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This month in mathlib (Oct 2021)

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

Highlighted PRs

  • In algebraic geometry, there has been a surge in activity. After the definition of schemes entered mathlib, the development stalled for a while, because the Liquid Tensor Experiment took up a lot of energy and attention. But new contributors showed up, and are pushing the library forward. Two highlights this month include:
    • PR #9416 Global sections of the spectrum of a ring are isomorphic to that ring.
    • PR #9861 Closed points in the prime spectrum correspond to maximal ideals.
  • In commutative algebra:

    • algebraic indepence and transcendence are introduced in PR #9229 and PR #9377 proved the existence of transcendence bases. See the stacks project for an informal account.
    • PR #9817 and PR #9834 fills a glaring hole in the field theory library: finite fields of the same cardinality are isomorphic.
  • In group theory:

    • PR #8976 proves the Jordan Hölder theorem. It also proves a version for submodules.
    • PR #9662 proves Frattini's Argument: If $N$ is a normal subgroup of $G$, and $P$ is a Sylow $p$-subgroup of $N$, then $\text{N}_G(P)*N=G$.
  • In number theory, PR #9646 and PR #9702 prove that Liouville numbers form a dense $G_δ$ set which has measure zero.
  • In combinatorics, PR #7825 proves a generalized version of Hall's marriage theorem using a compactness argument. Using the language of category theory, there is an inverse system of solutions to the marriage problem when restricted to finite subsets of the domain, and the limit of this kind of inverse system is nonempty.
  • In measure theory, the series of PRs PR #9462, PR #9461, PR #9576, PR #9679 and PR #9680 prove the Vitali and Besicovitch covering theorems (with the optimal constant for the Besicovitch covering theorem, following Füredi and Loeb, On the best constant for the Besicovitch covering theorem). These theorems ensure that, from a covering of a (not necessarily measurable) set $s$ by nice enough sets (e.g. balls), one can extract a disjoint subfamily covering almost all $s$. They are instrumental in forthcoming results on differentiation of measures.
  • In probability:
    • PR #9323 defines probability density functions,
    • PR #9378 proves that conditional expectation on real functions equal Radon-Nikodym derivative
    • PR #9469 brought us notations that really demonstrate we're now doing probability in mathlib!
  • In algebraic topology, PR #9252 and PR #9141 set up some basic homotopy theory.
  • In topological algebra, some foundational parts of the perfectoid project were moved to mathlib after a long time of inactivity:

    • PR #9521 defines the adic topology associated to an ideal
    • PR #9589 defines the topology associated to a valuation on a ring and proves its fundamental properties. In particular a valuation on a field 𝕂 is extended to the completion of 𝕂 with respect to the valuation topology.
  • In differential calculus,

    • PR #9496 and PR #9811 prove the divergence theorem on a box, for a Henstock-style integral and the Bochner integral respectively. This is a foundational result that will be used in complex analysis and differential topology.
    • PR #9440 defines superpolynomial decay of a function
  • In functional analysis:

    • PR #9540 proves the open mapping theorem for maps between affine spaces
    • PR #9924 states the Riesz representation theorem uniformly over $ℝ$ and $ℂ$. In 2020, Frédéric Dupuis proved this theorem over both $ℝ$ and $ℂ$ (see PR #4379), but, since mathlib lacked conjugate-linear maps at that time, the $ℂ$-version was stated in a rather awkward way. Conjugate-linear maps were added to mathlib last month and now the $ℝ$ and $ℂ$ versions of the theorem can be stated uniformly.
  • The core version of Lean was bumped twice in the past month. Most of the changes are in preparation for porting mathlib to Lean 4.