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January 31, 2017

The Brunn-Minkowski Inequality

Posted by Tom Leinster

Let AA and BB be measurable subsets of n\mathbb{R}^n. Then

Vol(A+B) 1/nVol(A) 1/n+Vol(B) 1/n Vol(A + B)^{1/n} \geq Vol(A)^{1/n} + Vol(B)^{1/n}

where A+B={a+b:aA,bB}A + B = \{ a + b : a \in A, b \in B \}. This is the Brunn-Minkoswki inequality. It’s most often mentioned in the case where AA and BB are convex, but it’s true in the vast generality of measurable sets (except for \emptyset, where it obviously fails).

Here I want to explain just a few basic things about this inequality and its consequences. For instance, it easily implies the famous isoperimetric inequality: that among all sets with a given surface area, the one that maximizes the volume is the ball.

Posted at 10:48 PM UTC | Permalink | Followups (8)

January 24, 2017

Papers Written While Drunk

Posted by Tom Leinster

I’m currently reading a preprint by a deservedly very well-respected and highly-reputed mathematician. It’s enjoyable, inspirational, and wonderful. The ideas that it expresses have been haunting and taunting me for years.

For various reasons, I have the impression that it was not wholly written while the author was wholly sober. That’s OK; I’ll judge the paper for what it is, not on how it was written. But it leads me to wonder: how common is this? In literature, it’s a well-established tradition to the point of cliché. For instance, here’s Ernest Hemingway —

Hemingway in Cuba

— giving a cocktail recipe for difficult political times (1937), “to be enjoyed from 11:00am on”. You can find countless examples of fiction writers enthusing about chemically-assisted escape from the so-called real world.

But mathematics prides itself on sharpness and precision in counterpoint to creativity. We love to say that we’re more creative than poets, but a piece of mathematics is in deep trouble if it’s logically wrong. So where does drugged, drunk or hallucinatory mathematics fit into our mathematicians’ culture?

Posted at 11:44 PM UTC | Permalink | Followups (21)

January 10, 2017

Category Theory in Barcelona

Posted by Tom Leinster

I’m excited to be in Barcelona to help Joachim Kock teach an introductory course on category theory. (That’s a link to bgsmath.cat — categorical activities in Catalonia have the added charm of a .cat web address.) We have a wide audience of PhD and masters students, specializing in subjects from topology to operator algebras to number theory, and representing three Barcelona universities.

We’re taking it at a brisk pace. First of all we’re working through my textbook, at a rate of one chapter a day, for six days spread over two weeks. Then we’re going to spend a week on more advanced topics. Today Joachim did Chapter 1 (categories, functors and natural transformations), and tomorrow I’ll do Chapter 2 (adjunctions).

I’d like to use this post for two things: to invite questions and participation from the audience, and to collect slogans. Let me explain…

Posted at 6:48 PM UTC | Permalink | Followups (81)

January 4, 2017

Globular for Higher-Dimensional Knottings (Part 3)

Posted by John Baez

guest post by Scott Carter

This is my 3rd post a Jamie Vicary’s program Globular. And here I want to give you an exercise in manipulating a sphere in 4-dimensional space until it is demonstrably unknotted. But first I’ll need to remind you a lot about knotting phenomena. By the way, I lied. In the previous post, I said that the next one would be about braiding. I will write the surface braid post soon, but first I want to give you a fun exercise.

This post, then, will describe a 2-sphere embedded in 4-space, and we’ll learn to try and unknot it.

Posted at 6:08 AM UTC | Permalink | Followups (9)

January 2, 2017

Basic Category Theory Free Online

Posted by Tom Leinster

My textbook Basic Category Theory, published by Cambridge University Press, is now also available free as arXiv:1612.09375.

Cover of Basic Category Theory

As I wrote when I first announced the book:

  • It doesn’t assume much.
  • It sticks to the basics.
  • It’s short.

I can now add a new property:

  • It’s free.

And it’s not only free, it’s freely editable. The book’s released under a Creative Commons licence that allows you to edit and redistribute it, just as long as you state the authorship accurately, don’t use it for commercial purposes, and preserve the licence. Click the link for details.

Posted at 5:43 AM UTC | Permalink | Followups (21)