Groupe d'étude
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.
Exposés
2019-2020
25 octobre: Pierre-Guy Plamondon, Introduction to scattering diagrams
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
[GHKK]).
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).
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.
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.
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.
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
Joyce.
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.
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.
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.
24 janvier 2020 : No session because of the Oberwolfach meeting on representations of quivers and finite-dimensional algebras
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.
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.
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.
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.
28 février 2020: No session
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.
27 mars: Bernhard Keller, Review and prospects (bilan et perspectives),
https://zoom.us/j/609347784.
The study group will take place remotely using Zoom. You are advised to download
Zoom beforehand from
https://zoom.us (it works on all devices, you do not need to open an account,
neither free nor paid). Shortly before the beginning of the meeting, click the above zoom link.
Your browser will open Zoom and give you access to the meeting.
Abstract: We will review some of the main constructions and results seen
so far and sketch what is yet to come.
3 avril 2020 : exceptionnellement en salle 2015
Tom Bridgeland, Scattering diagrams, Hall algebras and stability conditions,
arXiv:1603.00416
Ben Davison and Travis Mandel, Strong positivity for quantum theta bases of quantum cluster algebras,
arXiv:1910.12915
Mark Gross, Paul Hacking, Sean Keel, Maxim Kontsevich, Canonical bases for cluster algebras,
arXiv:1411.1394.
Jochen Heinloth, Lectures on the moduli stack of vector bundles on a curve,
available here
Sean Keel and Tony Yue Yu, The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus,
arXiv:1908.09861
Maxim Kontsevich and Yan Soibelman, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry,
arXiv:1303.3253
Lang Mou, Scattering diagrams of quivers with potentials and mutations,
arXiv:1910.13714
Christoph Sorger, Lectures on moduli of principal bundles over algebraic curves,
available here
Retour à la page de B. Keller
http://www.math.jussieu.fr/~keller/gdtcluster/
le 23 octobre 2019