As the title says. I’ll freely use the notation of my paper, although unfortunately I can’t figure out how to implement “mathscr” fonts on this blog, so objects denoted in mathscr fonts in the paper are denoted in mathcal fonts below – e.g. 680 more words

## Tags » Cohomology

#### Étale Cohomology

I have recently been somewhat obsessed with algebraic geometry (only the terminology makes it kind of confusing), and I revised what little I knew about étale cohomology. 866 more words

#### Topological Diophantine equations

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let be a commutative ring, which in the most classical case might be a number field like or the ring of integers in a number field like . 3,810 more words

#### Five proofs that the Euler characteristic of a closed orientable surface is even

Let be a closed orientable surface of genus . (Below we will occasionally write , omitting the genus.) Then its Euler characteristic is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be , roughly in increasing order of sophistication. 3,293 more words

#### The Unimaginable Mathematics of Borges' Library of Babel [book review]

**T**his is a book I carried away from JSM in Boston as the Oxford University Press representative kindly provided my with a copy at the end of the meeting. 435 more words

#### Explicit Eichler-Shimura for GL2 over imaginary quadratic fields

Let denote the Hamilton quaternions. Set

Let be the point with , . There is a transitive action of given by with multiplication taking place in . 529 more words

#### An abstract ergodic theorem, and the Mackey-Zimmer theorem

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has… 2,316 more words