The main piece of context is the local Langlands correspondence, a part of the Langlands program.
The Langlands program is fundamentally about roots of polynomials (in 1 variable) with coefficients in the fields which appear in number theory: number fields and p-adic fields.
Oct 31, 2023 • 8 tweets • 2 min read
Like the four colour theorem, the non-existence of a finite projective plane of order 10 is a nice example of a computer assisted proof which still does not have a human-readable counterpart. A modern implementation took (in 2020) ~24 months of compute on a desktop GPU:
arxiv.org/abs/2012.04715
Jan 27, 2022 • 24 tweets • 4 min read
Perverse sheaves are hard to visualise and this is (one reason) why they are awesome: a thread
Perverse sheaves are:
- hard to motivate and visualise
- extremely useful in algebraic geometry, arithmetic geometry and representation theory
This is a frustrating state of affairs!
Dec 13, 2021 • 11 tweets • 3 min read
While preparing some exercises for commutative algebra I read up on a funny class of rings: absolutely flat rings. As the name suggests, a (commutative, unital) ring R is absolutely flat if every R-module is flat. This is a strong condition with surprisingly many reformulations:
TFAE: 1) R is absolutely flat. 2) R is reduced of (Krull) dimension 0 (probably the clearest definition). 3) Every localisation of R at a prime ideal is a field. 4) Every localisation of R at a maximal ideal is a field.
Jan 26, 2021 • 39 tweets • 6 min read
Thread: one of my favourite circle of ideas in mathematics is Grothendieck's "six operation formalism", which roughly states that the cohomology of geometric objects (manifolds, algebraic varieties...) fits into a "sheaf-theoretic, functorial picture with Poincaré duality".
The prerequisites for this thread are a little steep, but you will hopefully get something out of it if you know about singular (co)homology, Poincaré duality for manifolds, homological algebra (including derived categories) and some sheaf theory.
Aug 10, 2020 • 24 tweets • 4 min read
Compact groups, unitary representations, the Peter-Weyl theorem and Tannaka-Krein duality:
I have just supervised a bachelor thesis on those topics, and I thought I'd write a few threads while it is fresh in my mind. First stop: compact groups. (1/n)
A compact group is a group which is compact ;-) .
For this to make sense, we'd better start with a *topological group*: a group G equipped with a topology such that the group law and the inverse map are continuous. Then we ask that G, as a topological space, is compact. (2/n)
Jan 31, 2020 • 19 tweets • 3 min read
People first encountering algebraic geometry are often disconcerted. What is all the fuss about? Systems of polynomial equations? XIXth century theorems about 27 lines?
That does not seem to jive with algebraic geometry being a core subfield of pure mathematics.
[1/n]
Here is one of the not-so-secret but under-emphatized reasons why algebraic geometry is great:
The mathematical world is surprisingly algebraic.
As in: there are many interesting mathematical objects which are non-obviously connected to algebraic geometry.
