Has anyone developed a 'theoretical minimum' for mathematics?

In 1991, Vladimir I. Arnold compiled a list of 100 mathematical problems in a paper titled "A mathematical trivium". (You can view all 100 problems in that pdf.)

In that paper he says:

The compilation of model problems is a laborious job, but I think it must be done. As an attempt I give below a list of one hundred problems forming a mathematical minimum for a physics student. Model problems (unlike syllabuses) are not uniquely defined, and many will probably not agree with me. Nonetheless I assume that it is necessary to begin to determine mathematical standards by means of written examinations and model problems. It is to be hoped that in the future students will receive model problems for each course at the beginning of each semester, and oral examinations for which the students cram by heart will become a thing of the past.

Although the problems are aimed at physics students, most of the problems are purely mathematical.

I don't know how famous these problems are, but there is an entire subforum on the Art of Problem Solving (AoPS) Fourms dedicated to these problems.

There is Garrity's All the Mathematics You Missed: But Need to Know for Graduate School, published in 2001. The author states in the preface:

The goal of this book is to give people at least a rough idea of the many topics that beginning graduate students at the best graduate schools are assumed to know.

Topics covered (rapidly) in the book include linear algebra, real analysis basics, vector calculus, point-set topology, Stokes' Theorem (classical and differential forms), curvature, geometry, complex analysis, countability, the Axiom of Choice, abstract algebra, Lebesgue integration, Fourier analysis, differential equations (ODEs and PDEs), combinatorics and probability, and algorithms.

At the same time that Lev Landau (and Yevgeny Lifshitz) were publishing the volumes of the course of theoretical physics, the Russian mathematician Vladimir Smirnov published the six volumes of his Course in Higher Mathematics. I think that, for those years, these volumes can be considered a good starting knowledge for a mathematician, and that even today are readable and interesting (if you can find the volumes, here an old French edition).

This edition is in four parts and six volumes. The first part is essentially about differential and integral calculus in one and two variables. The second part is more varied; it contains chapters on ordinary differential equations, multiple integrals, vector analysis and differential geometry, Fourier series and an introduction to partial differential equations. The third part (two volumes) is divided into two volumes: the first about linear algebra and group representations, the second about complex analysis and some special functions. The fourth part has a first volume about integral equations and the calculus of variations, and a second volume on partial differential equations.

Clearly it is an old setting of the matter, with an exposition strongly oriented to applications, expecially to physics. And certainly it is the result of a complex cultural phenomenon, related to the Cold War and the technological and scientific confrontation with the Capitalist World.

You can consider that, in the same years, in Europe, there was also the Bourbaki's ''school'' that published a series of volumes on the fundamentals of mathematics, with a completely different and more abstract approach. The ten volumes of the ''Éléments de mathématique'' published by this group of mathematicians was certainly very important and influenced many modern branches of mathematics, but the volumes are not didactical and really difficult to read.