# This month in mathlib (Jan 2022)

January 2022 saw 533 commits to mathlib. In this post we highlight some of these contributions.

• Analysis.

• PR number 10000 was saved for something special: the Cauchy integral formula on circles is in mathlib! This is a major milestone, and unlocks complex analysis. In particular the same PR deduces from it that complex differentiable functions are analytic. Then PR #11686 proved Riemann's removable singularity theorem.
• In functional analysis, PR #11604 introduces the topology induced by a family of seminorms and PR #11491 proves the Lax-Milgram theorem.
• PR #11707 constructs the Pontryagin dual of a topological group.
• PR #11015 defines $ℓ^p$ spaces and PR #11255 proves a Hilbert space is isometrically isomorphic to $ℓ^2(ι)$ for some type $ι$.
• PR #11320 builds Fourier series as an isometric isomorphism from $L^2(ℂ)$ to $ℓ^2(ℤ, ℂ)$. This includes Parseval's identity.
• Probability and measure theory.

• PR #11328 proves Egorov's theorem.
• PR #11554 proves almost every point of a set in a Besicovitch space (for instance a finite dimensional real vector space) is a Lebesgue density point.
• PR #11007 proves one direction of the optional stopping theorem.
• Algebraic geometry.

• PR #9802 built the the Gamma-Spec adjunction.
• A construction for gluing schemes was added in PR #11431, which was used in PR #10605 to show that the category of schemes has fiber products.
• The function field of an integral scheme is defined in PR #11147.
• In topos theory, PR #11252 shows that sheafification is left exact, and PR #11273 provides the pushforward-pullback adjunction.
• Algebra.

• PR #11139 proves Taylor's formula for polynomials.
• PR #11346 shows that Witt vectors of a domain are a domain.
• PR #11166 computes the exponents of the dihedral and generalised quaternion groups, and PR #11512 shows that nilpotent groups are solvable.
• PR #11422 shows that central extensions of nilpotent Lie modules / algebras are nilpotent.
• PR #11635 adds the proof of the solution of the cubic (Theorem 37 of the 100 Theorems List).
• PR #10730 shows that the dual numbers are a Clifford algebra.
• Combinatorics.

• PR #11390 defines synthetic projective planes, PR #11550 shows that the order of a projective plane is at least 2, and PR #11462 proves the familiar formula for the cardinality of a projective plane in terms of the order (i.e., the number of points on a line, minus 1).
• PR #10497 defines Freiman homomorphisms.
• PR #11372 applies a double-counting arguments to the edges of a bipartite graph.
• PR #10861 proves the rearrangement inequality.
• Tactics. PR #11646 adds a new tactic, linear_combination, that computes a weighted sum of equality hypotheses (with coefficients given by the user) and attempts to use this to close the goal. This tactic is useful on its own, and can also be seen as a "certificate checker" for a future Gröbner basis tactic that finds these coefficients automatically.

## New maintainers

Three people joined the mathlib maintainer team:

• Riccardo Brasca from Université de Paris in France
• Frédéric Dupuis from Université de Montréal in Canada
• Kyle Miller from University of California, Santa Cruz in the USA.