Future Directions for Research in Geometry:
A Summary of the Special Discussion Session
at the 2005 Midwest Geometry Conference
BY LAWRENCE J. PETERSON
The 2005 Midwest Geometry Conference took place at the Ohio State
University from April 29 to May 1, 2005. The conference included a
special session on the future directions of geometry. During this
session, the conference participants discussed some of the areas
within the field of geometry in which significant research activity is
likely to occur over the next few years. These notes summarize the
discussion during the special session.
One of the first topics that came up at the future directions session
was the connection between geometry on the one hand and particle
physics, gauge theory, Kaluza-Klein theory, and string theory on
other. As in the past, these connections are expected to generate a
significant amount of research activity in geometry in the coming
Participants at the special discussion session mentioned other areas
of geometry as well as some of their applications and their
connections with other fields within science and mathematics.
Research in geometry will in all likelihood continue to have
connections with research in ordinary and partial differential
equations. Participants mentioned Poisson geometry and noted that
this field is expected to have applications to plasma systems. In the
coming years, as in the past, connections between geometry, knot
theory, and the study of DNA molecules are expected to stimulate
research in geometry. Morse theory has been applied in the study of
computer imaging, and participants at the discussion session felt that
these applications were likely to continue.
Some participants at the discussion session indicated that they study
geometry for its own sake, without regard for its applications. And
yet geometry does indeed have many practical applications.
Participants noted that applications of geometric theory may occur
many years after the research that originally produced it. To find
examples of such geometric theory, one need only consider the field of
Riemannian geometry. Riemannian geometry served as a tool in the
development of Einstein's general theory of relativity, over fifty
years after the discovery of Riemannian geometry itself. It is also
worth noting that certain applications of Albert Einstein's work
occurred many years after the work itself. For example, the
construction of nuclear power plants and atomic bombs depended in part
on Einstein's idea of mass-energy equivalence, an idea that Einstein
formulated over thirty years prior to the applications themselves.
These examples illustrate the long-term value of continued basic
research in geometry and related fields.
Participants at the special session on future directions of geometry
also discussed graduate students and the question of suitable problems
for them to work on. The following geometry-related areas contain
many problems suitable for graduate students: symplectic geometry,
Poisson geometry, analysis on manifolds, heat asymptotics, quantum
gravity, conformal geometry, noncommutative geometry, and knot theory.
Success in working on many of the open problems in these fields will
require the student to have an extensive amount of background
knowledge in a wide variety of mathematical areas. For this reason,
geometry will continue to attract talented people who seek challenging
opportunities. On the other hand, participants also pointed out that
there are also open problems in geometry which are suitable for
graduate students with a more moderate level of experience and
background knowledge; one participant noted that certain problems in
noncommutative geometry are of this nature.
To summarize the discussion at the special session on future
directions, it suffices to say that geometry is a broad field with
many opportunities for research and many connections with other
disciplines. Geometry will likely continue to attract many graduate
students and to provide a wide range of benefits to society over the
This material is based upon work supported by the National Science Foundation
under Grant No. DMS-0509714 and by the Ohio State University. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the author and do not necessarily reflect the views of the National
Science Foundation or the Ohio State University. The author wishes to thank
Andrzej Derdzinski and William Ugalde for private communications during and
after the 2005 Midwest Geometry Conference.