Meetings for 2017-18
Meetings for the academic year 2017-18, including titles and abstracts of talks, can be found below.
Edinburgh, 3rd September 2018
Graduate Level Talk
Where: Room 5323, James Clerk Maxwell Building, King's Buildings, University of Edinburgh
Speakers: Peter Jørgensen (Newcastle)
The notion of $d$-abelian category was introduced by Jasso, where $d$ is a positive integer. Such a category does not have kernels and cokernels, but rather $d$-kernels and $d$-cokernels, which are longer complexes with weaker universal properties.
We will explain the definition and show a concrete example: Let $\Phi$ be the path algebra of $A_9$ modulo the relations that four consecutive arrows compose to zero. Inside $mod(\Phi)$, let $\cal F$ be the additive closure of the projective and injective modules. Then $\cal F$ is a 4-abelian category.
The notion of $d$-cluster tilting subcategories was introduced by Iyama. We will explain a theorem by Jasso, which states that a $d$-cluster tilting subcategory $\cal F$ of a module category $\mod(\Phi)$ is $d$-abelian. The example from the graduate talk is a special case of this.
We will introduce the notion of wide subcategories of a $d$-abelian category, and explain a theorem by Herschend-Jørgensen-Vaso which states that the wide subcategories of $\cal F$ are in bijection with certain $d$-rigid algebra epimorphisms $\Phi \rightarrow \Gamma$.
Glasgow, 19th June 2018
Sarah Kelleher - Geigle-Lenzing Weighted Projective Space
Where: Seminar Room 110, Glasgow University Maths Building
Speakers: Sarah Kelleher (Glasgow), Tim Weelinck (Edinburgh)
Weighted projective lines were first introduced by Geigle and Lenzing in 1987 and are important in representation theory of finite dimensional algebras. The result has recently been generalised to higher dimensional analogues in a paper of Hershend, Iyama, Minamoto and Oppermann. In this talk I plan to give an introduction to these spaces and link them to the study of singularities.
Tim Weelinck - Equivariant Homology Theories
Many invariants (various homology theories) have equivariant analogues when the object you consider has extra symmetries. For example, there is the Bredon cohomology of a space endowed with a finite group action. This homology theory generalises both group homology and singular homology. Another example is given by Hochschild homology for algebras with an involution. We describe these classical examples and then explain a unifying framework for equivariant homology theories bases on ideas from topological quantum field theory. If time permits we discuss some new categorical invariants that arise from this equivariant homology theory that are constructed out of quantum groups.
Edinburgh, 5th April 2018
Jenny August - Generalisations of Tilting Theory
Where: Room LG.06, David Hume Tower, University of Edinburgh
Speakers: Jenny August (Edinburgh), Billy Woods (Glasgow)
Tilting theory grew out of an attempt to generalise a result determining when two algebras have equivalent module categories. In this talk, I will give a brief introduction to the results of classical tilting theory and outline what was considered to be the 'problem' with this approach. I will then talk about two possible generalisations of the classical theory, both aimed at solving this problem, and show that although the approaches seem completely different, they are in fact strongly connected.
Billy Woods - Iwasawa algebras
Roughly speaking, an Iwasawa algebra is the completed group ring of a virtually pro-p group: its modules characterise (continuous) representations of the (topological) group. I'll take a brief historical detour to outline the number-theoretic context in which these representations first arose (featuring an erroneous proof of Fermat's Last Theorem), and sketch some of the current motivations for studying them. Then I'll survey some of what's known (and not known) about Iwasawa algebras themselves, including some of my own recent work, and draw some analogies between the theory of Iwasawa algebras and other, more classical algebraic objects.
Glasgow, 26th February 2018
Lukas Mueller - Algebraic Structures in Discrete Gauge Theories
Where: Seminar Room 116, Glasgow University Maths Building
Speakers: Lukas Mueller (Heriot-Watt), Ogier Van Garderen (Glasgow)
Topological quantum field theories in low dimensions give a geometric interpretation for a large list of algebraic structures. A particular accessible class of topological field theories are discrete gauge theories also known as Dijkgraaf-Witten theories.
In 2-dimensions they are closely related to twisted representation theory of finite groups. The 3-dimensional theory encodes the representation theory of the twisted Drinfeld double of a finite group. Their anomalies can be studied using non-abelian group cohomology and the Lyndon-Hochschild-Serre spectral sequence. In this talk I will introduce Dijkgraaf-Witten theories and explain their relation to the interesting algebraic structures mention above.
The talk is based on joint work with Richard Szabo and Lukas Woike.
Ogier Van Garderen - Combinatorial Approaches to Mirror Symmetry
In this talk, I will present an example of a mirror symmetry correspondence between the matrix factorisations of a singular 2-fold and the intersection theory of curves on a certain (non-compact, real) 2-manifold.
Each of the spaces involved has an underlying combinatorial structure which allows for an explicit classification of the indecomposable matrix factorisations up to quasi-isomorphism.
If time allows, I will outline how the matrix factorisations can be deformed to recover a classical construction by Seidel, which compactifies the 2-manifold.
Edinburgh, 7th December 2017
Simon Crawford - Azumaya Skew Group Algebras and Quantum Kleinian Singularities
Speakers: Simon Crawford (Edinburgh), Ruth Reynolds (Edinburgh)
In this talk, I will give a simple geometric criterion to determine when a skew group ring is Azumaya. Unlike a number of other results in the literature, this criterion is easy to check in concrete examples, as I will demonstrate. I will then outline how this result can be used to show that quantum Kleinian singularities, as defined by Chan-Kirkman-Walton-Zhang, are maximal orders. This allows for an alternative proof of Auslander's Theorem for these singularities.
Ruth Reynolds - The Noetherianity of Idealizer Subrings
The idealizer subring of a one-sided ideal I is the largest subring such that I becomes a two-sided ideal. They often exhibit interesting properties and pathological behaviour. In this talk I will describe some of their properties, in particular how their noetherianity in certain setups can be controlled by the orbits of points. In particular, I will describe how they were used by Sierra and Walton to prove that the universal enveloping algebra of the Witt algebra is not noetherian.