# Hodge Club

The Hodge Club is the seminar for Hodge Institute graduate students and postdocs. That means we're interested in Algebra, Geometry, Topology, Number Theory, and all possible combinations and derivations of the four. Before the 2016/17 academic year, the Hodge Club was known as the Geometry Club.

We meet every **Friday at 16:00**, where we take it in turns to present a topic of interest to the rest of the group. We hope to run this as a **hybrid seminar**, so in particular we will have both an in person and virtual audience. If you are attending in person, we will meet at **JCMB room 5323**. If you are attending virtually, we will **send an email with the Zoom link every week**. If you do not receive the weekly emails and would like to be added to the mailing list, please get in touch with one of the organisers.

Talks tend to be fairly informal and provide excellent practice for conference talks in front of a friendly audience. You can find our current schedule and a historical list of talks below.

The Hodge Club for the 2021/22 academic year is organised by Lucas Buzaglo and Hannah Dell.

### Current Schedule of talks for 2021/22

#### Semester 2

28th January | Lucas Buzaglo | Title TBC |

Abstract: TBC
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4th February | Šarūnas Kaubrys | Title TBC |

Abstract: TBC
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11th February | Speaker TBC | Title TBC |

Abstract: TBC
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18th February | Karim Réga | Title TBC |

Abstract: TBC
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25th February | Speaker TBC | Title TBC |

Abstract: TBC
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4th March | Theodoros Lagiotis | Title TBC |

Abstract: TBC
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11th March | Speaker TBC | Title TBC |

Abstract: TBC |

#### Semester 1

1st October (Bayes) | Sebastian Schlegel Mejia | An interactive stroll towards the E-polynomial of the moduli stack of rank two degree zero Higgs bundles |

Abstract: The subject of my talk is the calculation of the E-polynomial of the moduli stack of rank two degree zero Higgs bundles. However, the aim of the talk is not to reach the final calculation nor to give super fancy reasons why you should care about the calculation. Instead, we focus on giving a feeling for the concepts and techniques involved in the calculation. These include (moduli of) Higgs bundles, stacks, motivic measures, and lambda-rings. All kinds of questions and interruptions are strongly encouraged and will be seen as pleasant diversions on our field trip through the land of Higgs bundles and its surroundings.
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8th October (Bayes) | Augustinas Jacovskis | Geometry from derived categories |

Abstract: A lot of geometric information about a variety X can be recovered from its derived category D(X). If the variety is Fano, then X can in fact be reconstructed up to isomorphism from D(X). This begs the question of whether less information than D(X) can determine X up to isomorphism. In this talk I’ll discuss some known cases when “less information” means a certain subcategory of D(X) called the Kuznetsov component. Time permitting, I’ll discuss joint work with Zhiyu Liu and Shizhuo Zhang which describes the situation for index 1 Fano threefolds.
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15th October (Online) | Guy Boyde (University of Southampton) | Homotopy groups, and how they grow |

Abstract: Homotopy groups are an important invariant of topological spaces – loosely, the n-th homotopy group is a picture of the space taken from the point of view of the n-dimensional sphere. Unfortunately, they are incredibly hard to compute – even for a friendly example like the two-dimensional sphere, we do not know all of them. We might therefore like to ask a coarser question. The one I am going to ask in this talk is “what can be said about the behaviour of the homotopy groups as n goes to infinity?” Asking this question reveals lots of interesting structure, and that’s what I’ll aim to tell you about, starting with the basics and including lots of examples.
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22nd October (Bayes) | Patrick Kinnear | An Invertible Sheaf on the Character Stack? |

Abstract: It is known that the skein algebra of a surface defines a sheaf of algebras on the coordinate ring of the character variety of the surface, which is in a sense the moduli space of local systems on the surface. It is also known that this sheaf satisfies an invertibility property over an open and dense subset of the character variety. Another way to describe local systems on a surface is via the character stack, and we can ask if there is a stacky version of the above invertibility statement. In this talk we will outline how such a statement can (hopefully!) be obtained from the study of invertible morphisms in the Morita category Alg_3, and will mention recent results characterising invertibility in this category. Along the way we will describe the powerful technology of factorization homology, which connects these two viewpoints.
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29th October (JCMB) | Lucien Hennecart | Categorification of Hall algebras |

Abstract: Hall algebras are now ubiquitous objects of geometric representation theory. The rough idea is to build algebra structures on constructible functions, cohomology and K-theory of moduli spaces of objects in certain categories and to study the algebras one obtains. To define them (and to be able to study them), one needs to put strong conditions on the categories under consideration and the examples one should keep in mind are the categories of representations of a quiver or of coherent sheaves on a smooth projective curve (at the 1-dimensional level); modules over the preprojective algebra of a quiver or compactly supported coherent sheaves on a smooth quasiprojective surface (at the 2-dimensional level); modules over the Jacobi algebra of a quiver with potential or coherent sheaves on a Calabi-Yau threefold (at the 3-dimensional level). We will try to clarify these constructions and also show examples and modern questions related to this subject.
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5th November (Online) | Ben Brown | A Survey of Hyperkähler and Hypertoric Geometry |

Abstract: Hyperkähler manifolds are special classes of Kähler manifolds, with three complex structures obeying the algebraic identities of the quaternions, and compatible with a Riemannian metric. These constraints or symmetries give rise to special characteristics of the hyperkähler structure, e.g. the manifolds are 4n-dimensional, its holonomy lies within Sp(k), is Ricci-flat (thus Calabi-Yau), and it is complex-symplectic. It is therefore more difficult to find examples of hyperkähler manifolds than Kähler ones; indeed, there are only two known classes of compact ones, namely the 4-torus and K3 surfaces, yet any complex submanifold of CPⁿ is automatically Kähler. On the other hand, there are many non-compact ones, e.g. the moduli spaces of Higgs bundles, of Nahm equations, and of monopoles, as well as complex cotangent bundles, to name but a few. A construction that yields a vast array of examples of hyperkähler manifolds is that of the hyperkähler quotient, which is the hyperkähler analogue of symplectic reduction. Many of the previous examples can be constructed this way, and the quotient method also establishes various global properties of the quotient. There are several other interesting examples of hyperkähler manifolds that can be constructed just by simply taking the hyperkähler quotient of the flat quaternionic vector space, Hⁿ, with respect to the action of a compact Lie group, G. In this talk, I wish to provide an overview of hyperkähler manifolds and their respective quotient operation, as a prerequisite to introducing toric hyperkähler manifolds, a.k.a. hypertoric manifolds. Hypertoric manifolds are essentially the hyperkähler analogues of toric manifolds and can be studied both geometrically and combinatorially, via hyperplane arrangements in the image of the hyperkähler momentum mapping. This combinatorial description also provides a simple recipe for constructing hyperkähler manifolds (and orbifolds) and as well as identifying their Kähler submanifolds.
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12th November (Bayes) | Alyosha Latyntsev (University of Oxford) | BPS states, vertex algebras and torus localisation |

Abstract: To understand the ideas coming from string theory, mathematicians have noticed the importance of moduli spaces (of objects in a category). By this method the physics notions of 1. BPS states/D-branes, 2. a conformal field theory, 3. \Pi stability, ... have been turned into 1. cohomological Hall algebras, 2. vertex/chiral algebras, 3. Bridgeland stability conditions, ... , all of which are extremely rich mathematical objects. In this talk, I will a. explain what a cohomological Hall algebra and vertex algebra is (and why you should care), b. sketch their physics analogues (no physics knowledge required!), c. show how they are connected. The main tool used to prove c. is a new version of the torus localisation which works for singular/derived spaces. Loosely, torus localisation "turns geometry into combinatorics" and is an important tool in Gromov Witten theory, toric geometry, and Donaldson Thomas theory. If time allows I will explain how this works. This is a talk based on my recent paper.
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19th November (JCMB) | Andrew Beckett | Infinitesimal automorphisms of Cartan geometries and filtered deformations |

Abstract: A Cartan geometry is manifold which looks locally like a Klein geometry, ie. like a (connected) space of the form G/H, where G is a Lie group and H is a closed subgroup of G. Cartan geometries generalise a number of different types of geometric structure, including Riemannian and symplectic geometry. The automorphisms of a Cartan geometry can be described infinitesimally by a finite-dimensional Lie algebra, and in the case of a "regular" Cartan geometry, this algebra has a particular structure: it is filtered, and its associated graded algebra is a graded subalgebra of Lie(G) - we say that it is a "filtered subdeformation" of Lie(G). In this talk, I will give an overview of these structures and discuss how a particular cohomology of graded Lie (super)algebras, Spencer cohomology, can help us to understand filtered deformations. If there is enough time, I will say a little about my own work applying this to the infinitesimal automorphisms of some solutions of supergravity theories in physics.
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26th November (Bayes) | Vivek Mistry | Cohomological Hall algebras of character varieties |

Abstract: Cohomological Hall algebras (CoHAs) provide an interesting refinement of Donaldson-Thomas invariants for objects in a 3-CY category. In this talk I will give a brief introduction to CoHAs, focussing mainly on the case of the cohomology of the stack of representations of quivers and Jacobi algebras. I'll then explain a 2d version of this story for character varieties, with the end-goal to compare two CoHA multiplications and explain why they are in fact equal.
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10th December (JCMB) | Jeff Hicks | An introduction to tropical geometry |

Abstract: One of the main ideas of tropical geometry is that much of geometry of complex sub-varieties of the algebraic torus, (C*)ⁿ, can be reduced to understanding combinatorics of piecewise linear objects in Rⁿ. These are the central object of study in tropical geometry. In this talk, we'll see why we might expect such a degeneration from complex geometry to tropical geometry, and look at how (assuming that this degeneration holds) we can obtain the degree-genus formula for curves in P². Then we'll see in what ways tropical geometry fails to capture aspects of complex geometry. |

### Historical schedules

Hodge Club 2020/21

Hodge Club 2019/20

Hodge Club 2018/19

Hodge Club 2017/18

Hodge Club 2016/17

Geometry club 2015/16

Geometry club 2014/15

Geometry club 2013/14

Geometry club 2012/13

You can also visit the old Geometry club website for more historical schedules.