University of Oxford

How aware should we be of letting AI make decisions on prison sentences? Or what is our responsibility in ensuring that mathematics does not predict another global stock crash?

In this talk, Helena will outline how we can view ethics and responsibility as central to processes of innovation and describe her experiences applying this perspective to teaching in the Department of Computer Science. There will be a chance to open up discussion about how this same approach can be applied in other Departments here in Oxford.

Helena is an interdisciplinary researcher working in the Department of Computer Science. She works on projects that involve examining the social impacts of computer-based innovations and identifying the ways in which these innovations can better meet societal needs and empower users. Helena is very passionate about the need to embed ethics and responsibility into processes of learning and research in order to foster technologies for the social good.

C*-algebras provide non commutative analogues of locally compact Hausdorff spaces. In this talk I’ll provide a survey of the large scale project to classify simple amenable C*-algebras, indicating the role played by non commutative versions of topological ideas. No prior knowledge of C*-algebras will be assumed.

The d-dimensional bordism category Cob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. For d=1 and d=2 this category is well understood because we have a complete list of all 1 or 2-manifolds with boundary. In this talk I will argue that the categories Cob_1 and Cob_2 nevertheless carry a lot of interesting structure. 

I will show that the classifying spaces B(Cob_1) and B(Cob_2) contain interesting moduli spaces coming from the combinatorics of how 1 or 2 manifolds can be glued along their boundary. In particular, I will introduce the notion of a "factorisation category" and explain how it relates to Connes' cyclic category for d=1 and to the moduli space of tropical curves for d=2. If time permits, I will sketch how this relates to the curve complex and moduli spaces of complex curves.

In this session we will host a Q&A with current researchers who have recently gone through successful applications as well as more senior staff who have been on interview panels and hiring committees for postdoctoral positions in mathematics. The session will be a chance to get varied perspectives on the application process and find out about the different types of academic positions to apply for.

The panel members will be Candy Bowtell, Luci Basualdo Bonatto, Mohit Dalwadi, Ben Fehrman and Frances Kirwan. 

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.

Our perspective furnishes a common generalisation, essentially solving the problem. We also give a geometric criterion for Weyl group elements to yield strictly transverse slices.

We will discuss confinement in 4d N=1 theories obtained after soft supersymmetry breaking deformations of 4d N=2 Class S theories. Confinement is characterised by a subgroup of the 1-form symmetry group of the theory that is left unbroken in a massive vacuum of the theory. The 1-form symmetry group is encoded in the Gaiotto curve associated to the Class S theory, and its spontaneous breaking in a vacuum is encoded in the N=1 curve (which plays the role of Seiberg-Witten curve for N=1) associated to that vacuum. Using this proposal, we will recover the expected properties of confinement in N=1 SYM theories, and the theories studied by Cachazo, Douglas, Seiberg and Witten. We will also recover the dependence of confinement on the choice of gauge group and discrete theta parameters in these theories.

This is an elementary talk introducing the rational Cherednik algebra and its representations. Especially, we are interested in the case of complex reflection group. A tool called the Dunkl-Opdam subalgebra is used to decompose the standard modules into weight spaces and to construct the correspondence with the partitions of integers. If time allows, we might explore the concept of unitary representation and what condition a representation needs to satisfy to be qualified as one.

The well-known Schur-Weyl duality provides a link between the representation theories of the general linear group $GL_n$ and the symmetric group $S_r$ by studying tensor space $(\mathbb{C}^n)^{\otimes r}$ as a ${(GL_n,S_r)}$-bimodule. We will discuss a few variations of this idea which replace $GL_n$ with some other interesting algebraic object (e.g. O$_n$ or $S_n$) and $S_r$ with a so-called diagram algebra. If time permits, we will also briefly look at how this can be used to define Deligne's category which 'interpolates' Rep($S_t$) for any complex number $t \in \mathbb{C}$.