N3.12

Quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ are certain Drinfeld quantum affinizations of quantum groups associated to affine Lie algebras, and can therefore be thought of as `double affine quantum groups'.

In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. 

Utilising an extended double affine braid group action, Miki obtained in type $A$ an automorphism of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchanges these subalgebras. This has since played a crucial role in the investigation of its structure and representation theory.

In this talk I shall present my recent work -- which extends the braid group action to all types and generalises Miki's automorphism to the ADE case -- as well as potential directions for future work in this area.

The crepant resolutions of a singular threefold are related by a finite sequence of birational maps called flops. In the simplest cases, this network of flops is governed by simple combinatorics. I will begin the talk with an overview of flops and crepant resolutions. I will then move on to explain how their underlying combinatorial structure can be abstracted to define the notion of a cluster category.

Let $G$ be a $p$-adic Lie group. From the perspective of $p$-adic manifolds, possibly the most natural $p$-adic representations of $G$ to consider are the locally analytic ones.  Motivated by work of Pan, when $G$ acts on a rigid analytic variety $X$ (e.g., the flag variety), we would like to geometrise locally analytic $G$-representations, via a covariant localisation theory which should intertwine Schneider-Teitelbaum's duality with the $p$-adic Beilinson-Bernstein localisation. I will report some partial progress in the simplified situation when we replace $G$ by its germ at $1$. The main ingredient is an infinite jet bundle $\mathcal{J}^\omega_X$ which is dual to $\widehat{\mathcal{D}}_X$. Our "co"localisation functor is given by a coinduction to $\mathcal{J}^\omega_X$. Work in progress.

Perverse sheaves are an indispensable tool in geometric representation theory that can be used to construct representations and compute composition multiplicities. These ‘sheaves’ live in a certain $\ell$-adic derived category. In this talk we will discuss a beautiful construction of this category based on the pro-étale topology and explore some applications in representation theory.

We study an infinite family of Massive Type IIA backgrounds that holographically describe the twisted compactification of N=(1,0) six-dimensional SCFTs to four dimensions. The analysis of the branes involved suggests a four dimensional linear quiver QFT, that deconstructs the theory in six dimensions. For the case in which the system reaches a strongly coupled fixed point, we calculate some observables that we compare with holographic results. Two quantities measuring the number of degrees of freedom for the flow across dimensions are studied.

For a family of finite sets whose cardinalities are naturally called class numbers, the Gauss problem asks to determine the subfamily in which every member has class number one. We study the Siegel moduli space of abelian varieties in characteristic $p$ and solve the Gauss problem for the family of central leaves, which are the loci consisting of points whose associated $p$-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $4$. This geometric Gauss problem is closely related to an arithmetic Gauss problem for genera of positive-definite quaternion Hermitian lattices, which we also solve.

The classification of anti de Sitter black holes is an open problem of central importance in holography. In this talk, I will present new advances in classification of supersymmetric solutions to five-dimensional minimal gauged supergravity. In particular, we prove a black hole uniqueness theorem within a ‘Calabi-type’ subclass of solutions with biaxial symmetry. This subclass includes all currently known black hole solutions within this theory.

We will finish off the term with a discussion of allies and community. You are encouraged to bring a friend who has never been to Mathematrix before!

We will be joined by Frances Kirwan to talk about the various careers available for Mathematicians and how to be a successful applicant.

We will be joined by Charlotte Turner-Smith to discuss issues surrounding harassment and mental health, and how the department is helping to tackle these.