What is Wolfram Alpha good for?

It’s early days and this isn’t meant to be a carefully considered review of Wolfram’s “computational knowledge engine”. Rather, I just want to point out, for the benefit of anyone who might not yet know, that one small part of what it does is genuinely useful in a certain circumstance that comes up from time to time. Suppose that for some reason you want a list of primes, or to know to 100 decimal places, or the 100th power of 2. Previously I would have used Google for the first two, banking on someone somewhere having put the information online, and I might have struggled to understand just enough Mathematica to do the third. (However, I have just discovered that powers of 2 can also be found quite easily with the help of Google, so a more complicated example might be needed.)

Anyhow, with Wolfram Alpha one can type in some reasonable text such as “The first hundred powers of 2” or “pi to 100 places” and it works out what you mean and gives you the answer. That alone won’t change my life, but it is convenient and it will occasionally help me with things like preparing lectures for a general audience, which I think is just about enough to make it worth it to me to bookmark the site, though I haven’t yet done so. It will also sketch graphs and simplify mathematical expressions without one having to learn any special language to put them in — you just guess what to write and if your guess isn’t too perverse it can work out what you mean.

What else does it do? Typing in “father of Barack Obama” gives “Wolfram|Alpha isn’t sure what to do with your input”. Just typing “Barack Obama” gives you his full name and his date and place of birth. Typing “England” gives you various basic facts about England. Typing “capital of Uruguay” gives you Montevideo and various facts such as its population, current weather, etc. After noodling about like this for a short time, I did what any non-saint would do and typed in my own name. To be precise, I typed in “Gowers”. The result was “Wolfram|Alpha isn’t sure what to do with your input”, together with the helpful suggestion that perhaps I had meant “powers”.

I think that gives a fairly good idea of what it does and what it doesn’t do. Perhaps one should regard the latter as a truly positive and innovative aspect of Wolfram Alpha: a New Kind of Search Engine (or whatever it should be called) that doesn’t waste hours of your time by tempting you to look yourself up.

A solution to an exposition problem

Let me explain the title of this post by quoting from Timothy Chow’s highly recommended expository article A beginner’s guide to forcing: “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.” He goes on to claim that forcing was an open exposition problem, since there was no explanation in the literature that had these qualities.

I have just finished giving a graduate course in Cambridge on two highlights of theoretical computer science: Razborov’s lower bound for the monotone circuit complexity of the clique function, and Shor’s quantum algorithm for factorizing integers. Because nobody was taking an exam on the course, I was free to be somewhat informal in the lectures, but at one or two points it got slightly too informal so I rashly promised that I would produce notes on Razborov’s theorem. However, the document that resulted ended up being less a set of lecture notes in the usual sense and more an attempt to solve an exposition problem in Chow’s sense. Very briefly, Razborov produces a lattice of a certain kind, with a rather strange definition, and it goes on to do the job it is supposed to do in a seemingly miraculous way. What I have written is an attempt to solve the problem, “Where did Razborov’s definition come from?”

When normality is abnormal

Suppose you were reading a novel, or watching a play or film, that included a fictional mathematician …

My guess is that the moment you read the two words “fictional mathematician” a second or two ago, your mind leapt ahead and you had a pretty good idea of what he—yes he, since even if there are female fictional mathematicians out there, femaleness is unlikely to be part of your instant and not fully conscious reaction to the phrase—was like: a social misfit who is prone to flashes of extraordinary insight that completely baffle everybody else, or perhaps a social misfit who would like to have those flashes but doesn’t and goes mad instead, or perhaps a social misfit who does have the insights but with madness the huge price he has to pay.

So here is a question: is there any example of a mathematician in literature, theatre or cinema who is a fairly normal person socially, and pretty good at maths but not astoundingly so? Some examples that do not work are Uncle Petros, from Uncle Petros and Goldbach’s Conjecture, both the father and the daughter in Proof, and Will from Good Will Hunting: they’re all either ridiculously good at maths (usually without having to do all that routine stuff like learning the proof of Schur’s lemma, or the open mapping theorem, or the Gram-Schmidt orthogonalization process etc.) or mad, or both. I also don’t count characters if they are colleagues of a crazy genius and their main role in the book/play/film is to marvel at how clever the crazy genius is. Let’s say that the character has to be the main one, or at least the main mathematical one.

Cohomology for amateurs—by an amateur

I have just finished a Tricki article about how to recognise situations where homology and cohomology can help you. (It is aimed at people who might have seen the definitions but felt uncomfortable about how to apply them.) Ages ago, I sort of promised to write something on this topic on this blog, so I am posting the article. This will have the added, though no doubt painful, benefit that people who know far more about the topic will be able to point out mistakes, misleading statements, unnecessarily complicated ways of thinking about things, and so on. But the one thing I do like is the (admittedly old-fashioned) way of looking at homology and cohomology as “soap-bubble homotopy”. See below for an explanation.