This talk will introduce the notion of categorical rigidity and the automorphism class group of a category. We will proceed with calculations for several important categories, hopefully illuminating the inverse relationship between the automorphisms of a category and the extent to which the structure of its objects is determined categorically. To conclude, some discussion of what progress there is on currently open/unknown cases.

# Past Junior Algebra and Representation Seminar

For a prime number p, we will consider completed group algebras, or Iwasawa algebras, of the form kG, for G a complete p-valued group of finite rank, k a field of characteristic p. Classifying the ideal structure of Iwasawa algebras has been an ongoing project within non-commutative algebra and representation theory, and we will discuss ideas related to this topic based on previous work and try to extend it. An important concept in studying ideals of group algebras is the notion of controlling ideals, where we say a closed subgroup H of G controls a right ideal I of kG if I is generated by a subset of kH. It was proved by Konstantin Ardakov in 2012 that for G nilpotent, every faithful prime ideal of kG is controlled by the centre of G, and it follows that the prime spectrum of kG can be realised as the disjoint union of commutative strata. I am hoping to extend this to a more general case, perhaps to when G is solvable. A key step in the proof is to consider a faithful prime ideal P in kG, and an automorphism of G, trivial mod centre, that fixes P. By considering the Mahler expansion of the automorphism, and approximating the coefficients, we can examine sequences of bounded k-linear functions of kG, and study their convergence. If we find that they converge to an appropriate quantized divided power, we can find proper open subgroups of G that control P. I have extended this notion to larger classes of automorphisms, not necessarily trivial mod centre, using which this proof can be replicated, and in some cases extended to when G is abelian-by-procyclic. I will give some examples, for G with small rank, for which these ideas yield results.

In recent years, Ardakov and Wadsley have been interested in extending the classical theory of Beilinson-Bernstein localisation to different contexts. The classical proof relies on fundamental geometric properties of the dual nilcone of a semisimple Lie algebra; in particular, finding a nice desingularisation of the nilcone and demonstrating that it is normal. I will attempt to explain the relationship between these properties and the proof, and discuss some areas of my own work, which focuses on proving analogues of these results in the case where the characteristic of the ground field K is bad.

In this talk, I wish to address the problem of evaluating an integral on an n-dimensional complex vector space whose n-form of integration has poles along a union of (affine) hyperplanes, following the work of Heckman and Opdam. Such situation arise often in the harmonic analysis of a reductive group and when that is the case, the singular hyperplane arrangement in question is dictated by the root system of the group. I will then try to explain how we can relate the intersection lattice of the hyperplane arrangement with nilpotent orbits of a complex Lie algebra related to the root system in question.

Abstract: Triangulations of surfaces serve as important examples for cluster theory, with the natural operation of “diagonal flips” encoding mutation in cluster algebras and categories. In this talk we will focus on the combinatorics of mutation on marked surfaces with infinitely many marked points, which have gained importance recently with the rising interest in cluster algebras and categories of infinite rank. In this setting, it is no longer possible to reach any triangulation from any other triangulation in finitely many steps. We introduce the notion of mutation along infinite admissible sequences and show that this induces a preorder on the set of triangulations of a fixed infinitely marked surface. Finally, in the example of the completed infinity-gon we define transfinite mutations and show that any triangulation of the completed infinity-gon can be reached from any other of its triangulations via a transfinite mutation. The content of this talk is joint work with Karin Baur.

Abstract: We define the Hochschild complex and cohomology of a monoid in an Ab-enriched monoidal category. Then we interpret some of the lower dimensional cohomology groups and discuss when the cohomology ring happens to be graded-commutative.

Firstly I will outline Dirac cohomology for graded Hecke algebras and the branching rules for the projective representations of $S_n$. Combining these notions with the Jucys-Murphy elements for $\tilde{S}_n$, that is the double cover of the symmetric group, I will go through a method to completely describe the spectrum data for the Jucys-Murphy elements for $\tilde{S}_n$. If time allows I will also explain how this spectrum data gives rise to a a concrete description for the matrices of the action of $\tilde{S}_n$.