The affine and periodic Temperley–Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient, and construct a family of Wenzl–Jones idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that the uncoiled algebras are sandwich cellular and sketch some of the applications of the objects we defined. This is joint work with Alexi Morin-Duchesne.

Cluster algebras and their quantum counterparts were invented in the early 2000s in an attempt to construct elements of dual canonical bases. This turned out to be a harder goal than first realised. In this talk I will aim to give an introduction and overview of the theory and display the wide range of interesting maths which has gone into making steps in this area. I will try to assume as little possible prior knowledge and instead focus on interesting questions which remain open in this area.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Akshay Hegde, supervisor Dmitry Beylaev

Julius Villar, supervisor Dmitry Beylaev

Csaba Toth, supervisor Harald Oberhauser 

Let $(M, g)$ be a Riemannian manifold equipped with a calibration $k$-form $\alpha$. In earlier work with Cheng and Madnick (AJM 2021), we studied the analytic properties of a special class of $k$-harmonic maps into $M$ satisfying a first order nonlinear PDE, whose images (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith immersions, as they were originally introduced in an unpublished preprint of Aaron Smith. They have nice properties related to conformal geometry, and are higher-dimensional analogues of the $J$-holomorphic map equation. In new joint work (arXiv:2311.14074) with my PhD student Anton Iliashenko, we have obtained analogous results for maps out of $M$. Slightly more precisely, we define a special class of $k$-harmonic maps out of $M$, satisfying a first order nonlinear PDE, whose fibres (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith submersions. I will give an introduction to both of these sets of equations, and discuss many future questions.