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’m a little bemused by the popularity of the Galois theory notes. I’ve made quite a few sets of course notes public before, e.g.: Fourier analysis General topology Linear algebra Category theory But ...

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 ...

For questions 1 and 2, isn’t that true for any group G, not just the fundamental groups of a manifold? And moreover, I think of this as the definition of the profinite completion of a group: as an ...

Guest post by Christoph Dorn While the term “geometric higher category” is new, its underlying idea is not: coherences in higher structures can be derived from (stratified) manifold topology. This ...

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 ...

“Solid ring” sounds self-contradictory, since a ring should have a hole in it. But mathematicians use words in funny ways. Here’s one reason this is interesting: the category of affine schemes is the ...

Thanks for these reactions, Tim. I guess the first thing I’d like to know is the relationship between BBC’s tangent ∞ -categories and what nLab has as tangent (infinity,1)-category, as found in ...

This is the homepage for the UT Geometry and Quantum Field Theory Seminar. At the organizational meeting we will flesh out the details of our plans for the semester. Below are some suggestions to get ...