• Alexander–Beck modules detect the unknot 

      Szymik, Markus (Journal article; Peer reviewed, 2018)
      We introduce the Alexander–Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.
    • Algebraic invariants of links and 3-manifolds 

      Ødegaard, Reidun P (Master thesis, 2015)
      The goal of this thesis is to describe certain algebraic invariants of links, and try to modify them to obtain invariants of 3-manifolds. Racks and quandles are algebraic structures that were invented to give invariants ...
    • Drinfeld centers for bicategories 

      Meir, Ehud; Szymik, Markus (Journal article; Peer reviewed, 2015)
      We generalize Drinfeld’s notion of the center of a tensor category to bicategories. In this generality, we present a spectral sequence to compute the basic invariants of Drinfeld centers: the abelian monoid of isomorphism ...
    • Homotopy coherent centers versus centers of homotopy categories 

      Szymik, Markus (Journal article, 2018)
      Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also ...
    • Permutations, power operations, and the center of the category of racks 

      Szymik, Markus (Journal article, 2018)
      Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of ...
    • Quandle cohomology is a Quillen cohomology 

      Szymik, Markus (Journal article, 2018)
      Racks and quandles are fundamental algebraic structures related to the topology of knots, braids, and the Yang-Baxter equation. We show that the cohomology groups usually associated with racks and quandles agree with the ...
    • Rational Tambara functors 

      Ræder, Truls Bakkejord (Doctoral theses at NTNU;2017:95, Doctoral thesis, 2017)
    • Spectral sequences for Hochschild cohomology and graded centers of derived categories 

      Neumann, Frank; Szymik, Markus (Journal article; Peer reviewed, 2017)
      The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge ...
    • String bordism and chromatic characteristics 

      Szymik, Markus (Journal article; Peer reviewed, 2018)
      We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the ...
    • The homology of the Higman–Thompson groups 

      Szymik, Markus; Wahl, Nathalie (Journal article; Peer reviewed, 2019)
      We prove that Thompson’s group \mathrm {V} is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups \mathrm {V}_{n,r} with the homology of the ...
    • The rational stable homology of mapping class groups of universal nil-manifolds 

      Szymik, Markus (Journal article; Peer reviewed, 2018)
      We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, ...
    • Thesaurus Racks - Categorical racks and applications in the algebraic topology of Lie racks 

      Grøsfjeld, Tobias (Master thesis, 2016)
      Group objects of categories have been heavily studied in a general setting, but racks are mostly treated explicitly. Since rack structures are more general than groups, this thesis aims to explore the properties of general ...
    • The third Milgram--Priddy class lifts 

      Szymik, Markus (Peer reviewed; Journal article, 2020)
      Abstract: We show that the third cohomology of the finite general linear group with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element restricts to the third Milgram–Priddy class.