Like most things during the height of the pandemic, research that could be conducted virtually was conducted virtually. And that’s why, although juniors William He and Annie Wang have been working together on a research project since last September, they’ve never actually met in person.

He, a Math major from Houston, and Wang, a Computer Science and Math double-major from Raleigh, both work in the lab of Professor Debmalya Panigrahi, where the focus is on research in theoretical computer science, particularly graph algorithms. Wang and He did work on hypergraphs, and, after I asked them to explain what hypergraphs were in the most elementary terms (I am *not* a Math major), they went back and forth on how exactly to relay hypergraphs to a lay audience.

Here is what they landed on: hypergraphs are essentially generalizations of normal graphs. In a normal graph, there are edges –each edge connects two points. There are also vertices – each point is a vertex. But in a hypergraph, each edge connects multiple points.

He and Wang were looking at a generalization of graph reliability – if all edges disconnect at a certain probability, what is the probability that the graph itself will break down because crucial edges are disconnecting?

Their research adds to existing research on maximum flow problems, which Wikipedia tells us “entail finding a feasible flow through a flow network to obtain the maximum possible flow rate.” In a landmark paper written by T.E. Harris and F.S. Ross in 1955, the two researchers formulated the maximum flow problem using an example of the Soviet railroad and considering what cuts in the railroad would disconnect the nation entirely – and what cuts could be made with little impact to railway traffic flow.

Maximum flow problems are a core tenet of optimization theory, used widely in disciplines from math to computer science to engineering. You may not know what mathematical optimization is, but you’ve seen it in action before: in electronic circuitry, in economics, or unsurprisingly, used by civil engineers in traffic management.

It’s expected to be incredibly difficult to exactly calculate the target value of He and Wang’s question. They landed on an approximation that they know is far from the exact calculation, but still brings them closer to understanding hypergraph connectivity more fully.

So what draws them to research? For He, it’s like an itch. He describes that “sometimes I’ll be watching a movie, and then thirty minutes in I’m thinking about a possible solution to a math problem and then I can’t focus on the movie anymore.” You can’t get on with things until you scratch the itch, but the best part to him is when things finally start to make sense. For Wang, research is just plain fun. She enjoys learning about algorithms and theorems, and she loves the opportunity to work with professors who are at the forefront of their field.

After Duke, He wants to pursue a PhD, likely in theoretical computer science, while Wang is still weighing her options – whether she wants to go into academia or industry. While He came into Duke as a prospective Economics major, in quarantine especially he realized just how much he enjoyed math for the sake of itself.

Wang, similarly, thought she would want to pursue software engineering, but she’s slowly realizing that she likes “solving the problems within the field – problems that I need a PhD to solve.” The magic of research, for her, is that “you’re solving problems that no one has answers to yet.” And wherever the future takes both of them, she says that in doing research, even at the undergraduate level, “you feel like you’re pushing the boundary a tiny bit, and that’s a cool feeling.”

*Post by Meghna Datta, Class of 2023*