannon showed how random text generation could approximate natural language using statistical properties of existing texts. We tried a similar exercise in this talk by generating English text using trigrams (sets of three letters).

Fast forward to today, and the concept of random walks has evolved into something called large language models (LLMs), like the ones used in ChatGPT. These models use similar principles but at a much more advanced level, producing coherent text from given prompts.

I collaborated on a project with researchers from Google DeepMind to create programs that solve mathematical problems using large language models. While they can produce solutions for specific problems, they struggle to generalize across a broader set of problems.

To summarize, the central theme here is that random walks, initially a simple concept, have influenced diverse fields like epidemiology, finance, linguistics, and now AI. This story also underscores the collaborative potential between human creativity and machine-generated ideas.

This talk is based on material from my book, "Shape." You can see the different covers for the UK and US editions. The journey of the random walk illustrates how interconnected, and surprising the field of mathematics can be.

Thank you for listening, and I'm happy to take any questions.