# Math to the Nth Power

### The mathematics department quietly experiences dramatic growth in rankings and research prominence.

The next time you pull out your smartphone, take a moment to appreciate the tremendous amount of mathematics that it embodies.

Math is involved in converting the sound of your voice into radio signals that connect you to your friends. It is used to create the complex shapes of the fonts in your email messages. In fact, all the phone’s functions are performed by executing basic logical operations on binary code, strings of ones and zeros.

The sleek slab of glass, metal and plastic is an appropriate symbol of just how dependent modern society has become on its most complex art form. There are very few aspects of life today that can function efficiently without the liberal application of mathematics. At its base, mathematics is one of the truest creations of the human intellect. As Albert Einstein put it, “Pure mathematics is, in its way, the poetry of logical ideas.”

## Here’s the Proof

In the last 15 years, Vanderbilt’s mathematics department has played an increasingly prominent role in the world of mathematics. It has quietly transformed itself from a department whose majors were mainly concerned with getting teaching jobs in regional colleges into a leading math research department that turns out students who snag jobs at top universities.

“We have moved up substantially in the world,” says Dietmar Bisch, chair and professor of mathematics.

Mathematicians don’t make statements like this without proof. One of Bisch’s strongest pieces of evidence is the department’s performance in last fall’s evaluation of the nation’s graduate programs by the National Research Council.

In the NRC’s 1995 ranking, Vanderbilt’s math program was placed at 84, toward the bottom of the heap. According to department veterans, the old ranking didn’t accurately reflect its quality. But they are quite happy with the new report that places the program squarely among the top 20 percent of the 127 Ph.D. math programs that it analyzed.

The second piece of evidence Bisch cites is the recent hire of Fields Medal winner, Vaughan Jones, from the University of California, Berkeley. Awarded every four years, the Fields Medal is generally considered the Nobel Prize of mathematics.

Jones himself says that his move to the College of Arts and Science was due in part to “the positive atmosphere at Vanderbilt compared to all the negativity in California.” The new Distinguished Professor of Mathematics also cites the quality of the department and the greater ease with which he will be able to get things done here as major reasons for joining the school.

## Advanced Theories with Applications

When he arrived in August, Jones added considerable strength to one of the department’s theoretical research groups, the Center for Noncommutative Geometry and Operator Algebras. The center is directed by Bisch and includes Stevenson Professor of Mathematics Gennadi Kasparov, Assistant Professor of Mathematics Jesse Peterson, and Professors of Mathematics Guoliang Yu, Dechao Zheng and Daoxing Xia.

They study the properties of “non-commutative” spaces where, for example, 4 times 3 does not equal 3 times 4. These advanced theories describe the properties of subatomic particles and a number of other scientifically important spaces. Such spaces also play an important role in the latest manifestation of string theory, which is based on the idea that elementary particles are tiny vibrating strings instead of infinitesimal spheres.

Another theoretical group consists of Centennial Professors of Mathematics Alexander Olshanskiy and Mark Sapir, Professor of Mathematics Mike Mihalik and Associate Professor of Mathematics Denis Osin, recognized experts in group theory, which has its origins in geometry. Group theory is a powerful way of studying geometrical objects and has a number of applications ranging from crystallization to DNA replication to cryptography. Where geometry focuses on objects like the rectangle, group theory concentrates on operations like rotation and translation that these objects undergo.

## There are very few aspects of life today that can function effectively without…mathematics.

Some mathematical research is more down-to-earth, and the department works in several applied fields. One is constructive approximation, which specializes in finding simple techniques that approximate the behavior of complex mathematical expressions. In the Center for Constructive Approximation, Professors of Mathematics Ed Saff and Doug Hardin developed a new method for evenly distributing points on curved surfaces, a procedure with applications ranging from digitizing curved surfaces to modeling the coastal effects of tsunamis. The center, made up of Stevenson Professor of Mathematics Larry Schumaker, Professors of Mathematics Saff, Hardin, Mike Neamtu and Akram Aldroubi, and Assistant Professor Alex Powell, also publishes *Constructive Approximation, *one of the world’s most highly cited math journals.

## Talking Math

A key element in the department’s growing reputation has been an annual lecture honoring Professor Baylis Shanks, MA’40, and education administrator Olivia Shanks, MA’39, a couple who played major roles at Vanderbilt from the 1950s to the 1970s. The lecture, which emphasizes a different field of mathematics each year, allows the department to invite top mathematicians from around the world for an accompanying research conference that has developed a considerable following in mathematics circles.

It was an invitation to speak at the Shanks lecture that led to a collaboration with mathematician, Fields medalist Alain Connes of the College de France and the Institut des Hautes Études Scientifiques (IHES). For nine years, Connes has directed an annual spring institute that combines lectures and workshops, attracting both senior and junior mathematicians to Nashville. (Connes also serves as Distinguished Professor of Math-ematics here at Vanderbilt.) Because of these activities, the department now hosts 300 to 400 visitors annually, an exceptional number for a math department of its size.

Another factor in the department’s growth was Bisch’s proposal to replace the short-term lecturers who taught many of math’s 160 courses with post-doctoral researchers. This change freed up research time for graduate students, improved the quality of instructions and enhanced the research ambiance in the department significantly. It also brought the College of Arts and Science to the attention of departments nationwide looking for positions for their graduates.

This growing stature has attracted increasingly high quality students. At the undergraduate level, it recently added a new honors track specifically for students interested in pursuing careers in math research and its graduate students and postdoctoral fellows have been extremely successful in finding jobs despite the tough job market.

*photo credit: John Russell*