Lurie's research interests started with
logic and the theory of
surreal numbers when he was in high school. He is best known for his work, starting with his thesis, on
infinity categories and
derived algebraic geometry. Derived algebraic geometry is a way of infusing
homotopical methods into
algebraic geometry, with two purposes: deeper insight into algebraic geometry (e.g. into
intersection theory) and the use of methods of algebraic geometry in
stable homotopy theory. The latter area is the topic of Lurie's work on
elliptic cohomology. Infinity categories (in the form of
André Joyal's quasi-categories) are a convenient framework to do homotopy theory in abstract settings. They are the main topic of his book
Higher Topos Theory. Another part of Lurie's work is his article on
topological field theories, where he sketches a classification of extended field theories using the language of infinity categories (
cobordism hypothesis). In joint work with
Dennis Gaitsgory, he used his non-abelian
Poincaré duality in an algebraic-geometric setting, to
prove the
Siegel mass formula for
function fields. Lurie was one of the inaugural winners of the
Breakthrough Prize in Mathematics in 2014, "for his work on the foundations of
higher category theory and derived algebraic geometry; for the classification of fully extended topological quantum field theories; and for providing a moduli-theoretic interpretation of elliptic cohomology." Lurie was also awarded a MacArthur "genius grant" Fellowship in 2014. ==Publications==