• A homological approach to the Poincaré--Birkhoff--Witt theorem 

      Manum, Tallak (Bachelor thesis, 2021)
      Formålet med denne oppgaven er å utvikle noen verktøy deriblandt Hochschild cohomology, filtrerte og graderte algebraer og algebraisk deformasjonsteori for å kunne gjennomføre en konseptuel tilnærming til å bevise ...
    • Adams operations and symmetries of representation categories 

      Meir, Ehud; Szymik, Markus (Journal article, 2021)
      Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual λ–ring structure on these rings. From the representation-theoretical ...
    • 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 ...
    • Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories 

      Bohmann, Anna Marie; Szymik, Markus (Peer reviewed; Journal article, 2023)
      The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. ...
    • 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 ...
    • Finiteness Obstruction in Model Categories 

      Lien, Sverre Myhre (Master thesis, 2021)
      Vi viser in denne oppgaven at det er en naturlig måte og gjøre god for endelighets obstruksjon i en generell modell kategori. Operatoren $(Wa(-), w(-))$ vises å være en funktor fra modell kategorier hvor det eksisterer ...
    • 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 ...
    • Morse Theory applied to the Unitary Group 

      Evensen, Anders Krøger (Bachelor thesis, 2021)
      I matematikken har vi mange redskaper som kan brukes til å studere mangfoldigheter. Man kan for eksempel se på hvordan funksjoner på en gitt mangfoldighet oppfører seg. Ved å studere de kritiske punktene til en funksjon ...
    • Multicomplexes and their spectral sequences 

      Gardå, Odin Hoff (Master thesis, 2022)
      Multikomplekser generaliserer kjedekomplekser og dobbeltkomplekser. Vi ser hovedsaklig på spektralfølgen tilhørende et multikompleks over en kropp og gir betingelser for degenerasjon på første side i spektralfølgen. Videre ...
    • 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 ...
    • Power Quandles 

      Vik, Torstein (Bachelor thesis, 2021)
      Vi introduserer kategorien bestående av potensquandler og studerer den glemsomme funktoren Pq fra grupper til potensquandler sammen med dens venstreadjungerte, Gr. Vi formoder at hvis to endelige grupper har isomorfe ...
    • 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)
    • Representation Schemes for Finitely Generated Groups 

      Kjærbye Bagger, Gustav (Master thesis, 2022)
      Repræsentationsrummet for en endeligt genereret gruppe over et legeme k kan ses som et punkt på et affint k-skema. Der er en kategorisk ækvivalens mellem affine k-skemaer og kommutative k-algebraer, hvilket vil sige, at ...
    • 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 ...