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 ...

Galois Theory Posted by Tom Leinster I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023. I first shared the notes on my website some time ...

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 ...

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 ...

That is correct. There are finite index subgroups of profinite groups that are not open, i.e., there are profinite groups that do not equal their own profinite completion. However, by definition, the ...

At a first glance, it seems like finding concrete definitions for these directed geometric models would be a tall order. After all, the theory of stratified cobordisms itself has its mathematical ...

On the one hand, an ultrafilter on a set can be seen as a primitive sort of probability measure, in which every subset is assigned a probability of either 0 or 1 and the measure only has to be ...

That’s a nice question! We’d get that if the category of commutative rings had a factorization system which let us factor any morphism as an epimorphism followed by some other sort of morphism in a ...

This is very exciting. I remember hearing Michael Ching speak about this maybe a year or two ago and thinking that it sounded like such an ambitious project, I wouldn’t expect to see it completed for ...

Donaldson Theory and 𝒩 = 2 \mathcal {N}=2 SUSY Field Theory Alexandru Scorpan, “ The Wild World of 4-Manifolds ” J.D. Moore, “ Lecture Notes on Seiberg-Witten Invariants ” Gompf & Stipsicz “ ...