Algèbres amassées et représentations de carquois
(Cluster algebras and quiver representations)
A l'Institut de
Mathématiques de Jussieu-Paris Rive Gauche,
Bâtiment Sophie Germain, Place Aurélie Nemours, 75013 Paris
le vendredi de 14h à 16h en salle 1016
Organisé par Pierre-Guy Plamondon et Bernhard Keller
Ce groupe d'étude s'adresse au (post-)doctorants. Cette année,
nous étudierons les diagrammes de dispersion en théorie des
représentations des carquois suivant des travaux de Tom Bridgeland
et Fan Qin. Les exposés seront en anglais.
From March 27 onwards, the study group takes place remotely. Please contact
Bernhard Keller or Pierre-Guy Plamondon to gain access.
23 juillet: Léa Bittmann (University of Vienna),
Standard modules from the geometry
Abstract: After talking about the algebraic approach to standard modules of quantum
affine algebras during last talk, I will this time get into the geometric approach.
I will try to explain the link with quiver varieties, and how the geometry gives
insight and powerful results.
Kaveh Mousavand (Queens University): Hilbert Schemes of points, seen as Nakajima quiver varieties
This will be the first part of a lecture aimed at understanding the Hilbert schemes of
points as certain quiver varieties. We begin with a quick revision of our ultimate goals
for understanding such schemes and some fundamental results in the more general setting.
Then, we specialize to surfaces with isolated singularities-- in particular the
Kleinian singularities. The goal is to understand how the Hilbert schemes of points
on these surfaces are realized as the fine moduli spaces of semistable modules.
16 juillet: pas de séance
9 juillet: Léa Bittmann (University of Vienna),
Representations of quantum affine algebras, l-fundamental and standard modules
Abstract: I will present the basics of the theory of representations of
quantum affine algebras. We will see the quantum affine analogs of the
fundamental representations, called "l-fundamental", and more generally
we will focus on the so-called "standard modules", which play a key role
in the particular preprint which is the focus of this working group.
2 juillet: Pierre-Guy Plamondon (Orsay), Euler characteristics of Hilbert
schemes of points of simple surface singularities, after
Gyenge-Némethi-Szendröi and Nakajima
Abstract: In this introductory talk, we will define the Hilbert schemes of
points of simple surface singularities, state the theorem of Nakajima which
computes their Euler characteristics, and outline the strategy of his proof.
This will involve Nakajima's quiver varieties, and their links to
representations of quantum affine algebras. We will not go into any
kind of detail at each step; our aim is rather to have a more or
less complete picture before tackling the particulars of the proof
in the following lectures.
18 juin: Pierrick Bousseau (ETH Zurich), A scattering diagram for coherent
sheaves on the projective plane
Abstract: This talk is based on arXiv:1909.02985.
I will describe a scattering diagram computing Betti numbers of moduli spaces of
semistable coherent sheaves on the projective plane. The scattering diagram
lives in the space of Bridgeland stability conditions on the derived category of
11 juin: Pierrick Bousseau (ETH Zurich), DT invariants of 3-gon quivers
Abstract: This talk will be the exposition of a remark by Boris Pioline
(cf. section 3.3 of arXiv:12072230
and arXiv:2004.14466). I will describe
the computation of Donaldson-Thomas invariants for 3-gon quivers, consisting of 3
vertices connected by oriented cycles of arrows. As an application, we obtain examples
for which the stability scattering diagram differs from the cluster scattering diagram
by non-central elements (compare with Conjecture 3.3.4 in
28 mai: Alex Atsushi Takeda (UC Davis),
The wall and chamber structure of the real Grothendieck group IV, d'après Sota Asai
21 mai: Alex Atsushi Takeda (UC Davis),
The wall and chamber structure of the real Grothendieck group III, d'après Sota Asai
21 mai: Alex Atsushi Takeda (UC Davis),
The wall and chamber structure of the real Grothendieck group II, d'après Sota Asai
14 mai: Alex Atsushi Takeda (UC Davis),
The wall and chamber structure of the real Grothendieck group I, d'après Sota Asai
I will start presenting the results of
Sota Asai's paper
on the wall-chamber structures. This paper gives a complete description
of the stability chambers on the real Grothendieck group of any finite-dimensional
algebra, in terms of some equivalence classes of stability conditions under
some explicit relation (TF equivalence) and also 2-term silting objects.
In this first talk, I will begin by presenting an overview of the main results,
introducing the relevant notions and sketching some of the proofs in the first
parts of the paper.
7 mai: Lang Mou (UC Davis), Scattering diagrams of quivers with potentials, II
We will continue on discussing the cluster complex structure of cluster scattering
diagrams. Then we will study the mutations of stability scattering diagrams of QPs,
which will lead to the same cluster complex structure for non-degenerate QPs.
Connections to cluster algebras will be mentioned if time permits.
30 avril: Lang Mou (UC Davis), Scattering diagrams of quivers with potentials, I
Abstract: We will show how to obtain a canonical cone complex from any equivalence
class of scattering diagrams. We will then discuss the initial data that determine
cluster scattering diagrams, and how a cluster scattering diagram and its mutations
are related, leading to their cluster cone complex structures.
24 avril : Hülya Argüz, Equivalence of the cluster and stability scattering diagrams
Abstract: We will define the integration map which we will then use to obtain the stability
scattering diagram from the Hall algebra scattering diagram. We will then show that in some cases
the stability scattering diagram is equivalent to the cluster scattering diagram of
17 avril : Pierre-Guy Plamondon, Theta functions for the Hall algebra scattering diagram
Abstract: We will define the theta functions for the Hall algebra scattering diagram.
After looking at some examples, we will see some identities involving these theta functions. .
3 avril : Pierre-Guy Plamondon, Towards theta functions.
Abstract: We will review some more of the previous constructions,
in particular the Hall algebra scattering diagram and framed representations
of a quiver with relations. We will also see the notion of theta functions,
and sketch the main results for them in the Hall algebra scattering diagram.
27 mars: Bernhard Keller, Review and prospects (bilan et perspectives)
Abstract: We will review some of the main constructions and results seen
so far and sketch what is yet to come.
6 mars 2020: Peigen Cao, A family of fine moduli schemes from framed representations
Abstract: In this talk, we will introduce certain fine moduli schemes from framed
representations and give a characterization for the points of these fine moduli schemes.
These schemes will be used in Section 10 to describe the theta functions associated to the
stability scattering diagrams.
28 février 2020: No session
21 février 2020: Kaveh Moussavand, Moduli spaces of stable and semi-stable representations
Abstract: I will continue the previous lecture and aim to present some further results on the moduli spaces of (semi-)stable
representations which we need before returning to Bridgeland's work. After a quick
review of the main result stated in the second part of the preceding talk, I will
address some of the questions that were posed at the end. Then we see under which
conditions the moduli space of (semi-)stable representations satisfies a universality property.
14 février 2020: Kaveh Moussavand, Representations of framed quivers
Abstract: In this talk, we first recall some basic properties of representations of the framed quiver associated to a finite dimensional algebra A.
After a brief review of some notions from the theory of
moduli schemes, we will see how one can use representations of the framed quivers to construct certain moduli schemes.
These schemes are used in the description of theta functions associated to the stability
scattering diagrams of A with values in motivic Hall algebras introduced in the preceding lectures.
7 février 2020: Elie Casbi, Nearby stability conditions
Abstract: After some reminders about stability conditions on triangulated categories,
I will present the notion of nearby stability conditions following Bridgeland. This will lead
us to an interpretation of the walls of the Hall algebra scattering diagram in terms of stability conditions.
31 janvier 2020 : Elie Casbi, Slicings and stability conditions on triangulated categories
Abstract: The notion of slicing on a triangulated category has been introduced by Bridgeland as a natural
continuous analogue of the notion of t-structure (in the sense that the cohomology spaces of a complex shall
not be indexed by integers but by real numbers). I will present some elementary properties of slicings and
connect them to the notions of semistability and stability functions defined in Pierre-Guy's talks.
Then we will see how slicings provide a natural framework to study stability conditions on triangulated categories.
In particular, one can use them to endow the space of stability conditions with a structure
of metric space.
24 janvier 2020 : No session because of the Oberwolfach meeting on representations of quivers and finite-dimensional algebras
17 janvier 2020 : Pierre-Guy Plamondon, Stability conditions and the Hall algebra scattering diagram, III
Abstract: We will finally construct the Hall algebra scattering diagram.
10 janvier 2020 : Pierre-Guy Plamondon, Stability conditions and the Hall algebra scattering diagram, II
Abstract: We will show that representations with an HN-filtration with phases in a given interval define
an open substack of the stack of representations of Q,I. We will then deduce identities in the motivic Hall
algebra from the existence and uniqueness of the HN-filtration.
20 décembre : Pierre-Guy Plamondon, Stability conditions and the Hall algebra scattering diagram
Abstract: We will first review stability conditions for quivers with relations,
after A.King. Then we will see how they appear naturally in a canonical scattering diagram
with values in the (Lie subalgebra of) the motivic Hall algebra of the quiver with relations.
13 décembre : exceptionnellement en salle 2015 : Bernhard Keller, The motivic Hall algebra
Abstract: We will introduce the Grothendieck group of stack-parametrized
quiver representations, extend the scalars from Z to C(t) and define the
convolution product in order to obtain the motivic Hall algebra due to
6 décembre : Bernhard Keller, Algebraic stacks and the Hall multiplication
Abstract: We will end our discussion of algebraic stacks and
introduce the diagram that will allow us to define the Hall multiplication.
29 novembre: exceptionnellement en salle 2015 : Bernhard Keller, Stacks II
Abstract: In continuation of last week's talk, we will give more examples of
stacks, define representable morphisms and open substacks and
study our most important examples: stacks of representations of quivers.
22 novembre: Bernhard Keller, A primer on algebraic stacks
Abstract: After a brief overview of the context (cluster scattering diagrams versus stability
scattering diagrams), we will present the definition and basic properties of algebraic stacks (=Artin stacks).
The main examples will be stacks of finite-dimensional representations of finite quivers with relations.
15 novembre: Hülya Argüz, Scattering in higher dimensions
Abstract: We will complete the proof of the reconstruction theorem from last week, and afterwards will discuss
examples of consistent scattering diagrams in dimension three (this is part of joint work with Mark Gross).
8 novembre: Hülya Argüz, Reconstruction results for consistent scattering diagrams
Abstract: After briefly recalling the combinatorics of consistent scattering diagrams, we discuss reconstruction
results for such diagrams. In particular, we will overview a
theorem of Kontsevich-Soibelman, proving the
correspondence between consistent scattering diagrams and elements of an associated Lie group.
(For details of this proof, in the context of cluster algebras see also Theorem 1.17 in
25 octobre: Pierre-Guy Plamondon, Introduction to scattering diagrams
Tom Bridgeland, Scattering diagrams, Hall algebras and stability conditions,
Ben Davison and Travis Mandel, Strong positivity for quantum theta bases of quantum cluster algebras,
Mark Gross, Paul Hacking, Sean Keel, Maxim Kontsevich, Canonical bases for cluster algebras,
Jochen Heinloth, Lectures on the moduli stack of vector bundles on a curve,
Sean Keel and Tony Yue Yu, The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus,
Maxim Kontsevich and Yan Soibelman, Wall-crossing structures in Donaldson-Thomas
invariants, integrable systems and mirror symmetry,
Lang Mou, Scattering diagrams of quivers with potentials and mutations,
Hiraku Nakajima, Euler numbers of Hilbert schemes of points on simple
surface singularities and quantum dimensions of standard modules of
quantum affine algebras,
Christoph Sorger, Lectures on moduli of principal bundles over algebraic curves,
Retour à la page de B. Keller
le 23 octobre 2019