both the number of fields that now have overlaps with the mathematical sciences and the number of opportunities in each area of overlap.

The second of these trends was discussed in Chapter 3. This section elaborates on the trend toward greater connectivity on the part of the mathematical sciences themselves.

Internally driven research within the mathematical sciences is showing an increasing amount of collaboration and research that involves two or more fields within the discipline. Some of the most exciting advances have built on fields of study—for example, probability and combinatorics—that had not often been brought together in the past. This change is nontrivial because large bodies of knowledge must be internalized by the investigator(s).

The increased interconnectivity of the mathematical sciences community has led, as one would expect, to an increase in joint work. To cite just one statistic, the average number of authors per paper in the Annals of Mathematics has risen steadily, from 1.2 in the 1960s to 1.8 in the 2000s. While this increase is modest compared to the multiauthor traditions in many fields, it is significant because it shows that the core mathematics covered by this leading journal is trending away from the solitary researcher model that is embedded in the folklore of mathematics. It also suggests what has long been the experience of leading mathematicians: that the various subfields in mathematics depend on one another in ways that are unpredictable but almost inevitable, and so more individuals need to collaborate in order to bring all the necessary expertise to bear on today’s problems.

While some collaborative work involves mathematical scientists of similar backgrounds joining forces to attack a problem of common interest, in other cases the collaborators bring complementary backgrounds. In such cases, the increased collaboration in recent years has led to greater cross-fertilization of fields—to ideas from one field being used in another to make significant advances. A few recent examples of this interplay among fields are given here. This list is certainly not exhaustive, but it indicates the vitality of cross-disciplinary work and its importance in modern mathematical sciences.

Example 1: Cross-Fertilization in Core Mathematics

Recent years have seen major striking examples of ideas and results from one field of core mathematics imported to establish important results in other fields. An example is the resolution of the Poincaré conjecture, the most famous problem in topology, by using ideas from geometry and analysis along with results about a class of abstract metric spaces. As another



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