nLab Jacob Lurie


Jacob Lurie is a mathematician at the Institute for Advanced Study.

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. In 2014 Lurie was awarded a MacArthur Genius Grant and the Breakthrough Prize in Mathematics.


References (partly) indexed on the nnLab

Higher category theory

on higher category theory, internal (∞,1)-categories

Higher topos theory


Higher algebra


Higher geometry

The foundations of higher geometry:

Survey on the general program

A volume on E-∞ geometry (spectral algebraic geometry)

The basic definitions are in

Fundamental properties of E E_\infty-geometry are discussed in

Application to moduli stack of elliptic curves:

Elliptic cohomology

  • Elliptic cohomology I: Spectral abelian varieties (pdf)

  • Elliptic cohomology II: Orientations (pdf)

  • Elliptic cohomology III: Tempered Cohomology (pdf)

  • Elliptic cohomology IV: Equivariant elliptic cohomology, (announced)

Publications before thesis

  • M H Freedman, A Kitaev, J Lurie, Diameters of homogeneous spaces, Math. res. lett. 10:1, 11-20 (2003)

  • J. Lurie, Anti-admissible sets, J Symb Logic 64:2, 408-435 (1999)

  • J. Lurie, On a conjecture of Conway, Illinois J. Math. 46:2, 497-506 (2002)

  • J. Lurie, On simply laced Lie algebras and their minuscule representations, Comment. Math. Helv. 76:3, 515-575 (2001) doi

Other material

category: people

Last revised on November 13, 2021 at 10:15:49. See the history of this page for a list of all contributions to it.