How do you count rooted planar n -ary trees with some number of leaves? For n = 2 this puzzle leads to the Catalan numbers. These are so fascinating that the combinatorist Richard Stanley wrote a ...

These are some lecture notes for a 4 1 2 -hour minicourse I’m teaching at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens. To save time, I am ...

I’ve been blogging a bit about medieval math, physics and astronomy over on Azimuth. I’ve been writing about medieval attempts to improve Aristotle’s theory that velocity is proportional to force, ...

In Part 1, I explained my hopes that classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant k k approaches zero. In Part 2, I explained exactly what I mean ...

Posted by John Baez I keep wanting to understand Bernoulli numbers more deeply, and people keep telling me stuff that’s fancy when I want to understand things simply. But let me try again. The ...

When is it appropriate to completely reinvent the wheel? To an outsider, that seems to happen a lot in category theory, and probability theory isn’t spared from this treatment. We’ve had a useful ...

In this year’s edition of the Adjoint School we covered the paper Triangulations, orientals, and skew monoidal categories by Stephen Lack and Ross Street, in which the authors construct a concrete ...

Why Mathematics is Boring I don’t really think mathematics is boring. I hope you don’t either. But I can’t count the number of times I’ve launched into reading a math paper, dewy-eyed and eager to ...

I’ve long been fascinated by the relation between ‘classical’ and ‘quantum’. One way this manifests is the relation between cartesian monoidal categories (like the category of sets with its cartesian ...

I’ve been talking about Grothendieck’s approach to Galois theory, but I haven’t yet touched on Brauer theory. Both of these involve separable algebras, but of different kinds. For Galois theory we ...

The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a ...

The idea One of the standard constructions of the real numbers is as equivalence classes of Cauchy sequences of rationals. Let us consider a non-negative real number a. One way to say that a sequence ...