# Hodge Club

Here is the schedule of talks for Hodge Club in the 2023-2024 academic year. The main page for Hodge Club is here.

## Semester 1

29th September | Adrián Doña Mateo | Magnitude, magnitude homology, and Euclidean sets |

Abstract: Magnitude is an invariant of enriched categories that generalises some well-known concepts (such as Euler characteristic), but also specialises to previously unknown ones (such as the magnitude of a metric space). Magnitude homology is a homology theory that, under favourable conditions, categorifies magnitude, and hence provides an algebraic perspective of this invariant. I will begin by introducing both of these notions, with some examples. In the second half, I will talk about joint work with Tom Leinster about the magnitude homology of subspaces of Euclidean space. In particular, I will give a geometric characterisation of when two closed subsets of R^n are ‘magnitude homology equivalent’. | ||

6th October | Parth Shimpi | What does the A2 singularity have in common with an octopus? 🐙 |

Abstract: Bridgeland (2007) showed that the stability manifold of a Kleinian surface singularity is a Galois cover of the hyperplane complement of the same ADE type, thus adding another link in the network of ideas connecting algebraic geometry, homological algebra, representation theory, and geometric group theory. In this talk I will sketch some of these ideas, and exhibit a computation which adds marine biology to this network as promised in the title. | ||

13th October | Lucas Buzaglo | A classification of subalgebras of the one-sided Witt algebra |

Abstract: We give a classification of Lie subalgebras of the one-sided Witt algebra. As applications of the classification, we prove that universal enveloping algebras of these infinite-dimensional subalgebras are not noetherian, and one case of a conjecture of Kaiming Zhao with links to the Jacobian conjecture. This is joint work with Jason Bell. | ||

20th October | Harvey Yau | When every place lets you down |

Abstract: Given a variety over Q, oftentimes looking modulo various primes can give valuable information about the distribution of rational points. But sometimes, no prime is sufficient to explain why there are no points on the variety. This talk will introduce the Brauer-Manin obstruction, a method that can be applied to prove the non-existence of rational points, and give examples of its applications to curves and surfaces. | ||

27th October | Patrick Kinnear | An invertible sheaf on the character stack! |

Abstract: In this talk I will describe the construction of an invertible sheaf of vector spaces (i.e. a line bundle) on the character stack of a 3-manifold, which is the moduli stack of G-local systems. In fact, this is just one part of a larger functorial structure called a TQFT which has a nice invertibility property. At the level of surfaces we obtain an invertible sheaf of categories over the character stack, which is expected to relate to a sheaf of algebras whose global sections are the skein algebra of the surface. Invertibility of the sheaf of categories should relate to an invertibility property of the skein algebra called being Azumaya. I will explain how these constructions follow from algebraic statements about braided tensor categories which we regard as local statements, and can be "integrated" to statements for general manifolds via technologies such as the cobordism hypothesis and factorization homology. | ||

3rd November | David Cueto Noval | A Cluster-Theoretic Approach to Quantum Multiplicative Quiver Varieties |

Abstract: Quiver varieties have been studied for decades and have found numerous applications including to instanton moduli spaces. After reviewing some basic concepts in the classical setting, we will discuss the notion of quantum multiplicative quiver varieties introduced by D. Jordan. We believe these algebras can be studied via quantum cluster algebras and explain a possible approach. | ||

10th November | Luke Naylor | Conway's approach to symmetry |

Abstract: When I first saw the classification of wallpaper groups, it was a very algebraic approach. It involved considering the translation subgroup, the point group, and ways they can fit together. However the approach introduced by Conway in "the symmetry of things" is, instead of considering the groups themselves, to consider the space (which has the symmetries) quotiented by the group of symmetries giving so called orbifolds. This method generalises well to spherical and hyperbolic symmetry and gives clean picturesque ways of classifying certain types of tilings. This talk is a 50 minute picturesque exposition of the topics in this area. | ||

17th November | Hannah Dell | An etymological tour through mathematics |

Abstract: Have you ever wondered what a derived category has in common with a river? Or how corollaries and flower crowns are related? In this talk we’ll explore the meaning and origin behind some of the mathematical terms that Hodge club members use day to day. If you have any suggestions of terms you’d like me to look into, then please fill in this form: https://forms.office.com/e/CUS2HDXfvw. | ||

24th November | Shivang Jindal | TBD |

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1st December | Misha Schmalian | TBD |

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8th December | Simone Castellan | TBD |

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