original version of this story Appeared in Quanta Magazine.
In the 2020 Georgia gubernatorial election, some voters in Atlanta waited more than 10 hours to cast their votes. One reason for the long lines was that nearly 10 percent of Georgia's polling places have been closed over the past seven years, despite an influx of nearly 2 million voters. These closures were disproportionately concentrated in predominantly black neighborhoods that tend to vote Democratic.
But pinpointing the location of “voting deserts” is not as easy as it might seem. In some cases, a lack of capacity is reflected in long wait times at polling places, but in other cases, distance to the nearest polling place is an issue. It is difficult to combine these elements systematically.
In a paper scheduled to be published in a magazine this summer Siam review, Mathematician Mason Porter and his students at the University of California, Los Angeles, used the tools of topology to do just that. Abigail Hickok, one of the paper's co-authors, came up with the idea after seeing images of long lines in Atlanta. “I was really concerned about voting, especially since it was such an anxiety-provoking election,” she said.
Topologists study the underlying properties and spatial relationships of geometric shapes during transformation. Two shapes are considered topologically equivalent if one shape can be transformed into the other through continuous motion without tearing, gluing, or inserting new holes.
At first glance, the topology appears to be a poor fit for the problem of voting site placement. Topology concerns continuous shapes, and voting sites are in discrete locations. But in recent years, topologists have adapted their tools to handle discrete data by creating graphs of points connected by lines and analyzing the properties of those graphs. Hickok said these technologies can help not only understand the distribution of voting locations, but also examine who has better access to hospitals, grocery stores and parks.
That's where topology begins.
Imagine creating a small circle around each point on a graph. The circle starts with radius 0 but grows over time. Specifically, once the waiting time at a designated polling place is exceeded, the circle begins to widen. As a result, where the latency is short, the circles are larger and start growing first, whereas where the latency is long, the circles become smaller.
Some circles eventually come into contact with each other. In this case, draw a line between the centers of the points. If multiple circles overlap, connect all their points with a “simplice”. This is a general term for shapes such as triangles (2-simplex) and tetrahedra (3-simplex).