Index Theorem Wins Isadore AwardBy Gireeja V. Ranade
ASSOCIATE NEWS EDITOR
After half an hour of vigorously explaining just the statement of the index theorem on blackboard, Professor of Mathematics Richard B. Melrose asked with a flourish, “Now doesn’t that make you want to be a mathematician?” Obviously, the answer was yes!
The index theorem, which brings together topology, geometry and analysis was discovered and proved by MIT Institute Professor Isadore M. Singer and Sir Michael Francis Atiyah of the University of Edinburgh, who on Thursday were jointly awarded the Abel Prize for 2004 by the Norwegian Academy of Science and Letters.
“One of the things we have learned from the index theorem is the unity of mathematics,” Singer said.
The Abel Prize is a highly prestigious award in mathematics, and is awarded every year, starting in 2003.
The index theorem
“The index theorem is about the possibility of solving differential equations,” said the Head of the MIT Mathematics Department David A. Vogan. The theorem provides a formula to calculate the difference between the number of independent solutions and constraints of a system of differential equations, called the index of the system.
The role of the index is parallel to that which the difference between the row and the column ranks of a matrix plays in a system of linear equations. Through the theorem, Atiyah and Singer generalized the concept of the equality of the row-rank and the column-rank of a matrix, which for matrices would imply a unique solution, Melrose said.
The theorem helps us analyze the nature of the solutions of equations involving dirac operators, by calculating the index of the equation without actually solving it. Dirac operators are differential operators that arise in particle physics and operate on a mathematical representation of particles with spin known as spinor fields, Melrose explained. The wave function of an electron is an example of a spinor field.
It is almost impossible to solve the equations arising from applying a Dirac operator to a spinor and equating this with another spinor field, he said.
In a more precise language, the index of such an equation is given by the difference between the number of independent solutions to the corresponding homogeneous equation and the number of constraints on the input function on the right hand side of the equation, Melrose said.
Atiyah and Singer gave a formula for this index based on the coefficient functions of the dirac operator. These functions are based on the geometry and topology of the surrounding space, allowing one to calculate this index without actually solving the equation. An immediate use concerns homogeneous equations with dirac operators. A positive index for such an equation shows that it must have a solution, Melrose said.
Applications to theoretical physics
Gauge theory, monopoles, string theory and the theory of anomalies are among the various fields where the theorem is applicable.
Quantum theory and string theory do not work when there is more than one solution to certain equations, Singer said.
For example, a conformal anomaly in string theory only vanishes for a ten dimensional space, as can be seen from the index theorem. This explains why space-time is ten dimensional, Singer said, because such anomalies do not fit with current physical theory.
Singer applauded the theorem, saying that its breakthroughs will allow the next generation of mathematicians and physicists to explore new areas of research.