Joachim Kock

Joachim Kock

Joachim Kock
PhD in mathematics (2000) from Universidade Federal de Pernambuco, Recife, Brazil
Professor titular at the UAB

Personal webpage



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 Interests

  • 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. (2015, online first, DOI 10.1007/s11005-015-0785-7).
  • Hochster duality in derived categories and point-free reconstruction of schemes. With Wolfgang Pitsch. To appear in Trans. Amer. Math. Soc. (2015)
  • Univalence in locally cartesian closed -categories. With David Gepner. Preprint, arXiv:1208.1749