Much of Melanie Wood's research is motivated by questions in number theory, though the mathematics she studies also includes arithmetic and algebraic geometry, topology, probability, and random groups. She is interested in understanding the distribution of number fields and their fundamental structures, including class groups, p-class tower groups, and the Galois groups of their maximal unramified extensions. She works on questions including counting number fields, finding the average number of unramified G-extensions that number fields have, bounding the sizes of class groups, and function field analogs of all of these questions (which then leads to questions in topology about certain moduli spaces of curves). To understand the distribution of class groups and Galois groups of unramified extensions, she also studies random abelian and non-abelian groups to construct the random groups that are relevant for number theory and understand their properties. She has also been developing tools in probability theory to study randomly arising finite groups, such as the Jacobians of random graphs and cokernels of random matrices.
She completed her PhD at Princeton University in 2009 under the supervision of Manjul Bhargava, and was a Szego Assistant Professor at Stanford University from 2009-2011. She was an American Institute of Mathematics Five-Year Fellow from 2009-2017. She was faculty at the University of Wisconsin-Madison from 2011-2019. In Fall 2018, she was a Minerva Distinguished Visitor at Princeton University.