# Postgraduate Geometry & Algebra Seminar

## About

The purpose of this seminar is to allow postgraduate students in the Geometry and Algebra groups at Bath to give expositary talks on topics in these fields. The main theme is to present ideas well understood by experts in the appropriate communities, but perhaps less well understood by new graduate students (or experts in indirectly related areas), although talks with different aims may also be appropriate. For talks on current and ongoing research, see the Bath Geometry Seminar.

Each talk is approximately 50–60 minutes with time for questions.

## When and Where

Talks are held every other month or so in no fixed location. See the list of talks below for past times and locations.

## Upcoming Talks

Please email me if you are thinking of giving a talk!Bath is also a member of the TCC, which sometimes runs courses of interest to geometers and algebraists. The COW algebraic geometry seminar, and its postgraduate equivalent Calf, sometimes visit.

## Contact

If you have any questions about the seminar, would like to give a talk or would like to join the mailing list, please contact me.

## Past Talks

### 2017

10/05/17*Modular Tensor Categories in Conformal Field Theory*, by Leonard Hardiman. (14:15 in 1WN 3.11).

Much of my work is based on problems that arise out of Segal's mathematical formulation of conformal field theory. After briefly introducing this formalism I shall explain its relationship with modular tensor categories. As is traditional an A-D-E pattern will also be discussed.

### 2016

20/10/16*Monodromy of hyperkahler manifolds*, by Claudio Onorati. (11:15 in CB 4.5).

I will recall the definition of basic objects in the theory of irreducible holomorphic symplectic manifolds, including moduli spaces and monodromy operators. In particular the latter is the main character of this talk and I will try to give some evidence its importance. The last part of the talk is dedicated to explain the structure of the monodromy group of a particular example of irreducible holomorphic symplectic manifolds, namely the generalised Kummer varieties. If time permits I will do the same for the other main example, namely Hilbert schemes of points on a K3 surface.

07/03/16:*Einstein structures on manifolds and orbifolds*, by Christian Lund (Cambridge). (15:15 in 4W 1.7).

In this talk we study Einstein structures on manifolds and orbifolds. An Einstein structure is an equivalence class on the space of metrics. On a general compact manifold it is hard to say much about the structure of the moduli space, however, if we restrict our attention to K\"ahler manifolds, then it is locally an orbifold. We show how to get this from the study of Einstein deformations and we look at how to generalize this result to orbifolds.

### 2015

09/12/15:*Hyperkähler Manifolds*, by Claudio Onorati. (11:15 in 8W 1.33).

I will introduce Hyperkähler manifolds and give a geometric interpretation of such a class of manifolds. I will briefly recall some examples. Finally, I will introduce all we need to talk about the Local Torelli Theorem.

23/11/15:*In quest of a non-formal G_2 manifold*, by Alge Wallis. (13:15 in 8W 2.6).

Abstract: Does a manifold with special holonomy have to be formal? Answering the question to the affirmative has so far eluded mathematicians (to the author's knowledge). Thus the quest is on to find a counter example. Basics of formality and holonomy will be covered briefly. We examine the results that give us clues for where (mainly not) to look, such as Deligne et al that Kahler manifolds, which are precisely those with holonomy U(n), are formal. We introduce the technology available to us to conduct a meaningful search. That is, our constructor of manifolds with holonomy G_2, the twisted connected sum; and our formality detector, the Bianchi-Massey product. We conclude with our results to date.

02/11/15:*Geometry from a Holonomy Point of View*, by Claudio Onorati. (13:15 in 1W 2.01).

I will start by recalling the definition and basic properties of the holonomy group. After this I will explain why it is important to look at the holonomy group to get geometric information; this will include the decomposition theorem and the Berger classification. In conclusion I will give as many examples as possible (time permitting).

15/04/15:*Representation Theory of Associative Algebras: An Introduction*, by Matthew Pressland. (14:15, in 8W 2.5).

I will introduce some aspects of the representation theory of associative algebras. The main goal of the talk will be to introduce the \(\operatorname{Ext}\) functors (by popular request!) and use them to describe how any finite-dimensional unital associative algebra is Morita equivalent to a quotient of the path algebra of a uniquely determined quiver. I also plan to discuss some Auslander–Reiten theory for hereditary algebras, and explain how to compute the Auslander–Reiten quiver in the representation-finite cases.

13/01/15:*Combinatorial Reid's Recipe for Consistent Dimer Models*, by Jesus Tapia Amador. (15:15, in 1WN 3.24).

For \(G\) a finite abelian subgroup of \(\operatorname{SL}(3,\mathbb{C})\), Reid introduced a recipe to label the toric fan of the 'minimal' resolution \(Y\) of \(\mathbb{C}^3/G\) with the characters of \(G\). One of the main features of this recipe is that it encodes the relations of the line bundles in \(\operatorname{Pic}(Y)\). The singularity \(\mathbb{C}^3/G\) and its resolution \(Y\) are particular examples of three-dimensional singularities and resolutions coming from the study of consistent dimer models and moduli spaces of quiver representations. It is therefore natural to consider a generalisation of Reid's recipe to this larger set of examples. In this talk, I will introduce a combinatorial generalisation of Reid's recipe for consistent dimer models. If time allows, I shall also introduce a 'localisation' algorithm which could be used to prove that combinatorial Reid's recipe for consistent dimer models also encode the relations of the line bundles in \(\operatorname{Pic}(Y)\).

### 2014

10/12/14:*Mirror Symmetry, K3 Surfaces, and Modular Forms*, by Matthew Dawes. (14:15, in 6E 2.2).

The arithmetic mirror conjectures for K3 surfaces relate the algebraic geometry of families of K3 surfaces with arithmetic objects (lattices and modular forms). I will discuss some of these conjectures whilst giving an overview of mirror symmetry for K3 surfaces. The discussion will be motivated by specific examples and most of what I say will be based on work by Dolgachev, Gritsenko and Nikulin.

05/11/14:*Differential Geometry of Quaternionic Manifolds*, by George Frost. (14:15, in 6E 2.2).

In my PSS talk on Quaternionic Geometry, I defined quaternionic manifolds and remarked that they are locally isomorphic to the quaternionic projective space \(\mathbb{HP}(n)\). This time we will proceed in the "opposite direction", by defining an (almost) quaternionic manifold to be a so-called parabolic geometry with flat model \(\mathbb{HP}(n)\). This allows us to access to the large toolkit and representation theoretic language associated with these geometries.

In this talk I will outline the description of quaternionic geometry as a parabolic geometry, including the equivalence of categories between (almost) quaternionic manifolds and parabolic geometries modelled on \(\mathbb{HP}(n)\), before applying some general theory to examine some objects of interest in differential geometry: compatible connections and their curvatures and torsions.

*Abelian Varieties*, by Paweł Borówka. (14:15, in CB 3.16).

This talk will be an introduction to abelian varieties.

07/08/14:*Cluster Automorphisms and Homogeneous Spaces*, by Matthew Pressland. (11:15, in 6E 2.2).

This talk will explain how to view the set of labelled seeds of a cluster algebra as a homogeneous space for the action of a group of mutations and permutations. We describe a particular class of equivalence relations on homogeneous spaces, with the property that their equivalence classes are given by the orbits of a subgroup of the automorphism group of the space. In the labelled seeds setting, one subgroup arising in this way can be identified with the group of cluster automorphisms (in the sense of Assem–Schiffler–Shramchenko) and another with the group of direct cluster automorphisms. This is joint work with Alastair King (see Labelled Seeds and Global Mutations).

02/04/14:*Representations of Semisimple Lie Algebras*, by Amine Chakhchoukh. (16:15, in CB 3.15).

This is a simple introduction to the representation theory of semisimple Lie algebras. After giving the definitions needed, we will discuss the Cartan decomposition and the classification of complex semisimple Lie algebras with the help of their Dynkin diagrams.

27/03/14:*Projective Differential Geometry*, by George Frost. (15:15, in 8W 2.13).

Projective differential geometry is the study of manifolds with a preferred set of geodesics. After introducing necessary concepts such as connections and their geodesics, I will give an overview of some classical and contemporary results in projective differential geometry, including Levi–Civita's local classification theorem. Finally, I will discuss the invariant framework in which projective differential geometry sits, which allows complex, quaternionic and more exotic generalisations.

04/02/14:*Localising Varieties from Consistent Dimer Models*, by Jesus Tapia Amador. (16:15, in CB 3.15).

I'll review the construction of toric Gorenstein singularities and their crepant resolutions starting from a consistent dimer model. I'll also introduce two distinct algorithms that produce new consistent dimer models from old.

04/02/14:*Toric Fano Varieties and Tilting Bundles*, by Nathan Prabhu-Naik. (15:15, in CB 3.15).

We will describe what tilting bundles are, why they are interesting and give some concrete examples on toric Fano varieties.

21/01/14:*Quiver Representations, Cluster Algebras and Cluster Categories*, by Matthew Pressland. (16:15, in 4E 2.4).

I will discuss the representation theory of (bound) quivers, paying particular attention to why this is the same as the representation theory of finite-dimensional associative algebras with unit over an algebraically closed field. I will also describe the cluster algebra associated to a quiver, and connect its combinatorics to the representation theory via the cluster category. The discussion will mostly be restricted to the Dynkin (i.e. finite) case, where the correspondences are clearer.

### 2013

28/11/13:*Auslander's Theorem and The Algebraic McKay Correspondence*, by Matthew Pressland. (15:15, in CB 4.9).

Let \(K\) be a field, and let \(G\) be a (nice) finite subgroup of \(\mathrm{GL}(n,K)\), acting on \(S=K[x,y]\). Auslander's Theorem states that the endomorphism ring of \(S\), as a module over the \(G\)-invariant subring \(R\), is isomorphic to the twisted group ring \(S*G\). Using this isomorphism, we can set up one-to-one correspondences between the indecomposable summands of \(S\) as an \(R\)-module, the indecomposable projective modules of the endomorphism algebra, the indecomposable projective modules of \(S*G\), and the irreducible \(K\)-representations of \(G\). I will give a sketch of these ideas.

24/09/13:*Fano Varieties in Toric Geometry*, by Nathan Prabhu-Naik. (14:00, in 4W 1.7).

We will see how the combinatorial aspects of toric geometry allow us to understand Fano varieties. No knowledge of toric geometry or Fano varieties will be assumed (ideally, everything in the title will be explained).

19/04/13:*Dimer Models, Quivers with Relations, Resolutions of Toric Gorenstein Singularities and Moduli Spaces of Quiver Representations*, by Jesus Tapia Amador. (10:15, in 2E 3.1).

Given a consistent dimer model \(G\), its dual graph gives rise to a quiver with relations \(\Gamma\). The path algebra \(\mathbb{C}\Gamma\) modulo the ideal of relations \(I\) gives a non-commutative algebra whose centre 'is' a toric Gorenstein \(3\)-dimensional singularity. A resolution of this singularity coincides with the moduli space of quiver representations of \(\Gamma\). When the dimer model comes from a hexagonal lattice, \(\Gamma\) is the McKay quiver and the moduli space coincides with \(G\)-\(\operatorname{Hilb}\mathbb{C}^3\). In this case, the resolution of the singularity is well understood and may be easily computed using the Craw–Reid algorithm.

25/02/13:*A Brief Introduction to Curve Shortening Flow*, by Ben Lambert. (14:15, in 3W 4.7).

I will introduce curve shortening flow, geometrically one of the simplest examples of a flow (and historically the first heat flow like geometric flow to be studied). I will prove some basic properties (namely a maximum principle), discuss theorems on this flow (with animations!) and make analogies between the issues raised and problems on other higher dimensional flows. No knowledge of PDEs will be assumed.

15/01/13:*Kähler Geometry and the Kodaira Embedding Theorem*, by George Frost. (14:15, in 4W 1.7).

The aim of this talk is to give a statement of the Kodaira Embedding Theorem, which provides a link between complex differential geometry and complex algebraic geometry. To this end, we will give a differential geometric definition of a complex manifold, discuss the exterior algebra and holomorphic structures, and the Dolbeault cohomology. We will define Kähler manifolds and give several characteristic features, including the Hodge cohomology decomposition. Finally, some consequences of Kodaira's Theorem will be given. If there is time, we will discuss the (now proved) Calabi Conjecture.

The talk will assume a reasonable familiarity with basic differential geometry, including (real) manifolds, tensor algebra and vector bundles, and will be of a largely differential geometric flavour.

### 2012

30/11/12:*Orderings and Geometry*, by Acyr Locatelli. (14:15, in 4W 1.7). [notes]

I will introduce the real spectrum of a commutative ring, which generalize real algebraic sets and allows us to use real algebra tools to study geometric problems. They can be thought of as an analogue of the theory of schemes in relation to varieties in classical algebraic geometry. We will also see some applications of the real spectrum to semi-algebraic geometry.

*What Is a Cluster Algebra?*, by Matthew Pressland. (14:15, in 4W 1.7).

Cluster algebras, first described in 2000 by Fomin and Zelevinsky, occur naturally in various areas relating to geometry, topology and representation theory. This talk will give the definition of a cluster algebra and some of the first results and conjectures. The discussion will be motivated by the example of the homogeneous coordinate ring of the Grassmannian of planes in an \(n\)-dimensional complex vector space. General Grassmannians will also be discussed, with particular emphasis on the case of \(3\)-dimensional subspaces.