List of students

Current PhD students

1. Chris Milionis (started in September 2024)
   

Former MMath students in Newacstle

1. Rachel Foulkes: Knot invariants and braids (2025)
   
   The thesis gives a detailed account of the Jones polynomial via representations of the braid group on Temperly-Lieb algebras. The skein-theoretic description of the Jones polynomial is derived, and further properties of the Jones polynomial are discussed. The thesis concludes with a short section on the HOMFLYPT polynomial.
   

1. Ulukbek Akynbaev: Modular representation theory and the Verlinde category (2019).
   
   The thesis studies properties of the Verlinde category, the semisimplification of the representation category of the cyclic group Z_p over a field of characteristic p. At first the necessary background about monoidal categories and semisimplifications is developed. Among the main statements about Ver_p is a computation of the fusion rules and the conclusion that Ver_p does not admit a (super) fibre functor to vec or svec.


2. Elias Hedwig: The Picard group of a ring (2021).
   
   The thesis discusses the Picard group of a commutative ring R and its various incarnations: via the Grothendieck group, via Weil divisors or as the class group.


3. Benjamin Görg: Hopf algebras of rooted trees (2020).
   
   This thesis gives an overview about several Hopf algebras built from rooted (planar) trees, in particular the Connes-Kreimer Hopf algebra. After some preparations (e.g. the Milnor-Moore theorem) these Hopf algebras are defined and studied. The thesis concludes with the study of Pre-Lie structures on one of these Hopf algebras.


4. Nico Wolf: The big Witt ring functor (2020).
   
   The thesis presents the theory of the big Witt ring functor. Lambda rings and their properties are discussed, and then Witt polynomials and Witt vectors are studied. The constructions follows an idea of Lenstra. The link to Symm, the Hopf algebra of symmetric functions, is also explained.


5. Jan Thomm: Every abelian group is a class group (2020).
   
   The thesis gives a detailed proof of Claborn's classical theorem that every abelian group arises as the class group of a Dedekind ring. The main strategy is to first realize it as the class group of a Krull domain and then use some approximation argument to get to the Dedekind case.


6. Marena Richter: Classification of simple Lie superalgebras (2020)
   
   The thesis revisits and reproves Kac classification of basic classical Lie superalgebras from the 70's. To this end, Lie superalgebras are introduced. The main tool of the classification is a study of the action of the even part (a reductive Lie algebra) on the odd part of the Lie superalgebra.
   

Former Master students in Bonn

1. Lukas Bonfert: Translation Functors and Weight Diagrams for the Lie Superalgebra q(n) (2021) (joint with Catharina Stroppel).
   
   The thesis studies translation functors for the queer Lie superalgebra q(n). After some background like Brundan's description of translation functors via quantum group combinatoric, translation functors are studied in detail. At the end a description of the homomorphism spaces between indecomposable projective q(n) representations is derived.


2. Jonas Nehme: Tensor powers of the natural representation of OSp(r|2n) (2021) (joint with Catharina Stroppel)
   
   The thesis develops a theory of projective functors for the Khovanov algebra of type B. These functors correspond to translation functors when linking modules over the Khovanov algebra to representations of the algebraic supergroup OSp(m|2n). As an application one gets a description of the effect of translation functor on irreducible and projective representations and a description of the indecomposable summands in tensor powers of the natural representation.


3. Noah Schaumburg: An introduction to Nichols algebras (2022)
   
   The thesis gives an introduction to the theory of Nichols algebras. In the beginning background about braided monoidal categories and Hopf algebras in such categories is developed. Yetter-Drinfeld modules and Nichols algebras are introduced with a special focus on Nichols algebras over groups.


4. Georges Tyriard: Deligne categories of hyperoctahedral groups (2023)
   
   The thesis compares two different interpolation categories or Deligne categories which are attached to the hyperoctahedral group. One originates from Knop's approach and was further studied by Likeng and Savage, the other one is due to Flake and Maassen. The main theorem proves that there is an equivalence of monoidal categories between the two.


5. Bernhard Glauber: Atypical representations of Lie superalgebras (2024).
   
   The thesis gives a proof of Harish-Chandra's theorem in the setting of basic classical Lie superalgebras. After developing all the preliminary setup, the thesis follows loosely the original proof of Kac which was completed by Gorelik.