Section 1.1: Challenges in Foliation Theory

Thurston's foray into foliation theory began during his graduate studies, where he quickly started proving and publishing significant theorems. However, as his work advanced, many mathematicians began to drift away from foliation theory and explore other domains.

Thurston noted that some mathematicians were advised to abandon this field because he was effectively proving all relevant theorems, leaving little for others. Yet, he contended that this was not the primary reason for their departure. Instead, he attributed it to the complexity of foliation theory and the pressure mathematicians felt to accumulate “theorem-credits”—the accolades associated with being the first to publish new proofs. Acquiring these credits is crucial in academic mathematics, as failing to do so can jeopardize job security and career advancement.

The convergence of these factors made pursuing research in foliation theory particularly daunting. The difficulty of understanding the research made it nearly impossible for others to establish their own theorems and attain the necessary theorem-credits for academic success. Hence, it was no surprise that many chose to transition to other mathematical fields.

This exodus was detrimental to the advancement of foliation theory itself. With a dwindling number of mathematicians engaged in this area, numerous intriguing questions remained unaddressed. Thurston suggested that this stagnation not only hindered progress in foliation theory but also impeded developments in other mathematical disciplines.

Section 1.2: Learning from Experience with the Geometrization Conjecture

Later in his career, Thurston introduced the geometrization conjecture and proved it in a specific case. However, he recognized that his approach to mathematics differed from many of his peers, noting, “some concepts that I use freely and naturally in my personal thinking are foreign to most mathematicians I talk to.” This disparity complicated his ability to convey his work in an accessible manner.

Worried that his experience with foliation theory would repeat itself, Thurston resolved to take proactive measures. His generosity became evident in two key ways. First, he dedicated significant time and effort to elucidate the fundamental concepts and tools that would enable other mathematicians to grasp his work. Second, despite being capable of proving numerous theorems, he opted to delay publication to allow others the opportunity to gain recognition for their contributions.

Thurston's actions exemplified generosity in both time and credit. Rather than focusing solely on his research, he invested considerable effort to ensure that his colleagues could understand the foundations of the geometrization conjecture. Furthermore, by withholding his own publications, he created space for others to receive acknowledgment for their discoveries.

This benevolence ultimately benefited both individual mathematicians and the field of mathematics overall. By equipping others with the tools to explore the geometrization conjecture, Thurston facilitated their ability to achieve theorem-credits for their findings—credits he might have claimed himself but chose to forgo.

The impact of Thurston's generosity was profound; it enabled many mathematicians to comprehend the subject, leading to their own discoveries. As Thurston reflected, the collaborative work that emerged from this environment fostered the creation of “mathematical concepts which are quite interesting in themselves, and lead to further mathematics.”