# 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 2:45pm** at the **Bayes Centre** (Room **5.02** or **5.45**) 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 Ben Brown and Trang Nguyen.

### Current Schedule of talks for 2019/20

#### Semester 2

31st January 2020, 2:45pm | Sebastian Schlegel Mejia | What Is... A CoHA? |

__Abstract__: Kontsevich and Soibelman defined their cohomological Hall algebra (CoHA) as an attempt to find a mathematical definition of the BPS algebra in physics. The study of CoHAs has become a subject of its own with applications to representation theory. In this talk, I will define the CoHA of a quiver and, if time permits, I will work out some explicit examples.

7th February 2020, 2:45pm | Kai Hugtenburg | Holomorphic Curves in Symplectic Topology |

__Abstract__: Many essential theorems in symplectic topology rely on the study of holomorphic curves. I will set up the basic theory, and discuss some of the problems arising in defining these things rigorously. I will then exhibit one of the main examples: Gromov's non-squeezing theorem. This has an unexpected physical application: no physical setup can concentrate light from a cylindrical aperture into a sufficiently small rectangular aperture.

14th February 2020, 2:15pm | Augustinas Jacovskis | Introduction to Bridgeland Stability Conditions |

__Abstract__: Giving stability conditions for sheaves on a variety allows one to form well-behaved moduli spaces of sheaves on the variety. Bridgeland stability is a generalisation of earlier notions of stability, for example slope stability and Gieseker stability. In this talk, I’ll introduce the construction of Bridgeland stability conditions on a variety, and if time permits, I'll discuss some applications to problems in algebraic geometry.

28th February 2020, 2:45pm | Carlos Zapata-Carratala | Dimensioned Categories and Units of Measurement in Classical Mechanics |

__Abstract__: In this talk I will briefly present the problem of the mathematisation of scientific units of measurement to motivate the definition of Dimensioned Categories (first indirectly introduced by Baez and Dolan in the context of algebraic geometry) and the generalisation of conventional commutative algebra and differential geometry within dimensioned categories. I will then present some results about Jacobi manifolds reformulated in the dimensioned language to connect to the problem of introducing units into the mathematical framework of classical mechanics.

6th March 2020, 2:45pm | Harry Gingi | Introduction to Simplicial Sets and ∞-Categories |

__Abstract__: Simplicial methods have been the backbone of combinatorial methods in algebraic topology since the 1950s, and since then they have permeated every part of category theory and even large areas of algebra. In the past two decades, we have discovered why this is indeed the case. All of our favorite simplicial methods are in fact reflections of an encoding of the theory of (∞,1)-categories into the theory of simplicial sets, from construction of classifying spaces to construction of derived categories.

Today's talk will is an introduction to the basics of the simplex category and what Cisinski calls the 'simplicial yoga'. We will move to explain several of the models of the homotopy of (∞,1)-categories, and if time permits, we will cover some very simple applications of the theory.

13th March 2020, 2:45pm | Jon Eugster | TBC |

__Abstract__:

3rd April 2020, 2:45pm | Guy Boyde (University of Southampton) | TBC |

__Abstract__:

#### Semester 1

4th October 2019, 2:45pm | Dougal Davis | A tale of three curious quotients and other adventures in geometry and representation theory |

__Abstract__: Once upon a time, there was a group named SL_2. This group had three daughters (or "adjoint quotient maps") called Additive, Multiplicative and Elliptic. The littlest child, Additive, loved representation theory, and had just one singular fibre that everyone admired. The next child, Multiplicative, resembled her little sister Additive, but had two singular fibres that were very slightly harder to look at than Additive's one. The eldest, Elliptic, was a more difficult child, who stood a bit apart from her sisters. She had a greater taste for geometry, had four singular fibres, and always seemed just a little unstable.

11th October 2019, 2:45pm | Trang Nguyen | Moduli space of parabolic Higgs bundles |

__Abstract__: In this talk, we discuss parabolic Higgs bundles and the geometry of their moduli space. We also introduce a global analogue of the Grothendieck-Springer resolution of Lie algebras which arises from the study of parabolic Higgs bundles.

18th October 2019, 2:45pm | Ruth Reynolds | The classification of noncommutative projective curves and an important conjecture in noncommutative ring theory |

__Abstract__: In 1995, Artin and Stafford classified all noncommutative graded domains of GK dimension 2 (so-called "noncommutative curves"). In this talk we describe this result and the progress made to classify rings of higher GK dimension (noncommutative projective surfaces). We will also talk about Artin's conjecture which is the obstacle to obtaining this classification.

25th October 2019, 2:45pm | Will Reynolds | Weighted Projective Spaces: An Advert for Toric Geometry |

__Abstract__: Weighted projective spaces are simple generalizations of the well-known "straight" projective spaces. They find use, for example, as ambient spaces in which to embed and study other (projective) varieties via their accompanying weighted homogeneous coordinates. I will talk about weighted projective spaces as toric varieties. Along the way I will review the elements of toric geometry, and demonstrate how weighted projective spaces in return provide motivation for some further results about toric varieties more generally.

1st November 2019, 2.45pm, 5.45 | Ben Brown | Symplectic Reduction, Geometric Invariant Theory, and the Kempf-Ness Theorem |

__Abstract__: For a smooth, complex projective variety X inside P^{n} with an action of a reductive linear algebraic group G, we can consider an algebro-geometric quotient of X by G via the means of geometric invariant theory (GIT) to construct a quotient variety X // G, which parameterises the well-behaved closed orbits of X under the G-action. On the other hand however, X is also naturally a symplectic manifold, and since G is reductive it has a maximal real compact Lie subgroup K of G and we can also consider the symplectic reduction of X by K, with respect to an appropriate moment map. The Kempf-Ness theorem says that the results of these two constructions are homeomorphic. In this talk I will define symplectic reduction and the GIT quotient constructions and discuss a few examples of the Kempf-Ness theorem in action.

8th November 2019, 1.30pm, 5.02 | Vivek Mistry | The Grothendieck ring of motives and motivic vanishing cycles |

__Abstract__: The theory of motives was introduced to try to find a unifying theory for all the various cohomology theories that occur in algebraic geometry. In this talk I will introduce one of these motivic theories, borne from the Grothendieck group of naive motives over a complex scheme, and then look at an application of this theory in terms of vanishing cycles and how they can be used to define interesting invariants for our algebro-geometric objects.

15th November 2019, 2.45pm, 5.45 | Emily Roff | Hochschild homology for enriched categories |

__Abstract__: A 2017 paper by Leinster and Shulman describes a Hochschild-esque homology theory for enriched categories. The theory yields, as special cases, classical Hochschild homology of associative algebras, group homology, and the comparatively novel magnitude homology of metric spaces. I want to understand this construction; I plan to subject you to my current best attempt, and I invite your penetrating questions.

22nd November 2019, 2.45pm, 5.02 | Sebastian Schlegel Mejia | What are the dimensions of the tangent spaces of Hilbert schemes of points? |

__Abstract__: The title is the guiding question in this hands-on introduction to Hilbert schemes of points of smooth varieties. I will focus on the Hilbert schemes of affine spaces which are particularly nice to work with as they carry an action of the torus coming from the standard scaling torus action on affine space.

Spoiler: The answers in dimensions one and two are boring. Dimensions greater than four are way to wild to hope for an answer. Some clues point to a possible answer in dimension three...

### Historical schedules

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

### 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