N3.12

The Dedekind zeta function generalises the Riemann zeta
function to other number fields than the rationals. The analytic class number
formula says that the leading term of the Dedekind zeta function is a
product of invariants of the number field. I will say some things
about the class number formula, about L-functions, and about Stark's
conjecture which generalises the class number formula.

We will discuss the pros and cons of targets vs quotas in increasing diversity in Mathematics.

We will be joined for lunch by the undergraduate society for women and non-binary students in Maths. Sign up here: https://forms.gle/q3uywCN3Dn78Kvkq6. Note: this event is only open to women and non-binary students.

We will be joined by Professor Kobi Kremnizer, who is a trained mental health first-aider, to discuss ways to protect your mental health this season.

In the theory of relative algebraic geometry, our affines are objects in the opposite category of commutative monoids in a symmetric monoidal category $\mathcal{C}$. This categorical approach simplifies many constructions and allows us to compare different geometries. Toën and Vezzosi's theory of homotopical algebraic geometry considers the case when $\mathcal{C}$ has a model structure and is endowed with a compatible symmetric monoidal structure. Derived algebraic geometry is recovered when we take $\mathcal{C}=\textbf{sMod}_k$, the category of simplicial modules over a simplicial commutative ring $k$.

In Kremnizer et al.'s version of derived analytic geometry, we consider geometry relative to the category $\textbf{sMod}_k$ where $k$ is now a simplicial commutative complete bornological ring. In this talk we discuss, from an algebraist's perspective, the main ideas behind the theory of relative algebraic geometry and discuss briefly how it provides us with a convenient framework to consider derived analytic geometry. 

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.