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no one could figure out what a math lab was as in a laboratory have Erlenmeyer flasks and things where are the flasks where's the equipment in a in mathematics my contention is that this lab follows the scientific method students take many courses most of them have a set syllabus it's organized it goes from a beginning through definitions and theorems to an end they're asked to solve problems which are well posed and have a have an expected answer this course is completely different because here the questions are not well posed they have to decide what part of this question they're going to focus on they have to decide where the exciting mathematics is going to be it's actually very hard to get beyond the smooth outer shell of a problem we rarely give ourselves a specific target and aim only at that target it's very easy just to you know write down the statement of the problem and some equations that you know which of elevant and then just sit and ...

missed out on one that's quite fun uh and uh it leads to uh these fantastic pictures of fractals you may have seen I intended to bring in a book by pipin that has uh sort of the beauty of fractals is his title has pictures of the mandal Bro Set uh just amazing um and just it's interesting mathematics and it's mathematics of recent years so I'm going to put off the wave equation and heat equation to the second lecture today lecture 12 and discuss this one because it's sort of it's so here's the general plan you have some function and you start at some point u0 and you compute F of u0 and that gives the next point then you compute F of U1 and that gives U2 and then you keep going so so u n is so to speak F of f of f of it's where're we're uh we're taking well the mathematical word is composition so so let me just put up U2 so you see U2 U2 is f of f of U uo because that's U1 so we're we're repeating the same operation ...

PROFESSOR: I sent out a survey this morning about how the class is going, what you thought of the problem set. And I would appreciate if you provide me some feedback-- so things you like or don't like about the class or about the problem set that was just due last night. So I can try to adjust to make it more interesting and useful for all of you. Last time we talked about Szemerédi's graph regularity lemma. So the regularity lemma, as I mentioned, is an extremely powerful tool in modern combinatorics. And last time we saw the statement and the proof of this regularity lemma. Today, I want to show you how to apply the lemma for extremal applications. In particular, we'll see how to prove Roth's theorem that I mentioned in the very first lecture, about subsets of integers lacking three-term arithmetic progressions. First, let me remind you the regularity lemma. We're always working inside some graph, G. We say that a pair of subsets of vertices is epsi...

Now let's work out an example that shows how to use the pairwise independent sampling theorem to actually do some sampling and estimation. So let's remember that our basic theorem says that if we have n independent random variables-- pairwise independent, with the same mean and variance-- and we look at their average, the probability that their average differs from the mean by more than a given tolerance delta is less than or equal to this formula here. Which is the standard deviation over delta squared times 1 over n. Now we're just going to be plugging into this formula, but I want to state it here for the record to remember that this is the pairwise independent sampling theorem that we're given. Which is, what we've seen in general, allows us to calculate the degree of confidence we can have, the probability that we have given n or the n that we need given how confident we want to be. So let's go ahead and do the example. And I want to think abo...