# 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 2:30pm in JCMB 5323 or in the ICMS Lecture Theatre in the Bayes Centre where we take it in turn to present a topic of interest to the rest of the group. 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.

Future events are circulated on our mailing list and advertised on the Graduate School calendar. See instructions below on how to join our mailing list.

The Hodge Club is organised by Ruth Reynolds and Trang Nguyen.

### Current Schedule of talks for 2018/19

##### Semester 2

January 25, 2:30pm | Ruth Reynolds | Artin's Conjecture |

Abstract: One of the major achievements of noncommutative ring theory in the 20th century was the classification of all connected, graded domains of GKdim 2 (so-called noncommutative projective curves) by Artin and Stafford in 1995. In this talk we will describe this classification along with progress towards a broader classification of rings of higher GK dimension. In particular, we will describe Artin’s conjecture which, if proved, would significantly advance the classification. | ||

February 15, 2:30pm | Nivedita Viswanathan | Playing with Singular Curves |

Abstract:What are singular curves? Once we are convinced that a particular curve is singular, can we produce another curve with a singularity that is worse than what we started with? In this talk, I will explore various singular curves on a smooth surface and identify the worst possible such curves using an invariant called the log canonical threshold. I will firstly introduce you to the various definitions and notions of Algebraic Geometry that I would be needing for this venture of mine and then take you along this journey of classifying the singular curves using this invariant. | ||

March 1, 2:30pm | Ben Brown | Heisenberg Invariant Elliptic Curves |

Abstract: In this talk, we shall investigate some of the models for an elliptic curve by using theta functions.Particular families of theta functions generate an n-dimensional graded vector space, whose grading is determined by invariance of the n-torsion subgroup. For n > 2, the functions provide an embedding into projective (n-1)-space, whose image is either the Hesse pencil when n=3 or a set-theoretic intersection of n(n-3)/2 quadrics for n>3. The n-torsion subgroup acts on projective (n-1)-space via a projective representation, that can be lifted to a linear representation of the Heisenberg group in the Schrӧdinger representation. Finally, we shall look at how the automorphism group of the Heisenberg group acts upon the families of elliptic curves when n=3,5, where the singular members demonstrate tetrahedral and icosahedral symmetries, respectively. This talk is based on my Master’s dissertation. | ||

March 8, 2:30pm | Emily Roff | Diversity: developments in the biology of mathematics |

Abstract: In 1968, Gian-Carlo Rota wrote that "The lack of real contact between mathematics and biology is either a tragedy, a scandal, or a challenge, it is hard to decide which”. Over the past fifty years mathematicians and biologists have risen to this challenge (or scandal, or tragedy) and today the field of mathematical biology is extraordinarily active, meeting algebra, geometry and topology as well as analysis and statistics. This talk will focus on one particular point of encounter, arguing that "real contact" - that is to say, conceptual exchange - is now taking place between abstract mathematics and biology in the study of ecological diversity. Reviewing work by Leinster, Cobbold and Meckes, I will explain how the Euler characteristic of an enriched category is related to the diversity of an ecological community. Introducing recent developments in the theory of diversity, I'll show how an analogy between metric spaces and collections of species leads to a new numerical invariant of compact metric spaces, and a new candidate for the role of `uniform distribution' on an arbitrary compact metric space. | ||

March 15, 2:30pm | Will Reynolds | Lattice Polygons and the Number 12 |

Abstract: A curious class of convex lattice polygons in the plane seem to have a mysterious relationship with the number 12. A proof provided by Poonen and Rodriguez-Villegas in 2000 attempts to explain the mystery by introducing, of all things, a certain modular form (of weight 12)! In this talk I will explain this proof, tell you some reasons why lattice polygons should be on your radar, and discuss more recent generalisations. | ||

March 29, 2.30pm | Sjoerd Beentjes | Heisenberg algebra and Hilbert schemes of points on projective surfaces |

Abstract: Let S be a smooth projective variety. The Hilbert scheme Sn of n points on S parametrises n-tuples of points in S with multiplicities. In stark contrast with the arbitrarily singular Hilbert schemes of higher dimensional varieties, Sn is smooth when S is a surface. This fact is at the basis of a surprising connection between geometry and representation theory that emerges when studying the Sn for all n at the same time. Indeed, Göttsche showed that the Betti numbers of the Sn fit into a nice generating series. This is a reflection of an additional structure on the direct sum of the cohomologies of all the Sn: they form an irreducible representation of a Heisenberg algebra. Nakajima gave a beautiful geometric construction of this action via raising operators (adding a point to go from Sn to Sn+1) and lowering operators (acting as expected). In this talk, I will review these facts in some detail and describe the ingredients of Nakajima’s construction. | ||

April 5, 2.30pm | Vivek Mistry | An Introduction to Stacks |

Abstract: Stacks are an extension of the concept of a scheme in much the same way that schemes generalise varieties. Classifying geometric objects on a scheme is an important task, but the so called moduli problem is often not representable by a scheme. This is usually due to the objects having non-trivial automorphisms. But by using this more general notion of a stack we can sometimes represent these functors with an appropriate moduli stack instead. In this talk I will go over things like moduli problems, etale morphisms and Grothendieck (pre)topologies in order to introduce the notion of a stack, and then focus on examples of some of the more important stacks (hopefully with as little algebraic geometry as needed!). |

##### Semester 1

September 28, 1:45pm | Ruth Reynolds | Idealisers and Rings of Differential Operators |

In this talk we describe the ring of differential operators on an algebraic variety which was introduced by Grothendieck. We will also describe some of the theory of idealisers and how they link to ring of differential operators. In addition to this we will provide some illustrative examples from Weyl algebras. | ||

October 5, 1:45pm | Trang Nguyen | The Bruhat-Tits building of a p-adic Chevalley group. |

Abstract: In this talk we introduce the Bruhat-Tits building of a simply-connected Chevalley group defined over a finite extension of a p-adic field. We also discuss applications to representation theory if time permits. | ||

October 12, 1:45pm | Tim Weelinck | Fourier analysis and the Langlands program |

Abstract: We will take a representation theoretic perspective on Fourier analysis, discussing Pontryagin duality as well as some classical harmonic analysis. At the center of this we will find the question: what is the correct notion of `dual group'. With this viewpoint we then move to the Langlands program, and try to understand it as a non-abelian, algebraic and arithmetic version of Fourier analysis. | ||

October 19, 1:45pm | Carlos Zapata-Carratala | A Guided Tour to Local Lie Algebras |

Abstract: In this talk I will present an overview of local Lie algebras - a large class of Lie algebras containing some examples of great relevance in modern differential geometry and mathematical physics, e.g. Lie algebroids and Jacobi structures - motivating their definition, commenting on the problem of defining a well-defined notion of homomorphism and presenting some recent result about their integrability, in a Lie-theoretic sense. | ||

October 26, 1:45pm | Alexander Shapiro | A brief introduction to cluster algebras. |

I will start with discussing totally positive matrices, and how they lead to two different types of cluster coordinates. I will then say a few words on the relation of this story to Poisson geometry | ||

November 2, 1:45pm | NO MEETING | |

November 9, 1:45pm | Matt Booth | Deformation Theory |

Abstract: Deformation theory can be thought of as the study of infinitesimal thickenings of geometric or algebraic objects. Equivalently, it is the study of infinitesimal neighbourhoods in moduli spaces. In this talk, I'll provide an introduction to some deformation-theoretic concepts. I'll explain Deligne's philosophy that in characteristic zero, deformation problems are `controlled' by differential graded Lie algebras (dglas). I'll talk about derived deformation theory, and how it clarifies some concepts within the classical theory, in particular the proofs of Lurie and Pridham that dglas are equivalent to derived moduli problems via Koszul duality. Time permitting, I'll talk about an application of noncommutative deformation theory to birational geometry in the form of the Donovan-Wemyss contraction algebra, and I will mention how derived noncommutative deformations also fit into the picture. | ||

November 16, 1:45pm | Juliet Cooke | The Diamond Lemma and its Application to finding the Algebra of Invariants of the Factorisation Homology of the Four-Punctured Sphere |

Abstract: In 1978 Bergman wrote a paper which began 'The main results in this paper are trivial. But what is trivial when described in the abstract can be far from clear in the context of a complicated situation where it is needed.' I am going to talk about this trivial result, the Diamond lemma for ring theory, and how can be used in the complicated situation where I needed it: computing a presentation of the algebra of invariants of the factorisation homology of the four-punctured sphere. | ||

November 23, 1:45pm | Fatemeh Rezaee | A gentle introduction to "Geometric Invariant Theory" (GIT) : the notion of "quotient" in algebraic geometry |

Abstract: In this talk, I'm going to briefly introduce the notion of quotient in algebraic geometry with some examples to discover what kind of quotients could be interesting. If time permits, I will introduce the notion of quotient in symplectic geometry as well and compare it to the algebro-geometric one. |

### Historical schedules

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

### Mailing list

Announcements are handled by the mailing list hodgeclub. To subscribe, send a message to sympa at mlist.is.ed.ac.uk with the following content:

SUBSCRIBE hodgeclub QUIT