The 50th ARTIN meeting will take place on the afternoon of Friday April 21st and the morning of Saturday April 22nd at the University of Edinburgh. It is an early careers meeting, meaning that talks will be given by Ph.D. students in the ARTIN network. All members are invited to participate. It will be funded by the London Mathematical Society and the Glasgow Mathematical Journal Trust.

### Invited speakers:

Ibukun Ademehin (Manchester)

Jenny August (Edinburgh)

Cong Chen (Manchester)

Liam Dobson (Newcastle)

James Maunder (Lancaster)

### Schedule of events:

The provisional schedule is as follows:

Friday April 21st @ the Sydney Smith Lecture Theatre in the Old Medical School, Teviot (Central Campus)
**13:00-13:50 Jenny August (Edinburgh) - Cluster Tilting Theory and the Homological Minimal Model Program**

Abstract:In geometry, the Minimal Model Program (MMP) takes a variety, and aims to find a minimal model i.e. a “nicest approximation” of this variety. For 3-folds, these minimal models are not generally unique and it is often difficult to know how many there are or how they relate to each other. The Homological MMP, which has been developed by Michael Wemyss, associates a finite dimensional algebra to each minimal model, known as the contraction algebra of that minimal model. Each contraction algebra retains much of the geometric information including roughly what the corresponding minimal model looks like but also how many other minimal models there are and how other minimal models can be obtained from the given one. The latter two properties come from strong links to cluster tilting theory which will be discussed in this talk and used to show that the contraction algebras associated to the minimal models of a given variety are all derived equivalent.
**14:00-14:50 James Maunder (Lancaster) - Minimal models of quantum homotopy Lie algebras via the BV-formalism**

Abstract: It is well know that minimal models exist for many sorts of homotopy algebras. To prove existence and uniqueness of minimal models is usually fairly straightforward; for quantum homotopy Lie algebras the proof can be seen as a consequence of standard facts concerning Maurer-Cartan moduli sets of differential graded Lie algebras. Such arguments are, however, not constructive and one is often concerned not just with the existence of minimal models, but also wishes to have explicit formulae to hand. In this talk, I will discuss an approach to deducing the formulae for minimal models of quantum homotopy Lie algebras using integrals coming from the BV-formalism of mathematical physics. An advantage of this approach over other approaches (typically combinatorial in flavour) is that the proof is simpler and more conceptual. This work is joint with Christopher Braun.
**15:30-16:20 Ibukun Ademehin (Manchester) - Invariants in the exterior algebra of the little adjoint representation**

Abstract: The algebraic problem of decomposing the exterior algebra of a Lie algebra g-module is as old as Kostant’s work on the structure of the invariant ring of the exterior algebra of the adjoint representation of a complex simple Lie algebra g. In this talk I will show how the graded multiplicity of the invariants in the exterior algebra of the little adjoint representation can be obtained from Macdonald’s and Cherednik’s weight functions and inner products on irreducible characters of g-modules. I will also show how this result sheds some light on other submodules of the exterior algebra of the little adjoint representation. This links my work to the recent results of De Concini, Frajria, Papi and Procesi.
**16:30-17:30 Flash Talks Session **

**19:00** Conference Dinner at Kismot Indian & Bangladeshi Restaurant

Saturday April 22nd @ Room 5323 in the JCMB building (King's Buildings Campus)
**10:00-10:50 Liam Dobson (Newcastle) - Explicit constructions of quasi K-matrices**

Abstract: The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this talk, I will recall how the quasi K-matrix is constructed and show for certain classes of examples that it can be explicitly factored into a product of quasi K-matrices for sl2. This factorisation is similar to the factorisation of the quasi R-matrix for quantum groups.
**11:00-11:50 Cong Chen (Manchester) - The nilpotent variety of W(1,n)_p**

Abstract: Let L be a finite dimensional restricted Lie algebra over an algebraically closed field of characteristic p>0. In the late 1980s, Premet conjectured that the variety of nilpotent elements NL is irreducible. This conjecture is still open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Cartan series W, S and H. In this talk, I will introduce the Zassenhaus algebra W(1,n) and then show Premet's conjecture is true for its minimal p-envelope W(1,n)_p.

**Extra (optional) program on Saturday:**

14:00-16:00 Hike up Arthur's Seat (the ancient volcano in the city center)

There will also be a conference dinner on Friday, at a location to be announced.

The flash talk session will consists of short talks (3min) given by participants introducing themselves and their research.

### Registration and accommodation:

Please register if you plan to attend, using the following link:

To sign up, please use the link here: https://goo.gl/forms/nL0WnkZgeKf9IuEe2

This will help us to organise refreshments, plan the conference dinner, and manage the budget.

There is funding available for Ph.D. students for travel costs and accommodation, this will be awarded on a first come first serve basis (i.e. the first to book and declare costs, is the first to be awarded). Please do make use of the low(er) prices available for travel/lodging by booking early. In booking accommodation, it is recommended to search in the area between George Sq. and Kings buildings, e.g. Northumberland Hotel, Mayfield Lodge, or the International Guest House.

If you have any questions, please contact Tim Weelinck (T.A.N.Weelinck@ed.ac.uk).