Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
I'll briefly explain quantum groups and $R$-matrices and why they're the same thing. Then we'll see how to construct various $R$-matrices from Nakajima quiver varieties and some possible applications.
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.