Kirwan's research interests include
moduli spaces in
algebraic geometry,
geometric invariant theory (GIT), and in the link between GIT and
moment maps in
symplectic geometry. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and
topological properties. She introduced the
Kirwan map. From 1983 to 1985, she held a junior fellowship at
Harvard. From 1983 to 1986, she held a Fellowship at
Magdalen College, Oxford, before becoming a Fellow of
Balliol College, Oxford. She is an honorary fellow of
Clare College, Cambridge and also at Magdalen College. In 1996, she was awarded the
Title of Distinction of Professor of Mathematics. From 2004 to 2006, she was president of the
London Mathematical Society, the second-youngest president in the society's history and only the second woman to be president. In 2005, she received a five-year
EPSRC Senior Research Fellowship, to support her research on the
moduli spaces of complex
algebraic curves. In 2017, she was elected
Savilian Professor of Geometry, becoming the first woman to hold the post. While this entailed a move to
New College, Oxford, she was elected an emeritus fellow at Balliol. She was the convenor of the 2008–2009 meeting of
European Women in Mathematics and deputy convenor of the following meeting in 2010–2011. ==Personal life==