The 45th ARTIN meeting will take place at the University of Edinburgh on the 11th and 12th of September 2015. All talks will be in the James Clerk Maxwell Building, room 6206.

The theme of the meeting is noncommutative ring theory, with an emphasis on noncommutative algebraic geometry.

## Schedule:

Friday 11 September

 1300-1400 Dennis Presotto In the paper 'arXiv:1410.5207' (to appear in Journal of Noncommutative Geometry) Michel Van den Bergh and I generalize some classical birational transformations to the noncommutative setting. In particular we show that 3-dimensional quadratic Sklyanin algebras (noncommutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct an analogue of the Cremona transform for noncommutative projective planes. In this talk I will explain the approach and motivation for our paper. The motivation is based on the construction of the Cremona transform (in the commutative setting) using linear systems. I will also recapitulate some relevant facts about noncommutative surfaces and Artin-Schelter regular (Z-)algebras. 1430-1530 Charlie Beil Quantum nonlocality is the strange phenomenon where two entangled particles can affect each other instantaneously, regardless of how far apart they are. I will explain how this phenomenon can be described geometrically as a nonnoetherian singularity in spacetime. I will then introduce the notion of a noncommutative resolution of such a singularity, and describe how this notion fits into the quantum world. 1530-1630 Tea and coffee 1630-1730 Spela Spenko This is joint work with Michel Van den Bergh. We generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. We discuss a number of examples, both new and old, that can be treated using our methods; twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symme- tric matrices. Slides

1830- Conference Dinner, Kalpna (2-3 St. Patrick Square, Edinburgh EH8 9EZ).

Saturday 12 September

 0900-1000 Ana Rovi We describe Lie--Rinehart algebras in the tensor category of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras (in the category of linear maps) to Leibniz algebroids. 1015-1115 Martina Balagovic The construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation on tensor products of representations of that quantum group. This gives an action of the braid group of type A, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category. I will explain how the theory of quantum symmetric pairs allows an analogous construction of a universal K- matrix, which produces solutions of the reflection equation on tensor products of representations of that quantum group. This gives a representation of the braid group of type B, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category with a cylinder twist. Slides 1115-1200 Tea and coffee 1200-1300 Alexey Petukhov In my talk I will construct a functor from some class of skew fields to some class of commutative fields with map $$Q( C[x, \partial_x]) \to Q (C[x, y])$$ as the main example. I will also provide some link between this construction and some known results and conjectures on skew fields.

Escher exhibit

While not part of the official program, attendees might be interested in the M. C. Escher exhibit at the Scottish National Gallery of Modern Art. There may be a semi-organised outing to the exhibit on Saturday afternoon.

## Directions to School of Maths

The ARTIN meeting will take place in the James Clerk Maxwell Building at the University of Edinburgh King's Buildings. This is about 2.5 miles south of the city centre. Directions are here: http://www.maths.ed.ac.uk/contact/travel-directions

## Registration:

Please register here if you plan to attend: http://goo.gl/forms/4kFBgvXpc4