Last Wednesday, Bancroft Avenue was abuzz with mathematicians. Dozens of PhD students and professors crowded around the Earth Sciences Auditorium to attend a public lecture presented by world-renowned Fields Medalist, Professor Shing-Tung Yau.
The Fields Medal, often described as the Nobel Prize for Mathematical sciences, has earned Prof Shing-Tung Yau a great deal of respect within the academic community. One would think that having such a display of scholars in attendance would suggest an esoteric and elite discussion of mathematics. However, these attitudes are precisely what Professor Shing-Tung Yau’s new book, The Shape of Inner Space, sets out to prevent.
Yau explained in his lecture that mathematics does not need to be a strictly abstract discipline, contrary to popular characterizations of the field. Instead, it plays a central role in the understanding of our immediate universe. The Shape of Inner Space attempts to portray such ideas in a manner accessible to any general, yet curious, reader.
Yau is famous for his 1976 proof of the Calabi conjecture, a proposition offered by Eugenio Calabi over twenty years earlier, in 1954. Along with much of his additional work, this proof has provided a framework for attempts at unifying various theories concerning our physical world.
However, Yau explains that although his work has applications to string theory and theoretical physics, these applications were not something he directly intended. As a mathematician specializing in geometry, he is interested in mathematically describing a diverse collection of shapes and objects, an interest which took him to the study of “curvatures.” These curvatures are coincidentally what make up the geometry of physical space.
Yau admits that, although his studies in geometry were not wholly motivated by their relationship with the physical universe, he is nonetheless pleased and inspired by how his work was, and is, able to answer many fundamental questions in physics. He describes mathematics as a beautiful field of study, where discoveries or propositions have the potential to account for many different implications and results that extend beyond the original motivations.
In Yau’s opinion, major advances in physics can be partially credited to advances in geometry — in particular, classical geometry and calculus work behind classical physics, and Riemann geometry work behind Einstein’s theories of physics. Yau’s findings led him and his colleagues to open up a new avenue of geometry, an avenue known as “geometric analysis.” In response to a question from the audience, he also explained that the future will bring about a new kind of geometry — one that will be able to account for many theories in quantum physics.
The lecture also aimed to provide its audience with a sense of his individual experiences with mathematics, and how his investigations brought him through a personal journey spanning much of his career.
Yau recalled his first encounter with curiosity in geometry. He explained that with all the geometry and mathematics he had learned throughout high school, he still could not describe an object so common and seemingly simple as an apple. From there, and throughout college, Yau had a deep interest in being able to describe objects mathematically — objects common to everyday experiences, as well as abstract idealized objects.
Yau personally adopts an imaginative approach. He explains that his investigations have lead him to ponder the possibilities of most “extreme cases,” — that is, he wonders how a certain object would be affected if some property were ignored, or reduced to null.
Yau also explains that a geometer who relates his work to the realm of physics is much like a painter or artist who relates his work to reality. When a painter sets out to paint a landscape, he is first guided by his imagination. Only when he is satisfied does he look back to see how much of it is physically accurate.
