# 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 13:00pm**, where we take it in turn to present a topic of interest to the rest of the group via **Zoom**, in pertuity as the situation currently stands. 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.

We send out an initial email about the current week's talk, and follow it up closer to the time. **Details for accessing the virtual meeting will be contained in these emails**.

The Hodge Club for the 2020 - 2021 academic year is organised by Benjamin Brown and Sebastian Schlegel Mejia.

### Current Schedule of talks for 2020/21

#### Semester 2

15th January, 2021 | — | Social Meeting |

__Abstract__: N/A.

22nd January, 2021 | — | Social Meeting |

__Abstract__: N/A.

29th January, 2021 | Will Reynolds | Klyachko's Description of Toric Vector Bundles |

__Abstract__: Alexander Klyachko's ideas of the early '90s provide an equivalence of categories between toric vector bundles and a category of more combinatorial objects, namely vector spaces equipped with a finite family of filtrations satisfying a compatibility condition. This talk will review the basic ideas of toric geometry, and then explain the equivalence of categories as outlined in a later paper of Sam Payne.

5th February, 2021 | Hannah Dell | Counting Conics and Clemens’ Conjecture |

__Abstract__: We will start this talk by asking “how many planar conics are tangent to 5 given lines”. Despite being a simple question, trying to answer it will lead us to a complex problem called excess intersection that arises often in enumerative geometry. We will explore different ways to tackle this, focusing on the problem of counting rational curves on a quintic threefold, which is also a stunning example of an application of mirror symmetry.

12th February, 2021 | Kai Hugtenburg | Numbers bad, Algebra better, Categories best? |

__Abstract__: Hannah introduced us last week to Gromov-Witten invariants. I will pick up from there and show how these numbers can be packed into an algebraic structure called quantum cohomology. For a Fano projective variety X there is a striking relationship between semi-orthogonal decompositions of the category of coherent sheaves on X and the quantum cohomology of X. To excite the category theorists among us, I will talk a bit about the entropy of a functor. And if this all doesn’t make sense to you, fear not! I will spend a good chunk of the talk revisiting differential equations. Turns out there is more about linear, homogeneous differential equations than what they tell you in undergrad. Don’t believe your teachers when they say these are easy!

19th February, 2021 | — | Flexible Learning Week |

__Abstract__: N/A.

26th February, 2021 | Lucas Buzaglo | Moduli Spaces in Noncommutative Ring Theory |

__Abstract__: When studying noncommutative rings, it is often incredibly difficult to prove the most basic properties about them. We will see two related examples of moduli spaces and geometry appearing naturally, but perhaps unexpectedly, in the underlying structure of noncommutative graded rings.

Moduli spaces of point modules first came up when Artin and Van den Bergh got interested in Sklyanin algebras. They were suspected to be well-behaved rings, but Artin and Van den Bergh couldn't prove anything about them until they studied their point modules. A generalisation of these, called intermediate series modules, were introduced by Sierra and Špenko to prove that the universal enveloping algebra of the Witt algebra is not noetherian.

5th March, 2021 | — | Social Meeting |

__Abstract__: N/A.

12th March, 2021 | Ben Brown | Equivariant Localisation and Fixed-Point Formulae |

__Abstract__: Often in mathematics, we are tasked with the problem of evaluating an integral of a form over some space M. Depending on the form, this integral can be related to calculating volume, ﬁnding topological/enumerative invariants, or integrating characteristic classes. Such computations are often difﬁcult but two notions can simplify them, which are symmetry and localisation.

By symmetry, we mean that we have a group G acting on M, and by identifying orbits we reduce the problem to that over a smaller space, M/G; this comes up in symplectic reduction, gauge theory, and integrable systems. By localisation, this means that we reduce global calculations to local ones; for example, as in the Poincaré-Hopf theorem. Symmetry and localisation synergise together through the Atiyah-Bott-Berline-Vergne fixed-point formula - when we have a smooth manifold M together with the action of a compact, connected Lie group G, then the integral on M localises on the ﬁxed-point set of the G-action.

In this talk, I want to introduce the equivariant cohomology of a G-manifold via the Cartan model - that is, equivariant de Rham theory - before going through some example calculations including the equivariant Riemann-Roch-Hirzebruch, the Duistermaat-Heckman, and the Lefschetz fixed-point theorems.

19th March, 2021 | — | Social Meeting |

__Abstract__: N/A.

26th March, 2021 | Patrick Kinnear | Three views on 3-manifolds |

__Abstract__: While we are used to visualising 2-manifolds as embedded in 3 dimensional space, it is not clear how we can best view 3-manifolds. In this talk, we will survey three different approaches to working with 3-manifolds in a visual way. We will first introduce Heegaard diagrams, which describe 3-manifolds via curves in closed surfaces, and are the basis of Heegard-Floer homology. We will then discuss Dehn surgery, which relates 3-manifolds to knot theory. Finally, we will outline open book decompositions, which describe 3-manifolds using circle bundles and are closely related to contact geometry.

2nd April, 2021 | — | -- |

__Abstract__: --

9th April, 2021 | — | -- |

__Abstract__: --

#### Semester 1

2nd October, 2020 | — | Social & Introductory Meeting |

__Abstract__: N/A.

9th October, 2020 | Ben Brown | Compactifying Hypertoric Manifolds via Symplectic Cutting |

__Abstract__: A theorem of Delzant asserts a correspondence between a class of convex polytopes and symplectic toric manifolds, which lets us study them either geometrically or combinatorically. This construction can be extended to the hyperkähler case giving rise to hypertoric manifolds, first introduced by Bielawski and Dancer, and their analogues to the toric case forming the subject of Proudfoot’s thesis. A key difference however between the two cases is that the hypertoric ones are non-compact, which is reflected combinatorially too, with the half-space arrangements delimiting the polytopes are replaced by hyperplane arrangements. This non-compactness creates new problems if, for example, one wishes to consider the vector space of holomorphic sections on a hypertoric manifold for the purposes of geometric quantisation, given that this vector space is now infinite-dimensional.

This talk will go through how one can compactify these hypertoric manifolds via a construction called symplectic cutting, thus circumventing this dilemma. Firstly though we shall review the construction of the toric and hypertoric manifolds considered, along with their respective half-spaces and hyperplane arrangements. The compactification is also reflected in the hyperplane arrangement, by introducing half-spaces in a combinatorially manner, forming an assortment of polytopes that we call a polyptych. Now finally compact, we can apply localisation formulae to extract topological information about these compactified manifolds.

Slides for the talk can be found here.

16th October, 2020 | Augustinas Jacovskis | Cubic hypersurfaces and rationality |

__Abstract__: To ask whether a cubic hypersurface is rational (i.e. birational to projective space) is a natural and very deep question. We will survey the dimension 1,2, and 3 cases where the answer is known, as well as the elusive dimension 4 case which remains open to this day. Only basic algebraic geometry knowledge will be assumed.

23rd October, 2020 | Shizhuo Zhang | Low degree curves on GM-threefold and Kuznetsov's conjecture |

__Abstract__: Let X be a Gushel-Mukai threefold, we study the fano scheme of lines, conics and twisted cubics on X.

In particular, we show that the Fano scheme of twisted cubics on general special Gushel-Mukai threefold is an irreducible smooth projective threefold \cH. We show that the moduli space of Bridgeland stable objects of -2-class in the Kuznetsov component of X is a divisorial contraction of \cH. As a result, we show that the Kuznetsov components of quartic double solid is not equivalent to special GM threefold. If time permits, I will also show that the Kuznetsov component of quartic double solid is not equivalent to ordinary Gushel-Mukai threefold. This means that the Kuznetsov conjecture is disproved.

30th October, 2020 | — | Social Meeting |

__Abstract__: On The University of Edinburgh's GatherTown instance.

6th November, 2020 | Kai Hugtenburg | Hamiltonian Floer Homology |

__Abstract__: The Arnol’d conjecture predicts a lower bound for the number of 1-periodic orbits of a time dependent Hamiltonian on a symplectic manifold M. Floer proved this conjecture by considering the Morse homology of the loop space of M. I will give a brief overview of the ingredients in this theory. The main question I hope to answer is: why do symplectic topologists care about holomorphic curves?

13th November, 2020 | Matt Booth | An Introduction to Topological Hochschild Theory |

__Abstract__: I'll give an introduction to topological Hochschild (co)homology. Loosely, this is the same thing as usual Hochschild (co)homology, but one replaces the base ring with the sphere spectrum, which is a commutative ring in an appropriate sense. Time permitting, I'll indicate why you might be interested, with motivation from K-theory and from deformation theory. I'll assume a minimal familiarity with Hochschild theory or stable homotopy theory; in particular we'll spend some warm-up time discussing (usual, non-topological) Hochschild theory, and I'll also try to give a quick introduction to modern categories of spectra.

Slides for the talk can be found here.

20th November, 2020 | Federico Trinca | Riemannian Holonomy and the Gibbons-Hawking ansatz |

__Abstract__: The holonomy group of a Riemannian Manifold determines the geometrical structures on the manifold compatible with its Riemannian metric. In 1955, Berger classified all the groups that can appear as the holonomy of a Riemannian manifold. Kähler and Calabi-Yau manifolds are examples in this classification.

After a brief survey of Riemannian holonomy, we describe the spaces constructed via the Gibbons-Hawking ansatz, which form a family of non-compact Calabi-Yau 2-folds, together with the relative compact special Lagrangians.

27th November, 2020 | Jon Eugster | Condensed Mathematics |

__Abstract__: Condensed abelian groups are sheaves on the category of profinite sets. This additional structure allows algebraic objects like abelian groups to carry a topology and meanwhile form an abelian category. We give an introduction into this theory that combines topological and categorical phenomena and show some useful applications to algebraic geometry.

4th December, 2020 | Vivek Mistry | Calabi-Yau algebras, Donaldson-Thomas theory and fundamental group algebras |

__Abstract__: Classical Donaldson-Thomas (DT) theory was developed to count coherent sheaves on a Calabi-Yau 3-fold. Over the years it has been generalised to so called 3 Calabi-Yau categories giving invariants of moduli spaces of objects in these categories. I shall describe this Calabi-Yau notion in the case of algebras, give a brief overview of DT theory, and then look at the case of fundamental group algebras to see how one can approach DT theory for them.

### Historical schedules

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.