Noncommutative Analysis

Month: February, 2013

Mathematics on mathematics

 

This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective students in our department. This is supposed to be a story, it is intended to give a flavor, and neither the history nor the math are 100% precise, because it is a 15 minute talk! The big challenge is to take some rough ideas from this post, throw away the rest, and make that into an interesting quarter of an hour lecture. Comments are very welcome.

1. Introduction

What is mathematics? I am not going to answer that. You have all met mathematics in your life, in school, but also in other places as well, because math is everywhere. So you have some kind of idea what mathematics is about. However, I suspect that the most profound aspects of math have been hidden from you. I am here today to try to give you a taste of this mathematics which you have not yet seen. It is only fair to let you know that — for better and for worse — the mathematician’s mathematics, the mathematics that you will study if you do an undergraduate degree in math, is of a dramatically different nature from the math you learn in high-school or the math-is-everywhere kind of math which you meet in various popular accounts.

You have met various different kinds of mathematics: combinatorics, geometry, algebra, integral and differential calculus (aka HEDVA — which literally means “joy” in Hebrew). It seems as if mathematics splits into various branches, where in each branch there are different tasks that one should do. The objects of study of geometry are triangles, circles, trapezoids, etc; one has to prove that a certain triangle has this or that property, or one has to compute some angle or length or area. The objects of study in algebra are certain symbolic expressions or equations; one has to find the root of an equation, or to simplify an expression. In HEDVA the objects of study are functions; one has to compute the minimum of a function, or its anti-derivative, and so on.

The theme of this talk is that the objects of study in mathematics do not have to be only triangles or functions or equations, but they can also be geometry or analysis or algebra. Mathematics can also be used to study mathematics itself. This is profound. But perhaps more surprisingly, this has practical consequences.

Of course, I have no time to tell you precisely how this works. For this, I recommend that you come here and study mathematics. Read the rest of this entry »

Partial results

What is more wonderful: something wonderful, or the moment before the wonderful something happens?

Recently I proved a result that can best be described as a partial result. This is certainly not what I planned to obtain. My goal was to prove C. The plan was as follows:

  1. Prove A
  2. Prove B
  3. Prove that A and B together imply C

Step 3 is easy and I had it from the start. I’ve been trying to prove A and B for a several months now. I was sure that step 1 is the easier part and that step 2 is harder. I was quite hoping that step 1 follows from a general theorem which I also wanted to prove, but would have been just as happy to find it in the books.

Last week somebody showed me a counter example to the general theorem that would have implied A. Two days later I proved B. Now it remains to prove A, but A is not going to be true for the reasons I thought it would. Have to try a different approach…

But wait! I don’t want to prove A yet (not that I believe that there is a real “danger” in that happening). I want to enjoy B. B is beautiful. If I prove A, then C follows, and B is a triviality compared to C.

Worse, if A proves to be impenetrable, then as far as C goes, B is useless.

The role of B as a step in the proof of C can be appreciated a little more if we wait. Even if A is never proved, I wish to take some time enjoy B.

Alright, time out over! Back to A.