Talks by Wilkinson, Harris, Stanley, Blum, Bhargava, Shor on Wednesday.<\/p>\n
by\u00a0Nicholas Neumann-Chun<\/a><\/p>\n I am now one full day of math lectures older.\u00a0 The invited addresses are fairly accessible; that is, they’re aimed at a large audience (quite literally!).\u00a0 That doesn’t mean I understood all, or even most, of what was said, but I was able to engage in bits of the six talks I saw throughout the day.\u00a0 I’ll try to give a feeling for my immediate reactions to each of these talks.<\/p>\n <\/p>\n —<\/p>\n First, I got to hear Professor Amie Wilkinson, from Northwestern University, talk about Chaos and Symmetry in Partially Hyperbolic Systems<\/span>.\u00a0 At first, I was somewhat intimidated by the title, but I soon found there were several ideas that I could latch on to, and so get a general understanding of the lecture.\u00a0 Basically, the subject was the three-body problem<\/a>, which I had heard of before.\u00a0 Professor Wilkinson began by talking about the 1885 contest to solve a certain case of this problem, the winner of the contest, Poincar\u00e9, and his discovery of chaos.\u00a0 Then followed a lot of more recent mathematics, especially work by Smale<\/a> in the mid-20th century.\u00a0 I came away with a vague, pictorial sense of what “hyperbolic,” “ergodic,” and “partially hyperbolic” mean.\u00a0 The talk ended with a discussion of the Pugh-Shub Conjecture<\/a>, and a brief explanation of how symmetry relates to all of this.<\/p>\n —<\/p>\n Encouraged that I wasn’t completely lost in the first talk, I eagerly awaited the next speaker: Professor Joseph Harris from Harvard University.\u00a0 He spoke on The Interpolation Problem<\/span>.\u00a0 I was unable to guess at the nature of the talk from the title, but the key is just to go in with an open mind.\u00a0 I even understood his very first slide, which I believe was just a slightly different statement of the Fundamental Theorem of Algebra<\/a> than I am used to.\u00a0 This deals with the zeros of polynomials in one variable.\u00a0 I think that the rest of the talk was an attempt to explain an attempt to generalize this statement to polynomials in multiple variables.\u00a0 Professor Harris was an exceptionally engaging speaker, in that he was lively and animated, so it’s a shame that I didn’t understand anything past that first slide!<\/p>\n —<\/p>\n To start off the afternoon, Professor Richard Stanley, from MIT, delivered the first of three talks that he’s giving on permutations.\u00a0 This first was on Increasing and Decreasing Subsequences<\/span>. \u00a0I’m getting used to the pattern of these talks: I’ll understand the first couple of minutes very well, then a combination of new notation and concepts will leave me grasping only the bare outline of the talk, and finally a few key theorems will be introduced, of which I’ll understand only small bits and pieces.\u00a0 I gather that this is as it should be!\u00a0 Learning mathematics, I have heard, is in many respects like learning a language.\u00a0 I have learned many words already, and this allows me to partially understand many of the talks, but I will always have more to learn; I am familiar with some common syntactical constructs, allowing me to understand some common mathematical arguments, but, again, there is always more to learn!<\/p>\n Back to Professor Stanley’s talk on subsequences.\u00a0 First, he defined increasing and decreasing subsequences, and how these notions can be used to na\u00efvely model situations such as passengers boarding an airplane.\u00a0 Then he introduced SYT (standard Young tableaux).\u00a0 I think I understand what these are, but I didn’t see what purpose they serve in studying subsequences, nor did I follow the explanation of the RSK algorithm.\u00a0 Interestingly, we somehow wound up with two new interpretations of Catalan numbers<\/a>.<\/p>\n —<\/p>\n Next came a talk by Professor Lenore Blum from Carnegie Mellon titled The Real Computation Controversy: Is It Real?<\/span>.\u00a0 This was about computer science, about which I know very little; as far as I could tell, the subject was basically the difference between computation based in discrete mathematics (i.e. Turing Machines) and computation based in analysis.<\/p>\n —<\/p>\n My favorite talk of the day was from Professor Manjul Bhargava (of Princeton University), entitled The Factorial Function, Integer-valued Polynomials, and p-adic Analysis<\/span>.\u00a0 The basis of the talk was work that Professor Bhargava has done himself on generalizing the concept of factorial to rings other than the usual integers.\u00a0 This generalized factorial function is achieved in a non-trivial, non-obvious way, but is not too difficult to understand, and works as one would hope.\u00a0 That is, all the theorems from number theory and so forth that involve factorials extend in a simple way to general rings (or subsets of rings) using this new factorial function.<\/p>\n