Joachim Kock

Associate professor at UAB
Research area: Topology / Algebra

PhD in Mathematics obtained at Universidade Federal de Pernambuco, Recife, Brazil (2000)


PhD from UFPE, Recife, Brazil (2000); held postdoc positions in Stockholm, Nice and Montréal; professor titular at the UAB since 2009.

In his thesis (2000) he used gravitational quantum cohomology à la Witten to solve the characteristic number problem for rational curves in projective space.  With Toën (2005) he proved a non-linear version of the Deligne conjecture in homotopical algebra, and with Joyal (2007) he proved Simpson’s conjecture in dimension 3, connecting homotopy theory with higher category theory. He is currently developing the theory of polynomial functors, a categorical toolbox unifying structures found in operad theory and quantum algebra, algebraic combinatorics, constructive type theory, and perturbative renormalisation.

Research lines

  • Category theory
  • Homotopy theory
  • Algebraic geometry
  • Structural aspects of combinatorics, mathematical physics, logic and theoretical computer science

Selected publications

  • Polynomial functors and trees. Int. Math. Res. Notices 2011 (2011), 609-673.
  • Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees. With Imma Gálvez and Andy Tonks. Adv. Math. 254 (2014), 79–117.
  • Perturbative renormalisation for not-quite-connected bialgebras. Lett. Math. Phys. 105 (2015), 1413-1425.
  • Hochster duality in derived categories and point-free reconstruction of schemes. With Wolfgang Pitsch. Trans. Amer. Math. Soc. 369 (2017), 223-261.
  • Univalence in locally cartesian closed ∞-categories. With David Gepner. Forum Math. 29 (2017), 617-652.