In recent years a lot of people have asked me just what the phrase analytic combinatorics actually means. In this lecture I'm going to describe what the field is, and tell the story of how it came into being. I think this context is important for anyone who's interested in learning anything about the field. this lecture is dedicated to the memory of my friend and colleague Philippe Flajolet, who really was the driving force behind the development of the field and who died suddenly in 2011. I'll start off with a brief history to try to give some context for how we got there. I first met Philippe in 1977. my first research paper that I wrote, after getting my PhD, was on an algorithm called odd even merging. and I, I went to in those days you would get your paper typed by a secretary and go to a conference and present it. and I was very proud to have developed this formula that involves the gamma function and the zeta function and gives a precise description of the performance of this particular algorithm. and just a few months later we had a conference in Providence, Rhode Island where I was at the time and in, in those days you go to the conference and the first thing you do in the proceedings is go and and look at the table of contents of the proceedings, see what's there. And there was another type paper and I was amazed to see a formula very much like mine involving the zeta function and the gamma function, even though it was studying a completely different problem. and just as I was realizing that Philippe came up to me and said I believe that we have a formula in common and both of us were very surprised to see the similarities among these formulas. in, it might be said we spent the rest of our careers trying to understand why. now it's worth it to think about wh, what the world was like at the time, that we started our research careers. we were both, at that time, in that fa, just the earliest part of our research careers. and the world was changing in very important ways, all around us. Without going into too much detail, it really was the case that when we started school people wore coats and ties, coats and ties to dinner, and so forth. But by the time we get our PHD's there was Woodstock and hippies and, and so forth. And, but, but, with re, respect to the technology there were huge changes. when we started school computers were big expensive rare, there were physical devices for every switch, or for every bit. but not that much longer when we started research and teaching, we had integrated circuits and computers were becoming ubiquitous, and fast and cheap. another big thing was the access to computers. most of the time that we were in college and in graduate school you would get to develop a program. You had to put each line of the program on a punched card, and you had to give a box of punched cards to a computer operator. And you would get to run your program once a day. not that much longer, we had later when we su, started research in teaching, we had time shared terminals and we're always connected and have been connected ever since. and as I mentioned, when we started school my thesis was typed by a secretary. so you present the result and six months later you sort of see what it looked like and submit it. It might be might be a year between the time that you get the result and somebody sees it. but not that much longer we had word processing and and mathematical type setting and we can have much quicker and much broader communication of our, of our research results. And another important thing is that when we were in school and graduate school the curriculum was about math. Everybody learned lots of math. And I learned PDEs and abstract algebra and probability and topology. that's what that, that's what people with an interest in working in technical fields did. but the, by the time we started researching teaching, there was computer science. And people had to learn about compilers and algorithms and data structures and graphics and operating systems programming languages, numerical analysis. and all kinds of fields related to computer science. So these are huge differences in a relatively short amount of time and in thinking about it when preparing this talk, I really came to understand and believe that this was a really profound change in the way the world worked. Maybe even more profound then thee evolution of PCs personal computing, or, or even the internet. the world was a vastly different place, when we started to get to work. so that's a context where, where this story starts. now analysis of algorithm so, that's the, field of study that, both Phillipe and I, were engaged in. and it's actually, natural and each, questions. and it actually started with Babbage. So, this is a quote from, from Babbage, who's widely attributed to have one of the, maybe the first, designed the first computational engine. It was a mechanical device that could do, arithmetic computations. in what he said, even before building the thing, as soon as an analytic engine exists that will necessarily guide the future course of the science, because you'd be able to do computations. but he said whenever any result is sought, the question will arise, by what course of calculations can these results be arrived at, by the machine, in the shortest time? That's in 1864, and you can see why it was important to Babbage, this thing actually had a crank, and the only way that it could compute things, was by somebody turning the crank. Obviously, you want to minimize the number of times that you need, turn the crank. And computers were, expensive, and slow and, used energy and so forth, so minimizing the cost of computation was always very important. even Turing who many who is, [COUGH] , the founder of theoretical computer science could see the importance of these kinds of practical questions. We want to have a measure of the amount of work involved in the computing process even though it might be a crude one. We count up the number of times that elementary operations are applied in the whole process. and in order to figure out how much work it's going to take before to help in designing efficient computation. But the field of analysis of algorithms was really initiated by Knuth in the 1960's. and what Knuth told the world and there was some debate about it at the time, was that classical mathematics is really got the necessary tools that we need for understanding the performance of algorithms. there's things like recurrence relations, and generating functions, and asymptotic analysis. That has the benefit of giving a scientific foundation for the analysis of algorithm. And Knuth wrote a series of four books so far, it, and the first one came out in the, in the late 60s and two more came out in the early 70s. they really set out this scientific foundation, that we really can use classic mathematics to understand the performance of algorithms. and, with those mathematical models, we can go ahead and accurately predict performance, and compare the efficiency of algorithms. and that's what we found exciting. we could use classical mathematics to understand the cost of a computation, and then test out those results. And, formulate hypothesis about how long it would take to do something. And then validate those hypothesis by actually implementing and running the programs and checking them against the math. There were many, many practical applications where people needed to have these kinds of accurate math, mathematical models and, and predictions. and Knuth's books were very densely filled with information that helped us advance this science. so that's a brief history of where we got started with analysis of algorithms.