# This month in mathlib (Sep 2021)

This post summarizes some of the activity that happened in mathlib in September.

## Highlighted PRs

• Number fields. The Dedekind saga continues with PR #8847, PR #8701 (define number fields, rings of integers), PR #8949, PR #8964, and PR #9063, culminating in PR #9059 which shows that, in the presence of an admissible absolute value, the class group of an integral closure is finite.
• Probability theory. Foundations of probability theory are (finally!) coming to mathlib. With PR #8939 and PR #8920 (conditional expectation of an indicator) the way was open for PR #9114 which defines the conditional expectation of a function.
• Algebraic closures. Over two years ago, PR #1297 was opened, showing that algebraic closures are unique (up to non-unique isomorphism). Due to various issues, the material was not yet in shape for inclusion in mathlib. Over summer, the material got a facelift, leading to PR #9110, PR #9231, and PR #9376. There probably are no other commits that had to wait so long before being merged into mathlib.
• Riesz theorem on compact unit ball and finite dimension. Mathlib now knows the main difference between the topology of real normed spaces in finite and infinite dimensions: the closed unit ball in such a space is compact if and only if the space is finite dimensional. The implication from finite dimension to compactness was proved in PR #1687 and PR #9147 proves the converse.
• Measure theory. PR #9325 shows that any additive Haar measure on a finite-dimensional real vector space is rescaled by a linear map through its determinant, and computes the measure of balls and spheres. PR #9065 adds Radon-Nikodym and Lebesgue decomposition for signed measures, see this blogpost for more details.
• Henselian local rings. PR #8986 sets up the theory of Henselian rings. A ring $R$ is Henselian at an ideal $I$ if the following conditions hold:
• $I$ is contained in the Jacobson radical of $R$
• for every polynomial $f$ over $R$, with a simple root $a₀$ over the quotient ring $R/I$, there exists a lift $a ∈ R$ of $a₀$ that is a root of $f$.
• Semilinear maps. Several people undertook a massive refactor to generalize linear maps to semilinear maps. This opens the door for the use of semilinear maps in functional analysis, but also Frobenius semilinear maps in characteristic $p > 0$. PR #8857 introduces notation for linear_map.comp and linear_equiv.trans which makes it easy to work with identity-semilinear maps (aka, linear maps) without getting distracted by the redundant semilinearity conditions. With this notation in place, PR #9272 performs the actual redefinition of linear_map and linear_equiv to be semilinear.
• Convexity refactor. Another ongoing refactor aims to massively generalize and restructure material on convex sets/functions. One of the gems in a long stream of PRs shows that it is now trivial to deduce concave results from their convex counterparts: PR #9356 generalizes lemmas about convexity/concavity of functions, and proves concave Jensen.

## Other mathematical contributions

The following PRs are ordered by the date that they were merged into mathlib.

• PR #8894: add topological localization
• PR #8962: Prove (co)reflectivity for Kan extensions
• PR #8801: class formula, Burnside's lemma
• PR #8947: Define homotopy between functions
• PR #8946: rigid (autonomous) monoidal categories
• PR #8579: one-point compactification of a topological space (the Stone-Cech compactification had been around for ages, and now its little brother joined the gang)
• PR #9165: upgrade to Lean 3.33.0c
• PR #9288: Sylow's theorems for infinite groups