Jacob Lurie is a mathematician at Harvard University.
After an early interest in formal logic (specifically notions of computable surreal numbers, see Notices of the AMS vol 43, Number 7) Lurie indicated in his PhD thesis how the moduli stack of elliptic curves together with the collection of elliptic cohomology spectra associated to each elliptic curve is naturally understood as a geometric object in a homotopy theoretic refinement of algebraic geometry that has come to be known as derived algebraic geometry. He then embarked on a monumental work laying out detailed foundations of the subjects necessary for this statement, which is homotopy theory in its modern incarnation as higher category theory, higher geometry in terms of higher topos theory and finally higher algebra in terms of higher operads, all in principle very much along the lines originally developed by Alexander Grothendieck and his school for ordinary algebraic geometry, but now considerably further refined to the general context of homotopy theory. While some developments in these topics had been available before, Lurie’s comprehensive work has arguably led these subjects to an era of reinvigorated activity with a variety of further spin-offs. Among these most notable is maybe the formalization and proof of the cobordism hypothesis, which lays higher monoidal category theoretic foundations for (local, topological) quantum field theory.
(∞,2)-Categories and the Goodwillie Calculus (pdf, arXiv)
on (∞,1)-categories of (∞,n)-categories, specifically on internal (∞,1)-categories, and on Goodwillie calculus
on higher category theory, internal (∞,1)-categories
On the Classification of Topological Field Theories (arXiv)
on functorial extended topological quantum field theory classified via the cobordism theorem by symmetric monoidal (∞,n)-categories
subsuming
on stable (∞,1)-categories stable homotopy theory and homological algebra
Noncommutative Algebra (arXiv)
and
on the notions of Kähler differential and cotangent complex and their generalization from the ordinary context of algebra to that of higher algebra.
Moduli Problems and DG-Lie Algebras
on ∞-Lie algebras as formal neighbourhoods of point in ∞-stacks.
The foundations of higher geometry:
Survey on the general program
PhD thesis, derived algebraic geometry (pdf)
The basic definitions of E-∞ geometry (spectral geometry) are in
Fundamental properties of $E_\infty$-geometry are discussed in
Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems
Jacob Lurie, Proper Morphisms, Completions, and the Grothendieck Existence Theorem
Application to moduli stack of elliptic curves:
A Survey of Elliptic Cohomology (pdf)
on the generalized (Eilenberg-Steenrod) cohomology theory tmf, the gluing of all elliptic cohomology theories