What is a good outline of a mathematical article?

Outlines describe the content of an article and serve as a guide. They are similar to a table of contents, but can be much more expressive.

We consider publications with the highest standards for publication, including “Bulletin of the AMS” and “Mathematical Association of America”. From all publications, we look into those awarded a prize for best publication (Levi L. Conant Prize and Paul R. Halmos - Lester R. Ford Awards). How are the outlines in these publications?

Not all publications have a clear outline: some introduce the topic and simply continue with each section of the article never giving a clear overview.
Of those with an outline, they have widely different lengths: from 4 sentences to a full subsection.
Dependency between sections is key.
Not all sections must be mentioned. Overall: there is no best style, develop your own. Read the examples and arrive to your own conclusions.

There is an outline model where you go section by section describing their content. For example, “Section 2 presents the model. Section 3 states the theorems. Section 4 contains the proofs.”. Althought this fills the role of an outline, the authors miss the chance to give more insight, for example about the dependency between sections.

Examples

DOI: 10.1080/00029890.2021.1933834

Because a triple’s ratio pair determines the type of packing, the picture answer developed in this article will be a plot of the unit square of ratio pairs, indicating which ratio pairs correspond to which of the four types of packings in Figure 2. Presented here are two complementary approaches to the development of the plot, which is displayed in Figure 6. The first approach —presented in Sections 3 and 4— is the more constructive one, requires no previous background, and suggests new avenues for exploration. The second approach—outlined in Section 5—comes from the viewpoint of projective geometry, and provides an understanding of the general structure of the picture answer. It is possible to skip Section 5, jumping straight into the exploration in Section 6, which takes a closer look at the result, and gives ideas for further exploration. Prior to both approaches, Section 2 provides relevant background.
DOI: 10.1080/00029890.2020.1690387

Outline of the article. The rest of this article is organized as follows. In Sections 2–4, we present the numerical data alluded to above, we formalize several notions of “surprising” accuracy, and we pose three questions suggested by the numerical observations that will serve as guideposts for our investigations. The remainder of the article is devoted to unraveling the mysteries behind the numerical observations and uncovering, to the extent possible, the underlying general phenomena. We proceed in three stages, corresponding to three different levels of sophistication in terms of the mathematical tools used. The three stages are largely independent of each other, and they can be read independently. In the first stage, consisting of Sections 5 and 6, we use an entirely elementary approach to settle the mystery in a particularly interesting special case. In the second stage, presented in Sections 7–9, we draw on results by Ostrowski and Kesten from the mid-20th century to obtain a general solution to the mystery in the “bounded Benford error” case. In the third stage, contained in Section 10, we bring recent groundbreaking and deep work of J´ozsef Beck to bear on the remaining—and most difficult—case, that of an “unbounded Benford error,” and we present the surprising denouement of the mystery in this case. The final section, Section 11, contains some concluding remarks on extensions and generalizations of these results and related results.
DOI: 10.1080/00029890.2020.1785253

The structure of the article is as follows. In Section 2, we assemble a history (far from complete) of the theorem on circles that opened this article and provided Price with his inspiration. In this section we also distinguish five steps in the proof above, which are used both to speak precisely about the historical development, and to establish the framework in which the ellipse problem will be solved. In Section 3, we carry the five steps of the circle proof over to the ellipse, reducing the problem to the evaluation of the derivative of a certain polynomial \Omega_n (z) at a certain real number a + b. Then we explicate Price’s own proof of his beautiful formula, i.e., his evaluation of \Omega’_n(a + b), and outline the rest of the article. Sections 4–7 present our reorganized evaluation of \Omega’_n (a + b). Each of these sections presents one step in the argument near the beginning—Propositions 1–4, respectively—and then goes on to discuss connections with known theory, motivation, and relations with Price’s argument. Section 8 develops an additional connection revealed by Sections 4–7. A more detailed outline of the contents of Sections 4–8 is given at the end of Section 3, after the framework has been set up.
DOI: 10.1090/bull/1552

Lyapunov exponents make multiple appearances in the analysis of dynamical systems. After defining basic concepts and explaining examples in Section 1, we describe in Sections 2–4 a sampling of Avila’s results in smooth ergodic theory, Teichmüller theory and spectral theory, all of them tied to Lyapunov exponents in a fundamental way. We explore some commonalities of these results in Section 5. Section 6 is devoted to a discussion of some themes that arise in connection with Lyapunov exponents.
DOI: 10.1090/S0273-0979-2014-01471-5

Mathematics aids the pursuit. The carbon cycle is an unwieldy beast, but when one strips away insignificant complications, manifestations of elementary mathematical concepts emerge. Our focus here is phenomenological. We feature observational data and its mathematical interpretation, with the objective of providing targets for advancing theoretical understanding. Two types of problems receive special attention. Section 2 is devoted to the problem of decomposition: the processes by which organic matter is converted to CO2. We illustrate ways in which the heterogeneity of the problem can be understood, and we feature scaling laws that appear to result from this heterogeneity. We then discuss problems of dynamics (Section 3), using historical records of past changes to illustrate the scope of the problem. We conclude with an appraisal of the lessons learned and the challenges ahead.
DOI: 10.1090/S0273-0979-2013-01402-2

1.1. Outline. We begin in §2 with Zaremba’s Conjecture. We will explain how this problem arose naturally in the study of “good lattice points” for quasi–Monte Carlo methods in multi-dimensional numerical integration, and how it also has applications to the linear congruential method for pseudo-random number generators. But the assertion of the conjecture is a statement about continued fraction expansions of rational numbers, and as such is so elementary that Euclid himself could have posed it. We will discuss recent progress by Bourgain and the author, proving a density version of the conjecture.

We change our focus in §3 to the ancient geometer Apollonius of Perga. As we will explain, his straight-edge and compass construction of tangent circles, when iterated ad infinitum, gives rise to a beautiful fractal circle packing in the plane, such as that shown in Figure 1. Recall that the curvature of a circle is just one over its radius. For special configurations, all the curvatures of circles in the given packing turn out to be integers; these are the numbers shown in Figure 1. In §3 we will present progress on the problem, which integers appear? It was recently proved by Bourgain and the author that almost every admissible number appears.

In §4, also stemming from Greek mathematics, we describe a local-global problem for a thin orbit of Pythagorean triples, as will be defined there. This problem is a variant of the so-called Affine Sieve, recently introduced by Bourgain, Gamburd, and Sarnak. We will explain an “almost” local-global theorem in this context due to Bourgain and the author.

Finally, these three problems are reformulated to the aforementioned common umbrella in §5, where some of the ingredients of the proofs are sketched. The problems do not naturally fit in an established area of research, having no L-functions or Hecke theory (though they are unquestionably problems about whole numbers), not being part of the Langlands Program (though involving automorphic forms and representations), nor falling under the purview of the classical circle method or sieve, which attempt to solve equations or produce primes in polynomials (here it is not polynomials that generate points, but the aforementioned matrix actions). Instead the proofs borrow bits and pieces from these fields and others, the major tools including analysis (the circle method, exponential sum bounds, infinite volume spectral theory), algebra (strong approximation, Zariski density, spin and orthogonal groups associated to quadratic forms, representation theory), geometry (hyperbolic manifolds, circle packings, Diophantine approximation), combinatorics (sum-product, expander graphs, spectral gaps), and dynamics (ergodic theory, mixing rates, the thermodynamic formalism). We aim to highlight some of these ingredients throughout.

Lastly, while I was writing this article, I was preparing an article for the journal “Mathematics of Operations Research” One of its most read article has the following outline.

DOI: 10.1287/moor.2023.1354

This paper is organized as follows. Section 2 presents the model of financial networks, the various forms of monotonicity of division rules, and the definition of a clearing payment matrix. Section 3 is devoted to the lattice structure of the set of clearing payment matrices. Section 4 presents an example to show that under priority division rules, multiplicity of payment matrices can occur even when all endowments are strictly positive. This section also presents the main result of the paper: the sufficient condition for clearing payment matrices to be unique. Section 5 discusses the relation to other conditions for uniqueness that are found in the literature. We examine the connection between the uniqueness of clearing payment matrices and continuity of bankruptcy rules in Section 6. Finally, Section 7 presents the conclusion. All proofs except those related to Section 5 are relegated to Section 7.