4 Edinburgh Hodge Institute | ARTIN-52

The 52nd ARTIN meeting will be held at the Institute of Mathematics, University of Aberdeen on March 22-24, 2018. This meeting is supported by the London Mathematical Society, the Glasgow Mathematical Journal Trust and the Anglo-Franco-German Representation Theory network. It is being organized by Alexey Sevastyanov.

It is planned that speakers and participants will arrive on Thursday, Mar 22 in the morning or early afternoon and the talks will start at around 2 p.m. There will be a conference dinner on Friday night and the last talk should finish on Saturday, Mar 24 around noon. The full schedule will appear here shortly.

Schedule of talks:

Tea and coffee will be served before the first talk in the common space of Math Department, Fraser Noble building. All talks are in room FN156, Fraser Noble Building, Mathematics Department (note this is different than what was first announced)

  • Peter Jorgensen (Newcastle): c-vectors of 2-Calabi-Yau categories
    22nd March 2018, 2:00pm to 3:00pm -- Show/hide abstract
    Abstract: We develop ageneral framework for c-vectors of 2-Calabi–Yau categories withrespect to arbitrary cluster tilting subcategories, based on Dehy andKeller's treatment of g-vectors. This approach deals with clustertilting subcategories which are in general unreachable from eachother, and does not rely on (finite or infinite) sequences ofmutations. We propose a general program for decomposing sets ofc-vectors and identifying each piece with a root system.
  • Tea/coffee
    22nd March 2018, 3:00pm to 3:45pm
  • Luis Paris (Dijon): Commensurability in Artin groups
    22nd March 2018, 3:45pm to 4:45pm -- Show/hide abstract
    Abstract:  Recall that an Artin group is a group defined by a presentation withrelations of the form sts… = tst…, the word on the left hand sideand the word on the right hand side having the same length.

    These groups were introduced by Tits in the 1960s and they are involved in several fields such as singularities, low-dimensional topology, or geometric group theory. There are very few results valid for all Artin groups and the theory mainly consists on the study of particular families. The most studied and better understood are the family of right-angled Artin groups (RAAG) and the family of spherical type Artin groups. The latter is the topic of the talk.Following a result from 2004 where I proved that two spherical type Artin groups are isomorphic if and only if they have the same presentation, the project of classifying these groups up to commensurability was born. This relation is a kind of equality up to finite index. Our aim will be to give a general presentation on this question and to explain recent advances. This is a joint work with Maria Cumplido Cabello.

  • Dmitry Rumynin (Warwick): Localisation and duality for Kac-Moody Groups
    23rd March 2018, 9:30am to 10:30am -- Show/hide abstract
    Abstract: We will look at representation theory of a complete Kac-Moody group G over a finite field. After defining the group we discuss localisation of its category of smooth representations. We also discuss homological duality for this category.
  • Tea/Coffee
    23rd March 2018, 10:30am to 11:00am
  • Kevin McGerty (Oxford): Kriwan surjectivity and moduli spaces
    23rd March 2018, 11:00am to 12:00pm -- Show/hide abstract
    Abstract: A classical result of Kirwan proves that cohomology ring of a quotient stack surjectsonto the cohomology of an associated GIT quotient via the naturalrestriction map. In many cases the cohomology of the quotient stackis easy to compute so this often yields, for example, generators forthe cohomology ring of the GIT quotient. In the symplectic case, itis natural to ask whether a similar result holds for (algebraic)symplectic quotients. Although this surjectivity is thought to failin general, it is expected to hold in many cases of interest. Inrecent work with Tom Nevins (UIUC) we establish this surjectivityresult for Nakajima's quiver varieties. An important role is playedby a new compactification of quiver varieties which arises from thestudy of graded representations of the preprojective algebra. Timepermitting we will discuss work in progress on the case ofmultiplicative quiver varieties.
  • Lunch
    23rd March 2018, 12:00pm to 2:00pm
  • Eric Vasserot (Paris VII): Yangians and cohomological Hall algebras
    23rd March 2018, 2:00pm to 3:00pm -- Show/hide abstract
    Abstract: Using symplectic geometry and Nakajima's quiver varieties, Maulik and Okounkov have associated an R-matrix to any quiver. For quivers of Dynkin types, we recover the rational R-matrices, corresponding to Yangians. For general quivers, there is no algebraic description of the quantum groups corresponding to these R-matrices. There are expected to be closely related with the Hall algebra of the quiver. We'll explain a conjecture relating these new quantum groups to a new family of algebras called cohomological Hall algebras. We'll also describe some recent progress toward the proof of this conjecture.
  • Ian Grojnowski (Cambridge): TBA
    23rd March 2018, 3:00pm to 4:00pm
  • Tea/coffee
    23rd March 2018, 4:00pm to 4:30pm
  • Andrey Mudrov (Leicester): Extremal twist and complete reducibility of tensor product of highest weight modules.
    23rd March 2018, 4:30pm to 5:30pm -- Show/hide abstract
    Abstract:  Consider a reductive quantum group and its two irreducible modules of highest weights, of which one is infinite dimensional. We give acriterion for complete reducibility of their tensor product.  It is controlled by a linear operator that we call extremal twist. This operator is defined on an "extremal" subspace of either of the modules determined by the other module. Then the tensor product is completely reducible if and only if either of these operators (and hence both) has zero kernel. Extremal twist is related to the Zhelobenko cocycle and dynamical Weyl group of Etingof-Tarasov-Varchenko.
  • Dinner
    23rd March 2018, 6:00pm to 8:00pm
  • Andrey Lazarev (Lancaster): Derived categories of 2nd type and theorems of Riemann-Hilbert type.
    24th March 2018, 9:30am to 10:30am -- Show/hide abstract

    A twisted module over a differential graded (dg) algebra A is a dg A-module that is free over A after forgetting the differential. The homotopy category of twisted A-modules is an analogue of the derived category of A but is a finer invariant, in particular it is not a quasi-isomorphism invariant of A. It is sometimes called a derived category of second type of A (or a contraderived category of A). I will explain how the contraderived category of A is a homotopy invariant of A. Taking A tobe the de Rham algebra of a smooth manifold X or a singular cochain algebra of a topological space X leads to categories that could be called derived categories of X, in analogy with derived categories of algebraic schemes. These categories turned out to be equivalent to the categories of cohomologically locally constant sheaves on X and thus carry a lot of information about its homotopy type.This is a joint work with J. Chuang and J. Holstein.

  • Tea/Coffee
    24th March 2018, 10:30am to 11:00am
  • Philip Boalch (Orsay): Twisted wild character varieties
    24th March 2018, 11:00am to 12:00pm -- Show/hide abstract
    Abstract: I will describe all the symplectic leaves of the space of GL_n(C) representations of the wild fundamental group of a smooth complex algebraic curve (i.e. the Tannaka group of the category of connections on vector bundles). In the compact case the symplectic form is due to Atiyah-Bott/Goldman 1983/4, the tame case is in Atiyah's 1990 knot book, the untwisted wild cases were done by the speaker (analytically 1999, algebraically 2002-14), and finally the twisted cases were completed in work with D. Yamakawa (2015), for any complex reductive group. I will try to describe the spaces using the "Deligne-Lusztig" approach (proved by Malgrange 1982) involving flags on sectors in fixed relative positions (i.e. Stokes filtrations), as well as the equivalent "wild monodromy" approach used in the speaker's papers.
(Open in Google Calendar)

There are some additional funds to support participants from within the ARTIN network. If you plan to attend, please fill out the form here.

If you need an overnight accommodation you may try to contact Aberdeen Northern Hotel https://www.aberdeennorthernhotel.com .

If you have any questions, please contact Alexey Sevastyanov, a.sevastyanov at abdn.ac.uk .