Melody Chan, assistant professor of mathematics, is proud to say that her Erdös number is three.
That number means that she’s published a paper with someone who has published with someone who has published with Paul Erdös, the legendary Hungarian mathematician. As an undergraduate, Chan studied for a semester in Budapest, Erdös’ birthplace and a stronghold of sorts for mathematical research.
“My first summer research adviser had Erdös number two,” she said. “So right away after that summer my Erdös number was three.”
But Erdös was much more than just the Kevin Bacon of mathematics. He was one of the most important mathematicians of the 20th century and a pioneer in the field of combinatorics.
Chan’s research area combines Erdös’s field of combinatorics (the study of things that can be counted in a finite manner) with algebraic geometry (the study of geometric objects derived from polynomial equations). It’s a relatively new field, dubbed tropical geometry, that has just come into being in the last decade or so.
“Tropical geometry is, approximately, a translational engine that takes old problems in algebraic geometry and reformulates them into problems about combinatorial objects,” Chan said.
The idea combining geometry with combinatiorics isn’t new. “If you wanted to count the squares in a chess board, you could simply count them 1, 2, 3 and so on," Chan said. "But you’re not taking advantage of the inherent geometric structure of the board, which is that the boxes are laid out in an eight-by-eight grid. So you develop new methods for counting that are intrinsically tied to underlying geometry.”
What is new is bringing this approach to bear on the complex objects studied in algebraic geometry. That complexity is precisely why a combinatorial approach to understanding them can be so useful. For example, one type of object that algebraic geometers study is known as an algebraic curve. To study them, researchers use what’s known as degeneration — essentially breaking them apart into smaller pieces.
“Those pieces then interact in a combinatorial ways,” Chan said. “You use finite mathematics and its tools to analyze the ways that pieces fit together. That in turn tells you information about the curve.”
Chan says it’s exciting to be a part of such a new field. And she sees it as a priority to help other mathematicians to understand how it works.
“You can imagine us as explorers trying to explore a new land,” she said. “For some people, it’s the most important thing to march up the tallest mountain and plant a flag to prove they were there. It may not be as important to them to help other people understand the terrain. To me, if you get to the highest mountain, you should write exactly how you got there. You should explain the paths you found along the way that could be of interest to other people. You should help others follow.”
For Chan, that means writing clearly, giving good talks, mentoring students and providing support for her colleagues.
Chan, who earned her Ph.D. from the University of California–Berkeley, comes to Brown following a postdoctoral position at Harvard. She says she’s looking forward to working with her new colleagues, several of whom she’s already worked with.
“Brown is an amazing school and the department is friendly and supportive,” she said. “The students who come through here just rave about their experience — and that's not the case everywhere.”