By Lisa Mohan and Itay Neeman
 
Zaher Hani's plan to pursue an undergraduate degree in physics at the American University of Beirut (AUB) in Lebanon quickly changed when he discovered an unexpected passion. Says Zaher, "I realized that mathematics is a very important tool in physics, so I started to take some advanced math classes. I enjoyed that more than the physics I was hoping to learn after I had mastered those mathematical skills." An introductory course in analysis particularly engaged him, and Zaher knew that he had found his mathematical focus. By his third year, Zaher had taken all of the undergraduate analysis courses AUB had to offer and furthered challenged himself by taking a graduate course on harmonic analysis.

Across the world at UCLA, Terence Tao, professor of mathematics and the recent recipient of a 2006 Fields Medal (the mathematics equivalent of the Nobel Prize), began publishing his notes on his graduate harmonic analysis course via his faculty web page. Says Zaher, "Everyone knew how nice and important Terry's notes are and my professor was following them more or less, so I actually read Terry's notes." Zaher knew then that he wanted to pursue his graduate studies in harmonic analysis and quickly set his heart on UCLA and the chance to study with a group of strong research analysts, including Tao, John Garnett, Christoph Thiele and Rowan Killip.

After being accepted in to UCLA's graduate mathematics program, Zaher's interests broadened when he took a course on partial differential equations (PDE), which involve a function depending on several variables and its (partial) derivatives with respect to those variables. Says Zaher, "PDE is one of the broadest fields of mathematics, and the techniques used to study different classes of PDE are quite distinct despite a few common ideas. For example, certain fields in PDE rely heavily on tools from functional analysis and measure theory while others dispense completely of those tools for the sake of others, such as complex analysis, dynamical systems, harmonic analysis, etc."

Next came a reading course with Tao, which covered his 2006 book, Nonlinear dispersive equations: local and global analysis. The field of nonlinear dispersive PDE uses quantitative estimates and ideas from harmonic analysis as a fundamental tool. Says Zaher, "This was one of the initial attractions that made me get into this field. However, I soon started to appreciate the ideas and tools developed in the field and more importantly its connections to other disciplines of mathematics like dynamical systems, geometry, and even analytic number theory."

Not only had Zaher found a research focus, he had also found his PhD mentor in Tao who had inspired him as an undergraduate. Tao suggested a research direction studying non linear dispersive PDE that are posed on manifolds. "My research has mostly been concerned with the asymptotic behavior – behavior for very large time – of solutions of some nonlinear dispersive PDE posed on manifolds other than Euclidean spaces," explains Zaher. "I find this line of questioning particularly interesting for several reasons: for one, working on manifolds with different geometries makes the linear theory more subtle and rich with its connections to geometry and even analytic number theory as in the case of the torus. Moreover, the asymptotic theory on general manifolds can be considerably richer than that on Euclidean space, and we tend to see more genuinely nonlinear phenomena in cases when asymptotically linear behavior is seen in the Euclidean analogue. However, our understanding here is still rather poor in comparison to the Euclidean setting."

Zaher credits his successful research journey to Tao's inspirational and practical guidance, as well as the open atmosphere in the department's weekly participating analysis seminars. When Zaher's progress slowed on the initial problem that Tao had posed for his thesis, Tao advised him to turn to another, related problem, which he was able to solve completely and was the first part of his PhD thesis. The second part was the progress that he made on the first problem, which he found more intractable but is still trying to solve. "I didn't give up on it," says Zaher. "That's part of the advice that Terry gave me. Sometimes things get easier with time because the tools that get developed either by you or by your colleagues in the field allow us to attack problems in a new and smarter way." The weekly participating analysis seminars also provided a productive, but relaxed research environment. "There was a family atmosphere almost that I found very comfortable," he says. "Even when someone was giving a talk, it was very interactive between the audience and the speaker, both exchanging ideas." He also benefitted from daily faculty interactions, particularly with Rowan Killip and later Monica Visan, who joined the analysis group during Zaher's third year. Says Zaher, "One of the things I really liked is that everyone kept their doors open and was happy to talk to me and answer questions. Having Terry on one side as my advisor, and Rowan and Monica, who also work in the field, on the other, made my stay at UCLA a very pleasant one." Informal research discussions with his fellow graduate student analysts, who Zaher initially bonded with in the department's basic exam boot camp, also provided intuition into his own research.

Now a postdoctoral research fellow at the Courant Institute of Mathematical Sciences at NYU, Zaher has the opportunity to work with other pure PDE researchers who also have their hand in applied PDE, an area that Zaher finds increasingly important. Advice from his UCLA PhD advisor stays with him. Says Zaher, "Terry realized how important it is to be broad, to have an open mind, not to finish my PhD only knowing or only interested in what I did in my thesis. But rather explore new directions and new problems in my field and in related fields. That's very important."

Published in Winter 2012