## Advice to freshmen math majors: Learn other subjects, such as physics and computer science

Zhiwu Lin is an associate professor in the School of Mathematics. He comes to Georgia Tech with an international education in mathematics, earning a B.S. from Peking University, in China; an M.S. from Tokyo University, in Japan; and a Ph.D. from Brown University, in the U.S. He joined Georgia Tech in 2008.

I work on partial differential equations (PDE), as applied to fluids, plasmas, and other physical problems involving wave phenomena. Many physical problems can be described by PDEs. For example, fluid motion is modeled by the Navier-Stokes equation, and gravitational waves, by the Einstein equation.

In many physical phenomena, one can observe coherent structures such as traveling surface water waves propagating at a constant speed, the large scale vortex structures in the atmosphere and ocean, the great red spot on Jupiter, and the elliptical or spiral galaxies in the universe. It is interesting and important to explain the formation of these coherent structures. A first question is their stability, that is, whether they persist under small perturbations. Stable structures are more likely to be observed.

Understanding the mechanisms of instability is also important. One example is fusion reaction devices such as tokamaks, which are designed to get sustainable and cheap energy by the fusion of a gas of charged particles called plasmas. However, fusion is very difficult to achieve because plasmas are highly unstable. An important problem in plasma physics is to understand the mechanisms of instability. They might help us design better fusion devices to contain the plasmas for a longer time.

I have been working on finding stability and instability criteria for coherent structures from various physical phenomena, by mathematical analysis of the PDE models. For example, it has been suggested that the highest traveling surface water waves have a 120-degree angle at the crest. However, in reality such high waves are not observed. I gave an explanation by proving the instability of traveling waves with heights close to the assumed highest one.

Another recent focus of my research is to understand the dynamical roles of coherent structures. For example, how do we explain the appearance of the large-scale structures observed in the atmosphere and oceans? What is the mechanism for an initially unstructured state to approach a final coherent state? These problems are far from being solved. I believe that mathematics would play an important role in finding the answers.

What has been the most exciting time so far in your research life?

One of the most exciting time in my research career was in 2001, during my Ph.D. at Brown University. I proved that any periodic BGK wave of 1D Vlasov-Poisson equation modeling electrostatic plasmas is unstable under perturbations of multiple periods (such as a double periods). This proof confirmed the conjecture in the physical literature based on numerical computations. The new approach developed has been used later in many other problems. It also gave me great confidence to pursue an academic career. It is interesting that the first stable BGK wave under perturbations of the same period was constructed only last year, by myself and Yan Guo. Fifteen years had passed!

Another exciting moment has just occurred. I just finished a 175-page paper with my School of Mathematics colleague Chongchun Zeng to give a general theory to study linear Hamiltonian PDEs. This theory can be used to study the stability of coherent states of general non-dissipative models whose energy functional only have finitely many negative directions. So it can be applied to a wide range of problems from fluids to plasmas and nonlinear wave models. We have been working hard on this project for almost three years.  It is a big relief to get it done.

How did you find your way to mathematics research?

In my high school, I did pretty well on science subjects, among which chemistry was my favorite.  However, I could not choose chemistry as my major when I applied for colleges because I failed to pass some screening test on vision. So I chose mathematics which is my second favorite.

Today I feel very lucky for this choice. I am not a handyman in life and quite possibly will also not perform well in labs. Also, you can do mathematical research at any time and any location, unconstrained by labs or equipment.

What advice would you give to a college freshman who wants to be a mathematician?

Besides getting a solid training in mathematics, it is beneficial to broaden your interests by learning other subjects such as physics and computer science. The research today tends to be more interdisciplinary, and computers are used more.

If you could not be a mathematician, in what line of work would you be now?

Maybe a historian. I have always been interested in history, particularly the clues behind historical facts.

What is the most exciting thing about being a part of Georgia Tech?

The campus has a very diverse culture. Our faculty are outstanding. Even better, I share research interests with quite some of them.
What are you most surprised about in your encounters with Georgia Tech students?

Before joining Tech, I had been in several universities that are quite different from technological institutes. Not surprisingly, Tech students are more motivated in learning math and have better math background. What surprised me is the diversity and good behavior of the students. They care more about grades but they also work harder.
What three destinations are still in your travel to-do list?

Australia, South America, and Africa. I have never been to these continents. It is always fascinating to experience different cultures.

If you could have dinner with any person in history, whom would you invite?

Newton. I have been teaching calculus for many years. It would be interesting to know the stories behind the formulas as told by their inventor.

#### Related Media

Click on image(s) to view larger version(s)

• Zhiwu Lin