Duchin's mathematical research has focused on
geometric topology,
geometric group theory, and
Teichmüller theory. In 2022 Duchin appeared in the
Netflix documentary
A Trip to Infinity, discussing the mathematical implications of
infinity. Duchin's expertise in geometry has led her to conduct research on the mathematics of
gerrymandering. A key aspect of this research is the geometric notion of the
compactness of a given political district, a numerical measure that attempts to quantify how extensively gerrymandered it is. “What courts have been looking for is one definition of compactness that they can understand, that we can compute, and that they can use as a kind of go-to standard”, she said in an interview with
The Chronicle of Higher Education. As a part of this project, she founded a summer program to train mathematicians to become expert witnesses in related legal cases. In 2016, she founded the
Metric Geometry and Gerrymandering Group (MGGG) which is a nonpartisan research group that coordinates and publicizes research on geometry, computing, and their application to the
redistricting process in the US. In 2018, Governor of
Pennsylvania Tom Wolf enlisted Duchin to help him evaluate newly drawn redistricting maps for fairness. This happened as a consequence of the
Supreme Court of Pennsylvania decision which declared the state's 2011 US congressional districting map to be unconstitutional. Duchin prepared a report published on February 15, 2018. In 2022, a panel of judges threw out Alabama's soon-to-be-used congressional maps, citing the fact that the percentage of black people in the state had risen to about a quarter of the population. To draw some new, fairer maps, they turned to Duchin, who came up with 4 nearly-similar maps that would put the Black and Democratic-leaning cities of
Mobile and
Montgomery together, therefore complementing the one Black and blue-leaning district in the state with a second one. ==Awards and honors==