The Proper Abstract

Mathematicians rarely have the opportunity to present their research directly to a broad scientific audience. One of the few venues to do so is publication in the Proceedings of the National Academy of Sciences (PNAS).

Published weekly and online, this prestigious journal features papers covering a wide range of disciplines, from applied mathematics and computer science to molecular biology and the social sciences. PNAS also has a strong media relations effort, which encourages wide dissemination of research results to the general public.

Unfortunately, even when mathematicians do take advantage of this opportunity, they often fail to communicate well and end up reinforcing stereotypes of mathematical research as a highly specialized pursuit of limited interest to all but a few.

Article abstracts are a crucial first step in garnering attention. Here’s the beginning of an abstract for a mathematics paper published some years ago in PNAS:

We introduce cohomology and deformation theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. . . .

Given the journal’s diverse audience, this abstract does nothing to suggest why someone other than, perhaps, a mathematician working in this specific area of mathematics ought to consider delving into the subject. It provides no context. It assumes knowledge that most readers would not have. And it goes on and on.

It’s a wasted opportunity, and the rest of paper gets even more technical. If this research is deemed of broad interest, then it should be worth the effort to present it in a way that reaches a wide audience.

It can be done. The following example illustrates how an abstract, also published in PNAS, can present an exciting mathematical finding concisely and in an appealing fashion.

Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ramanujan-type congruence.

Notice how the abstract establishes the historical and mathematical context of the finding without getting overly technical (and without the use of any symbols). Moreover, the careful choice of words such as “striking” to highlight noteworthy features of the finding adds to its appeal.

This abstract serves as a model of what a mathematical abstract should be like, not only in PNAS but also in the mathematical literature, whether for a paper or an oral presentation. For some informal advice to students on preparing abstracts, see the article “A Guide to Writing an Abstract” by Robert W. Vallin.

Mathematical articles can succeed in PNAS. The Cozzarelli Prize honors papers, selected from the 3,000 or so articles published annually in PNAS, that reflect scientific excellence and originality. The inaugural award in 2005 went to Karl Mahlburg, then a doctoral candidate in mathematics at the University of Wisconsin, Madison, for his paper “Partition Congruences and the Andrews-Garvan-Dyson Crank.”

Here’s the abstract:

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial proof of Ramanujan’s congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

The paper itself and an accompanying commentary by George E. Andrews and Ken Ono provided the context and the details.

Mathematicians should take publication in PNAS more seriously than they seem to, and they should make more of an effort to show the relevance of their research to other disciplines. Learning how to compose accessible, meaningful abstracts is one piece of the puzzle of spreading the word about mathematical achievement.—I. Peterson