 Not all departments do. I remember as I had post-doc at Rice, at the time I had a very strong lunch crowd tradition. I learned so much about mathematics from the older faculty. Not that we sat at lunch and said, we're now going to talk about contravariant functions. No, we were just chit-chatting. But I learned how the math world worked. But I also learned what it was like to be an adult mathematician. I learned how to integrate mathematics with personal life. And there's two stories I want to say, because it's about Bill Veitch. How many people have heard of Bill Veitch? He was a big-time analyst. There was this group of people who I thought of as the old guys. I was 27. They were probably about 50. But I thought of as the old guys. I liked them. One of them was Bill Veitch. Bill Veitch is the kind of person who published a serious research paper every year of his career. For the research map pages, we're talking annals levels. I mean, we're talking serious stuff. I remember two things about him. One is some of the young faculty were talking about, what do you do when you finish a paper? And a research paper. And he just said, well, I know what I do. And we were, what, Bill? Panic! And we're, what? I just don't know where the next one's going to come from! Oh my God, a 50-year-old is, I mean, I was worried about that 27. Am I going to produce enough papers to get 10? Am I going to keep a job? Oh, I have a baby! Are we going to be on the streets to starve to death? Oh my God, I might have to become, oh my God, I might have to become a lawyer! You know, there's all kinds of internal fears. And here I was seeing a 50-year-old who had produced an amazing amount of work still had those fears. And his fears were not driven by crass materialism. I'm sure there's part of that. It was just because he knew that mathematics was hard, that it was, in that sentence, you could see that he thought of it as sort of an adventure and at the same level, one that you could fail at. And I said, that's the way I want to be when I'm 50. He also, I think it was in 1988, he came in for lunch once and said, I'm really tired. Well, why is that? I couldn't sleep last night. Why? Then he's a big guy who is not really a feeling, sharing his feelings kind of person. And I was like, why is that? And he said, I'm just too excited about my recent paper. Whoa! That's cool! I get excited about that kind of stuff. Actually, and that was really interesting because about six weeks ago, I was talking to Pierre Arnaud, who's a big name in dynamical systems. Things are named after him. And we started talking about Bill Veitch. Now, Veitch is someone who, yes, he published a serious paper every year of his career, but he wasn't that big of a name. But in recent years, I've noticed his name is catching on all over the place. The number of times I go to random colloquium, we talk to him to hear about Veitch surfaces or Veitch this, or Veitch this, or Rossy Veitch fractals. So Pierre was telling me, oh yeah, he published this amazing paper back in 1988. He said, I think I was working with a few people who read it. I really read it carefully. I got all kinds of results from it. Of course, now it's being referenced thousands of times every year. So Veitch knew how important that work was in 1988, and he was excited about it. And he was not excited because he knew he was gonna make a killing in the market. He was just purely intellectual. I was wonderful. Lunch crowded, let me do that. Makes sense? Sorry, I'm pricking on you. So in all of this in terms of, it's not, I mean, you can obsess over how much to think about mathematical maturity, but view it as an ongoing process. And you have to put your stuff in situations where you might fail. And so I tell students regularly that I hate mathematics. And I quickly say, I also love mathematics. Colin Adams and I, who disagree on many things, as some of you have seen DVDs about, he always kind of brings math. Oh, isn't math fun? And I always say, oh my God, math is living hell. We just have to do it by the nature of our souls. But we're both conveying the same thing that we think it's important, which is from our different personalities. And in trying to create that, there's other things departments can do. And here's two things that Williams has tried to do, two experiments. The other things I've talked about have been pretty standard. Talk about research, try to look at other areas in mathematics, listen to the older people and try to get, you know, feel the excitement, which it is exciting. But here's some things we've tried at Williams. Neither one really worked. But it's what we try. One is called Dada math. The name Dada is coming from the artistic movement after World War I, that eventually led to surrealism. Here's what we tried to do. We sat around a table, the department. It was private, we didn't let anyone come. Math reviews, which Ed Dunn talked about last week. Math reviews reviews all research papers and mathematics going back to 1939. They have a classification scheme. You know, there's, can someone throw out one of them who happens to know a classification scheme? 13 is commutative algebra, 14 is algebraic geometry. And it's not just that, 13A1, commutative algebra over rings of characteristic that are not prime. I'm making that one up, but there's all these numbers. So what we did is we printed out the classification, cut them up into little numbers, put them into a bowl, and then we had someone pick out three of the numbers. Then I think, it's usually you, Mihai, right? You're in charge of putting them on the board, I think you volunteer. And so we go, and so we look it up and go, okay, 15A13, what is that for? And we write down what it is. And then we got 73 seeded. We write down what it is. We're all three. Then we talk about, does anyone know what any of these words mean? Which we don't know. But then, and what we're doing is an experiment among friends, but still people who are willing to kind of risk ridicule, is we try to ask a question that unites all three. We're hoping that there's just some kind of randomness that will work. Do we spend our careers doing that? Absolutely not. It's something we do for an hour every two or three years, right? Yeah, it's kind of fun. The most, my dream was, we have a real research paper coming out. That has not happened. That's why I said it was sort of a failure. The first time was the most successful is we actually came up with the conjecture that we later discovered was a fairly well-known conjecture number theory. I said, that's not bad, bunch of amateurs doing it. The more reason once have not been so successful because beer has been involved. They were more fun. But we still learn things. So for example, one of the things we pulled out was reverse mathematics. Who here has heard of reverse mathematics? None of us had. One person way in the back, two people. Okay, none of us had heard of reverse mathematics. We could tell from the subject classification of something with logic, and we're going, I don't know what it is. Now Susan Lapp, who's a community of algebras, had the best idea of what reverse mathematics was. She said, I know, I know. It's when, and she was sitting in a chair, it's when you're doing proof, you go, beep, beep, beep, beep. In other words, we had no clue. But what we did do, every single person when they went back to their office looked up what reverse mathematics meant. We learned some. It's kind of cool. It's kind of like, here's a level of mathematical maturity. One reason why older faculty can pick up math more fast than younger people. And one reason is that we kind of know what results are important. So if you're learning measure theory, the Lebesgue dominated, for K through 12, it's an important theorem. Lebesgue dominated convergence theorem. We know that the Lebesgue dominated convergence theorem was not true. No one would study measure theory. Doesn't matter what the axioms are. If that theorem wasn't true, it wouldn't be a first year graduate course. And there's various results like that. We know that if that theorem wasn't true, no one would study it. What reverse mathematics does, is does that with a vengeance. It takes the results you want and then tries to find the appropriate definitions and axioms that will get it to you. It's a really clever idea. Did any of us change our research? Absolutely not. But it's fun. The second thing we do is every few years, we have our mathematical orgies, which for the younger people is not what you think it is. What it is is we choose a topic that we want to learn about. Now Williams is a fairly small department. There might be 12 or 13 mathematicians. But we still want to keep abreast of it. We want to know what's going on. So we choose a topic and spend a day talking to each other about it. And here's two examples of how it could work. The first time we did it, it was about, there was in 1996, there was the second string theory revolution of Cybert Whitten. If you'd asked me in 1999, what was the most important areas of modern mathematics, I would have said, the stuff from string theory involving algebraic geometry and in number three, Taniyama Shamora. I said, neither one, I'm never gonna touch. I have my own stuff going on, the world's best are doing that. But I was on leave at University of Michigan in 2000, 2001. I was hanging out with the algebraic geometers and they talk physics, physics, physics, physics, physics. And now one fear I have as an older mathematician is that I'll end up lecturing from yellowed notes. The same ones I had when I was 25 and having absolutely no change in them. That I missed something major happening because I'm just stuck in my rut. And it dawned on me listening to that in that leave year that string theory had something that should be talked to undergrad, it's not string theory itself and I'm not gonna explain what it is. It's mainly there was a new idea and it really comes back to the work of Simon Donaldson in the early 1980s engaged theory. Don't worry about what those words mean. But what really happened was the following. Traditionally there's math, physics. There's not all that much connection actually. If anything, math influences physics historically. What happened a large part with Simon Donaldson when you and I were his age unfortunately but in first or second year of graduate school. What he did is he took an area of physics that was important. He identified the important math in it. Then he took that math seriously in non-physical contexts and he discovered structure that was unbelievable. For the expert he technically took, he took Yang-Mills equations which was in elementary particle physics. He took those equations, he applied it to four manifolds and he looked at the space of all connections on four manifolds. In any dimensional space there's no structure but if you look at those that are Yang-Mills some under some symmetries, you end up with a five dimensional space that has a corboranism with the original four mandible plus a whole bunch of copies of CP2 and CP2 with reverse order. It was amazing. And in 1980, the study of this area math four manifolds was a very respectable area of mathematics. 1984 was the hottest area in the world. Something changed. The arrow, math now influences physics. That should be taught to undergraduates. So we were gonna do string theory and here's an example of how someone can learn something by doing that. People volunteered to give talks on this. We all knew we were amateurs. One of them was Frank Morgan. Oh, good, I can see people better. I should have walked up here earlier. Frank Morgan is a very serious differential geometry. He studies geometric measure theory, something called calculus of variations. In the early 90s I had been increasingly impressed by something called Nerther's theorem. I think it's one of the most important theorems of the 20th century. Is there an echo here? Oh, wow. What is this theorem? No, no, it's not a probital, I'll gladly explain it later. But what Nerther's theorem does is sort of links the differential equations or the integrals that describe the world with fundamental symmetries of the universe with things that are fundamentally conserved or invariant. I was amazing. So Frank, I knew when I lived down the street from him, we'd regularly walk home and I asked Frank, you do calculus of variations. Nerther's theorem must be important to you, and he would go, no, okay. Now Frank was very fast. But he is very fast, he's still alive. Very fast, it didn't make sense to me. Okay, a couple of years later I asked him again, he went, no, okay. So we're now gonna do the orgy on string theory. And he volunteered and I said, Frank, you're gonna talk on Nerther's theorem, he says that has nothing to do. He says, here's the references. He went and looked about the next day, he was in my house and going, my God, this is important. What I'm convinced is what happened was, is that he heard whenever I said the word Nerther, he thought Nertherian ring, same Nerther, totally different concept. And it had absolutely nothing to do with it, but he was very fast, he heard the Nerther, and he was like, no, no, no, no, no, Tom once again is not into what he's talking about. And he learned something and that was good and he gave a great talk. The other thing in terms of this, in terms of these orgies, is putting yourself at risk. And now I'm gonna talk about Olga Beaver. She went by Ollie. She was the oldest, she's passed away since this, it happened. She was a perfectionist. It pained her to make a mistake. I mean, you could see her physically crumble when she said something on the board that wasn't quite right and got mixed up. I don't care, I make mistakes all the time. But she volunteered to speak. And I think we were talking about early work of Terry Tau. It was before he went to Fields Medal. I think I kind of noticed, huh, there's this young guy who's showing up all over the place. Wow, Terry Tau. And he wrote this long survey about what he's doing. I'm like, oh my God, this is amazing. This guy's gonna win a field. This is amazing. We gotta learn about this. Oh my God, this is amazing. She volunteered. So she had to speak on a part of a paper of his that was far, far from her research specialty. She did not know what was going on. But she gave a great talk. We later heard the full story. Perfectionist. Here's the story. She couldn't figure it out. She did not know what was going on. We started, you know, she kept working on it. She was pulling all night. This is the oldest person in her department. Our administrative assistant says, why do you all work weekends? She says, look at Ollie. She should be home with her grandchildren. No, she was working, staying up all night working. She returned to smoking, which is not a good thing. But I mean, I know she took years to stop and she returned to smoking. Though I heard she later said whenever, she lived fairly near me and I would walk by her house. She said, yeah, didn't you notice that? Sometimes you saw my head. I went like this, but I still smoke and I didn't want you to see it. She was panicked. She couldn't figure it out. It was the day of our orgy. Someone was speaking from 10 to 1030. She was in the back. She did not know what she was gonna say. She was full of panic. If you looked at her, she looked pretty calm. Teen, she said, suddenly, boom. She saw exactly what was going on. She gave up a great talk. What's impressive about that is that here was someone at the time who was probably in her mid-60s. She was taking an amazing risk and she was willing to do that. And she had that phenomenal moment of why most of us do mathematics, that moment of insight. She became, even though she was already quite mathematically mature, she became more mathematically mature even at a late stage of her career. That's what we should all be doing all the time. That mathematical maturity is not a single thing. It's that every stage we're learning more and more. Sometimes just facts, but also ways of seeing the world. And that's what we should bring to our teaching at all times and that's what unites everyone in this room, even the photographer. Thank you very much. It's occasionally shocked at Williams during Gay Pride Week. I'm serious. I'm not quite sure what I should say, but thank you, Tom. No, that was wonderful. Thank you very much. Are there any questions, comments? Yeah, in a mature fashion. Oh, thanks, yeah. And geometry is what I primarily taught. In a geometry course, who has trouble at it, respect their mathematical maturity. So the question is, how do you bring mathematical maturity to the real world of K through 12 in terms of the standards? One, if you have someone in a geometry course, is it the level of mathematical maturity? Can't you do fractions? But it's still required to pass that geometry test. Your life is dependent, your school is dependent upon it. I mean, yeah. That's, I'm gonna punt that question. You know, when I hear that kind of stuff, I go, oh, thank God I teach at a small liberal arts college. I don't know how. I do think, and this is not answering your question, I'm doing the standard college professor of shifting the question to one that I can answer. One thing I like about the rhetoric of mathematical maturity is that it's not talking about talent. We're not supposed to talk about talent. But even though there really is talent out there, but somehow most Americans, I suspect in other countries too, but certainly in America, people think it's either you're good at math or you're not good at math. And we know that's totally false. But people are at just different levels of mathematical maturity. That's not answering your question. I don't know. I have seen, hence, that by the time people are getting to college now, their levels of mathematical maturity is not quite as strong as they were five or 10 years earlier. And I describe it possibly, and I'd be curious to see if we could actually prove it or disprove it from no trial left behind in all the testing. I'm not sure that's true, but it seems people now in secondary school and primary school is to have many, very simple tests. Other questions? Other questions? Let me bring the mic up. Is there anything you would do to improve the, I'm particularly thinking of the subject classification experiment, but you mentioned these things that you do and said they're sort of failures. Do you have any advice or initial thoughts on how they could be done better? Well, no, if I did, we would do it. But in thinking about, I think like the data, which I like the idea of just pulling out randomly and trying things, it would have to be done a lot. It would have to be almost like a weekly thing. And suddenly, the faculty at Williams are busy. They're very ambitious people. They work, I don't know, 16 hours. They work all the time and they love what they do and no one's going to be willing to do that. But it was like every week we'd be going more comfortable with it. We'd start learning what are reasonable questions, but that's not going to happen. What would you say, Mihai, who you've experienced, Dada? What? I think we should not do it every week. Yeah. Yeah. Good, other thoughts? Yep. So I am an undergraduate at UC Berkeley. Uh-huh. So as an undergraduate, I feel like there are two different ways to learn mathematics. I mean, as an undergraduate, you can either be in a big public school with a lot of mathematicians and a lot of different activities going on, or you can be in a small liberal arts college with more attention, which I guess you would know more about. But which one do you think is better for a young mathematician? Which one? That depends. I mean, I think for most human beings at a small liberal arts school, it's better. You've been paying attention to it. I talked to my best friend who's now on Wall Street, but he was a serious math person for a long time. And he told me when we were in mid-30s that if he had to go through college again, he'd want to go to Penn State. I went, what? He'd gone to Haverford, a small liberal arts college. Penn State? And he went, oh yeah, they have everything there. He said, but you regret going to Haverford? No, in 18, I went to Haverford. I wanted someone to be watching out for me, more guidance, but at 35, I was ready for a big state university. It depends. I went to a huge public institution, even the joke I made about it. I cannot imagine having gotten a better education than I got at University of Texas at Austin. Now, one of the advantages back in the mid-70s, no one took upper-level math. My classes were one, two, or three, most of them. I had phenomenal attention. The business school old classes were hundreds. So I didn't have the standard experience. And I felt a great deal of attention. Certainly the classes I had at University of Texas were much smaller than the classes I teach at Williams. So, but the thing at a large state university, you have to watch out for yourself more. No one, you just have to be more savvy. And that's where a lot of people get wiped out. They're not at a certain, maybe not mathematical maturity, but they're not at a certain intellectual maturity to be able to make reasonable choices. While at a liberal arts school, people will protect you much more. It's hard to screw up. Does that make sense? Yeah, yeah. It's also most human beings kind of like people to know who you are. I actually did not like that. I wanted to be as far away from my high school experience as possible. And I loved being a mere number. Now, since then, in terms of teaching, I love being a place where I know my students. So if you were to go back to being an 18-year-old, which would be a good place? I would still, I would go back to University of Texas. But both my kids, I encouraged to go to liberal arts schools when they did. I think for most people, it's the right move. But not for everybody. I'm really intrigued by the questions that you were asking some of the folks from different subject areas about whether there was something analogous to mathematical maturity and their being flummoxed or not able to come up with something. And I'd love to get your thoughts on the extent to which those conversations shaped or helped you think about the nature of mathematics and how it is fundamentally different than other subjects. Well, first of all, I think they just don't know what they're talking about. I mean, I think there really is the notion of historical maturity. I just couldn't convey it. So that was my actual, I have no evidence for that's what I believe. I also think though, in terms of mathematical maturity, there's a level of intellectual maturity. And that one of the difficulties when you're doing various colleges and universities is that people come in at different levels of intellectual maturity. For example, there was about 10 or 15 years ago, there was a major survey of various students. Could they recognize that they looked at a bibliography to recognize what was a good reference and what was a bad reference? This is being told to us at a faculty meeting at Williams and it's supposed students were horrible at it. I suspect many of you younger people, horrible at it. We were aghast and we asked, well, what schools were any of you? And they told us the name of the schools. Now, Williams is a very elitist, arrogant place. You could feel us at the falcon going, well, that wouldn't be it, Williams. Our students are some of the best, the creme de la creme of the world. And then the person reporting says, by the way, Williams was in this list. Our students sucked too. Not in terms of, and so I was thinking in terms and my colleagues did not use this rhetoric, I thought it's intellectual maturity. They don't know how to read a bibliography but they don't aren't aware of this is a reputable source. This is not a reputable source. If that kind of answer your question, that helped me. And I tell people, we have to tell them. Someone, do you have to teach students how to read? I regularly find myself teaching students how to read because they're taking a serious math class and they want to read the math text linearly line after line, like it's a novel. And I look, I'm like, you don't do that. You jump around, you do this, you try, I say, I get out of books sometimes, okay, look at the table of contents, try to find out what the correct question is. Go to the index, find out what the big words are. Find out what kind of chapters are they having. Try to recreate the argument as much as possible. Read the first chapter, read the last chapter. Then you might be ready to start going. Create the intellectual arm the entire time. That's kind of like the intellectual maturity, right? It can also be more, Frank Morgan described it, you should read a math paper like you would read People Magazine in a dentist office. So he had images of that, you have a People's Magazine, he's not the type of person who reads People's Magazine. So he had People's Magazine there, he's waiting for the dentist, and he kind of pages through it and he finds the articles he wants to read. He's there, stuck there for an hour or two. He kind of goes back and forth, by the end of the hour, he knows that People Magazine thoroughly. And he says, that's how you read a math paper too. And that's kind of also intellectual maturity. Anything else? Okay, well, let's thank Tom very much. Thanks.