# This month in mathlib (May 2022)

In May 2022 there were 606 PRs merged into mathlib. We list some of the highlights below.

• Model theory

• PR #13980 proves that any theory with infinite models has arbitrarily large models
• PR #13723 and PR #13982 prove the Löwenheim–Skolem theorem
• Category theory and homological algebra

• PR #13882 proves that simple objects are indecomposable
• PR #13707 proves that a left rigid category is monoidal closed and that functors from a groupoid to a rigid category is again a rigid category.
• Homology is still making progress, both in concrete setups such as PR #12622 which develops the API for complexes of modules, and in abelian categories with PR #14009 bringing short exact sequences from the Liquid Tensor Experiment.
• In algebraic topology, PR #14304 defines the nerve of a category as a simplicial set.
• Algebra

• PR #13414 starts the study of torsion ideals. This is a step towards the long awaited classification of finite type modules over principal ideal domains, stay tuned!
• PR #13784 starts the formalization of Laurent Polynomials. Many more PRs followed or are on their way.
• The work to remove unneeded commutativity assumptions continued, in particular in PR #13966 and PR #13865.
• Matrices also got some work, both on the algebraic side with PR #13845 proving that invertible matrices over a ring with invariant basis number are square, and the analytical side with PR #13520 defining the matrix exponential. In order to make this easier to work with, mathlib now has two different normed algebra structures on matrices, the $L^\infty$ operator norm from PR #13518 and the frobenius norm from PR #13497.
• In representation theory, PR #13685 proves the category of Rep k G of representations of a group G on k-vector spaces is symmetric monoidal, and PR #13782 proves it is linear. PR #13740 introduces the category of finite dimensional representations.
• Number theory

• The general theory of Dedekind domains progressed with PR #13067 about the Chinese remainder theorem and PR #13462 which extends the v-adic valuation on a Dedekind domain to its field of fractions K and defines the completion of K with respect to this valuation, as well as its ring of integers, and provides some topological instances.
• On the concrete examples side, PR #13585 computes the ring of integers of a $p^ n$-th cyclotomic extension of $ℚ$.
• Functional analysis and geometry:

• PR #9862 finished the Banach-Alaoglu theorem.
• A long series of PRs culminated in PR #7288 proving several versions of the geometric Hahn Banach theorem.
• Also related to convexity, there had been progress on locally convex topological vector spaces, including PR #13547 characterizing the topology induced by seminorms in terms of neighborhoods of zero, and the introduction of uniformly convex normed spaces in PR #13480.
• The theory of vector bundles over topological spaces progressed with PR #14361 and PR #8545 building the pullback of a vector bundle under a continuous map.
• Integration theory and calculus

• PR #13540 defines the convolution of two functions. It proves that when one of the functions has compact support and is $C^n$ and the other function is locally integrable, the convolution is $C^n$ and its total derivative can be expressed as a convolution (this requires to generalize the usual notion of convolution which would be enough only for partial derivatives). This PR also proves that when taking the convolution with functions that "tend to the Dirac delta function", the convolution tends to the original function. This all comes from the sphere eversion project.
• PR #13179 proves comparison lemmas between finite sums and integrals.
• PR #14129 ensure that the asymptotic relations is_o/is_O work in calc blocks.
• Complex analysis

• PR #13178 proves the Phragmen-Lindelöf principle for some shapes in the complex plane.
• PR #13000 continues the study of the $\Gamma$ function, proving it is complex analytic (away from its poles at non-positive integrers of course).
• PR #12892 introduces torus integrals, paving the way towards the higher-dimensional Cauchy formula.
• Probability theory

• In elementary probabilities, PR #13484 use the counting measure to reformulate earliers contributions.
• PR #13630 proves the optional stopping theorem from martingale theory.
• After a long series of PRs including PR #13912 which defines the variance of a random variable, and proves its basic properties, and PR #14024 which defines identically distributed random variables, we finally got PR #13690 which proves the strong law of large numbers!

Note also that, with PR #14237 we finally completed the long quest to provide module docstring for all math files in mathlib.