Photo by Bachrach
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
Email @tufts.edu: loring.tu
Algebraic geometry, topology, and differential geometry
Manifolds are the higher-dimensional analogues of curves and
surfaces. For more than twenty years I have been studying the
geometry and topology of manifolds.
In my younger days what attracted me to mathematics was its
universality, aesthetics, applicability, and certainty. It is
reassuring that the theorems discovered by the Greeks more than two
thousand years ago remain relevant and true today, in contrast to
the natural sciences where so many theories have been discarded over
the centuries. Today I think of mathematics as part of our
collective cultural heritage.
I'm particularly interested in the interplay between algebra and
geometry. Algebra makes precise what geometry can only intuit. Two
surfaces might look the same visually, but when described with
algebraic equations, their differences become apparent.
My current research focuses on localization theorems. What this
means is that often all the relevant information about a manifold is
encoded at a few special points, and I seek to discover situations
when this is true. For example, when the earth rotates, there are
two special points, the north and south poles, which remain fixed.
It turns out that the topology of the earth can be gleaned from
information coded at these two fixed points.
Higher-dimensional geometry is fundamental to our understanding of
the universe. After several centuries of religious and scientific
controversies, we all know now that the earth is round, but is the
universe flat or curved? That is still an open question. How many
dimensions has the universe? According to the latest physical
theory, the string theory, the universe might have many more than
four dimensions. What is the shape of the universe as a geometric
object? The mathematics behind these tantalizing questions continues
to serve as a beacon for my research.