Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups

In a compelling exploration of projective differential geometry, two prominent mathematicians, V. Ovsienko and S. Tabachnikov, delve into the intricate connections between classical concepts and modern mathematical practices. They guide the reader through the nuances of the Schwarzian derivative, unraveling its significance in the study of differential structures and their projective properties. The authors' deep understanding of geometry comes alive as they weave historical insights with contemporary developments, revealing the evolution of these ideas over time.

Readers are introduced to a wealth of mathematical tools and concepts that illuminate the interactions among geometry, algebra, and mathematical physics. Through rigorous yet accessible explanations, Ovsienko and Tabachnikov highlight the vital role that the cohomology of diffeomorphism groups plays in understanding the underpinnings of projective geometry. Their approach encourages a fresh perspective on traditional geometrical themes, fostering a rich dialogue between old and new methodologies.

The authors also emphasize the importance of projective differential geometry in broader mathematical contexts, showcasing its relevance in fields such as topology and physics. Their thoughtful exposition aims to inspire both established mathematicians and newcomers, offering insights that bridge various areas of study. Each chapter meticulously builds on the previous one, making complex ideas more digestible and engaging.

Overall, this work serves as a significant contribution to the field, blending historical context with modern mathematical advancements. Ovsienko and Tabachnikov's expertise and passionate narrative invite readers into the depths of projective differential geometry, promising not only knowledge but also inspiration for future exploration and discovery.