# 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 1

** 4th October, 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.

In this talk, I'll introduce you to these three sisters, and tell you about some of the adventures they went on together in the lands of algebraic geometry and representation theory.

** 11th October, 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, 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, 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, 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, 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, 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, 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