# The Evolution of Shape

New Professor of Mathematics Lu Wang says she first became interested in math in elementary school. Her sixth grade teacher, she says, was passionate about math and taught her that math could be a creative endeavor.

Today, Wang is just as enamored with math as her teacher. Her speciality is an area known as geometric flow, which involves analyzing the evolution of shapes. The field has known applications in computer graphics and image processing, but Wang says her drive is to "pursue the beauty."

Wang grew up in Beijing, China. She earned her bachelor's degree in mathematics from Peking University in 2006 and her PhD in mathematics from MIT in 2011. She became a J. J. Sylvester Assistant Professor at Johns Hopkins University in 2011, a Chapman postdoctoral fellow at Imperial College London in 2014, and a tenure-track assistant professor at University of Wisconsin-Madison in 2015. She joined Caltech's faculty in the fall of 2019.

#### What does the field of geometric flow entail?

Geometric flow is the study of the shapes of objects and how they can evolve or change. This evolution process is described by partial differential equations. Imagine the shape of a dumbbell with two lobes connected by a long neck. This shape can evolve in ways such that the neck gets smaller and then pinches down to a point, leaving two spheres. The spheres can also shrink down to points. Those points are basically called singularities, and that's when the topology of the shape has changed.

The topology of an object can be understood by thinking of a donut and a coffee cup. They both have one hole, so they have the same topology. But if one of these objects were pinched to a point, its topology would have changed. I'm interested in understanding how this topological change happens.

#### Are there practical applications for this kind of work?

Yes, people use the equations I study for computer graphics and image processing, for example to essentially sharpen the image. Personally, I am motivated by the beauty of math. I think of math as art in some senses. I don't think about how useful my research results are, but focus on the pursuit of beauty and coming up with simple and clean solutions to problems.

#### What are some of the reasons you chose to come to Caltech for math?

In the math department here, we have a lot of chances to teach and to interact with young people. Not many departments have these same teaching opportunities. Sometimes, I'll run out of ideas for a problem and get stuck, but I have found that working with young people helps freshen me up and makes me think about things differently. Teaching helps me become unstuck.

#### Can you tell us more about your math teachers growing up?

When I was in elementary school, I was very interested in Chinese literature, but then I was lucky to have a very passionate sixth grade math teacher who introduced me to math. My teachers at Peking University were also very goodâ€”they have one of the top math programs in the country. My sixth grade math teacher was a woman, while most of my teachers after that were males. One thing I like to do is to inspire more women to go into math. I have participated in various related activitiesâ€”for example, I co-organized the Women and Non-binary People in Mathematics at Wisconsin (WIMAW) lectures, and I taught in the Institute for Advanced Study Women and Mathematics Program.

#### What does a typical day look like for you?

I usually start with checking newly posted articles on the website www.arxiv.org. I like to listen to classical music, like Chopin, while I am drawing on paper and doing computations. I am constantly communicating ideas with collaborators over email or Skype, or meeting with my students to discuss research projects. Sometimes, when I feel stuck or tired, I like to take a walk around campus or go to the gym.

Written by Whitney Clavin

Whitney Clavin
(626) 395-1944