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 homotopy combinatorics.

• Category theory
• Homotopy theory
• Algebraic geometry
• Algebra
• Combinatorics

• MTM2016-80439-P: Homotopy theory of algebraic structures. MICINN. IP: N. Castellana and J. Kock. 30/12/2016-29/12/2020.
• MTM2013-42293-P: Homotopy theory of classifying spaces and function complexes. MICINN. IP: N. Castellana and J. Kock. 01/01/2014-31/12/2016.
• PICS264428: Combinatorial Hopf algebras and noncommutative probability. Projet International de Coopération Scientifique (France–Espagne), Centre National de Recherche Scientifique (France). IP: F. Patras (Université de Nice Sophia-Antipolis) and J. Kock. 01/01/2016-31/12/2018.

• Infinity-Operads as Analytic Monads; With David Gepner and Rune Haugseng; Int. Math. Res. Notices. (2020), DOI:10.1093/imrn/rnaa332
• Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations; With Kurusch Ebrahimi-Fard, Loïc Foissy, and Frédéric Patras; Adv. Math. 369 (2020), 107170.
• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory; With Imma Gálvez and Andy Tonks; Adv. Math. 331 (2018), 952-1015
• Polynomial functors and combinatorial Dyson-Schwinger equations; J. Math. Phys. 58 (2017), 041703, 36pp.
• Hochster duality in derived categories and point-free reconstruction of schemes; With Wolfgang Pitsch; Trans. Amer. Math. Soc. 369 (2017), 223-261.