Peter Ozsváth, Columbia professor of mathematics, and Zoltán Szabó, of Princeton University, have been awarded the prestigious American Mathematical Society 2007 Oswald Veblen Prize in Geometry.
Granted every three years, the Veblen Prize is one of the field's highest honors for work in geometry or topology, the study of the intrinsic properties of spaces. The prize ceremony took place Jan. 6 at the Joint Mathematics Meetings in New Orleans.
Ozsváth and Szabó were honored for their contributions to 3- and 4-dimensional topology (also known as low-dimensional topology), through a theory they developed called Heegaard Floer homology. This geometric tool is used for studying the properties of low-dimensional spaces. The pioneering mathematicians developed these new techniques in a highly influential series of more than 20 papers in the last five years.
A 3-dimensional space is one on which particles are constrained to move in three degrees of freedom, "such as the space in which we live," Ozsváth explained. A 4-dimensional space is one in which there are four degrees of freedom, such as space and time.
Topologists consider spaces with arbitrarily many dimensions, or degrees of freedom, but ironically, 3- and 4-dimensional spaces have proven to be the most difficult to understand.
"My goal is to understand spaces from a mathematical point-of-view, not just from what we have learned through equations from the field of physics," Ozsváth said, adding, "We want to be able to illustrate the content of the physical theories with concrete geometrical objects."
One very concrete manifestation of 3-dimensional topology is "knot theory," which deals with the easily stated but often deceptively difficult questions about knotted strings in 3-dimensional space. This subject dating to the early 19th century, is a standard benchmark for progress in three-manifold topology.
"One can envision Knot 8-10 (pictured) being unknotted after two unknotting operations," Ozsváth said, "and it seemed quite plausible that it could not be unknotted in only one. But the techniques available before Heegaard Floer homology were unable to demonstrate this point." As a byproduct of their general theory, Ozsváth and Szabó showed that the knot cannot be unknotted in one step, in a paper from 2003.
New findings in knot theory are already starting to find their way into mainstream science. For example, techniques from this subject are used in biology, where scientists continue to try to unravel the structure of DNA, which, in order to fit inside a small amount of space, is knotted.
"Mathematics is a is a vibrant and active field, driven by pure curiosity, and constrained only by intellectual rigor," Ozsváth said. "Practical applications of a given subject often appear long after the discoveries are made, but nonetheless often prove to be very fruitful."
Another pair of mathematical collaborators, Peter Kronheimer of Harvard University and Tomasz Mrowka of the Massachusetts Institute of Technology, also received the 2007 Veblen honor "for their joint contributions to both 3- and 4-dimensional topology through the development of deep analytical techniques and applications," according to the prize citation.
Peter Ozsvath, at the Department of Mathematics