The 44th ARTIN meeting will take place at the University of Manchester on the 8th and 9th of May 2015. It will be funded by the London Mathematical Society and Glasgow Mathematical Journal Trust. All talks will be in the Frank Adams Room, on the first floor of the Alan Turing Building.
The theme of the meeting will be broadly in the area of geometric and homological aspects of representation theory. A full programme in pdf form is available here: artin-programme.pdf
- Grzegorz Bobinski (Torun, Poland)
- Peter Jorgensen (Newcastle)
- Matthew Pressland (Bath)
- Julia Sauter (Bielefeld, Germany)
- Ed Segal (Imperial)
- Michael Wemyss (Edinburgh)
- 13:00 - 14:00 Matthew Pressland (Bath)
Title: Cluster Structures from Internally 3-Calabi-Yau Algebras
Abstract: I will introduce internally 3-Calabi-Yau algebras, which are algebras exhibiting 3-Calabi-Yau duality away from their 'boundary'. Such algebras play a significant role in the study of Frobenius categorification of cluster algebras. In particular, I will explain how an internally 3-Calabi-Yau algebra A gives rise to a Frobenius category containing a cluster-tilted object with endomorphism algebra A. I will also describe aspects of Geiss-Leclerc-Schroer's cluster structures on homogeneous coordinate rings of partial flag varieties, which provide motivation for the general theory, as well as lots of examples of internally 3-Calabi-Yau frozen Jacobian algebras.
- 14:15 - 15:15 Julia Sauter (Bielefeld)
Title: On Kato's standard modules (for nilpotent representations of the oriented cycle)
Abstract: We explain a cell decomposition of quiver flag varieties of nilpotent representations for the oriented cycle which is parametrized by (certain) multitableau. This is a generalization of work of Fresse on Springer fibres in type A and of the Schubert cell decomposition. The cells give a basis of the (co)homology groups of these quiver flag varieties, which can be identified with "standard" modules (introduced by Kato) for the associated quiver Hecke algebra. We introduce these modules and play around with the information we win from the cell decomposition.
- 16:00 - 17:00 Ed Segal (Imperial)
Title: All autoequivalences are spherical twists.
Abstract: Seidel and Thomas found a symmetry of a triangulated category, called a spherical twist, using the idea of a `spherical’ object. Their construction was swiftly generalized to produce spherical twists around `relatively-spherical’ objects, and from there to a completely abstract construction of a twist around a `spherical functor’. I’ll explain why this notion of a twist around a spherical functor is so general that any autoequivalence of a triangulated category can in fact be described as a spherical twist, for purely formal reasons.
- 09:15 - 10:15 Grzegorz Bobinski (Torun)
Title: Characterization of singular points of orbit closures for Dynkin quivers of type D
Abstract: When studying orbit closures of representations of quivers, it is a nontrivial task to describe tangent spaces and, in particular, to determine if a given point is nonsingular. The problem lies in the fact that, in general, there is no representation theoretic interpretation of equations describing orbit closures. On the other hand, there exist natural schemes, which are defined in terms of hom-spaces and whose reduced structures coincide, in the case of Dynkin quivers, with those of orbit closures. Moreover, Riedtmann and Zwara have proved that these schemes are reduced if a quiver is of type A. The main result of my talk says that in the case of Dynkin quivers of type D the singular points of the orbit closures can be detected by investigating the above schemes.
- 11:00 - 12:00 Michael Wemyss (Edinburgh)
Title: The combinatorics of flops, via cluster mutation.
Abstract: There is a homological version of the minimal model program which interprets the flop functor equivalence of Bridgeland-Chen as a mutation functor in cluster theory. This gives a surprising number of corollaries and new results, and in this talk I will focus on one, namely the braiding of flops in dimension three, and their combinatorics. Unexpectedly, the braiding of these derived functors is controlled not by the standard braid group of a Dynkin diagram, or even the braid group of a Coxeter group, but instead by a naturally occurring hyperplane arrangement. I will give lots of examples, and outline some open problems, particularly when we try to lift the action to affine braid groups. This is joint work with Will Donovan.
- 12:15 - 13:15 Peter Jorgensen
Title: SL_2-tilings, infinite triangulations, and continuous cluster categories
Abstract: This is a report on joint work with Christine Bessenrodt and Thorsten Holm. An SL_2-tiling is an infinite grid of positive integers such that each adjacent 2x2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes.
We will show a bijection between SL_2-tilings and certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation.
The infinite triangulations also give rise to cluster tilting subcategories in a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The SL_2-tilings can be viewed as the corresponding cluster characters.
Please register here if you plan to attend: http://goo.gl/forms/WJCHSqpkXh
Unfortunately, we are no longer able to arrange accommodation for participants. We recommend that people reserve accommodation at one of the following options within easy walking distance of the Alan Turing Building:
Ibis Princess Street
Holiday Inn, Oxford Road
Getting to the Alan Turing Building:
The nearest train station to the Alan Turing Building is Manchester Oxford road, closely followed by Manchester Piccadilly.
From Manchester Oxford Road (15 minutes, 0.8 miles): Upon exiting the station either go down the steps to the right or the sloped road to the left. Either way, you will find yourself on Oxford Road. Turn right (i.e. south) and follow Oxford Road. After you cross Booth Street West/East and before passing the Manchester Museum, turn left onto a path between a red brick building (Kilburn Building) and a modern glass/grey tin-can-like building (University Place). Continue on this path until you are under the solar panels joining two buildings. The one on the left is the Alan Turing Building. Alternatively, hop on almost any bus (42,142,43,143,41,147) heading south along Oxford road - get off at the Precinct Centre stop.
From Manchester Piccadilly (17 mins, 0.9 miles): Exit the station from the South entrance using the escalators between Platform 10 and Sainsbury's (go down two levels). This will bring you to the taxi rank. Cross the cross-roads in front of you diagonally and head under the railway bridge. Immediately after the railway bridge, turn right onto the former UMIST campus. Go straight to the end of this road (passing a barrier) At the end of the road there will be a security lodge on your left. Turn left; you will pass under a white building that goes over the road. Following the pavement will take you under an overpass; take the next left and you will go under the Mancunion Way overpass. Continue, and you will cross a single-way road. You will then come to a complex pedestrian crossing; go across the first crossing, then turn right and cross the multi-lane road. Once across, turn left and walk south. After about 2 minutes you will cross Booth Street East and then pass a multi-storey car park on your right. The silver-coloured building on your right after the multi-storey car park is the Alan Turing Building.