Schedule for: 18w5178 - Stability Conditions and Representation Theory of Finite-Dimensional Algebras

I will explain how geometric invariant theory gives rise to the numerical (and categorical) condition of theta-semistability for representations of quivers and describe its relationship to the classical slope semistability of Mumford and also to Schofield's construction of determinantal semi-invariants. I will also try to touch briefly on the conceptual evolution towards theta-torsion theories and the relationship with scattering diagrams (largely following Bridgeland).

Bridgeland proved that any triangulated category has a associated space of stability conditions which is a complex manifold. In general, such spaces of Bridgeland stability conditions are difficult to compute and relatively few examples are well understood. Discrete derived categories, as defined by Vossieck, form a class of triangulated categories which are sufficiently simple to make explicit computation possible, but also non-trivial enough to manifest interesting behaviour. For these examples, combinatorial techniques can be used understand the structure and, in particular, to prove the contractibility of the corresponding space of stability conditions. I will give an overview of this topic, introducing the key definitions. Finally, I will outline an approach to producing partial compactifications of the stability spaces, by considering generalised stability conditions. This is joint work with David Pauksztello, and David Ploog and Jon Woolf.

Gentle algebras can be realised as certain algebras associated to partial triangulations of unpunctured surfaces. Using this fact, we give a geometric model for the module category of any gentle algebra. This is joint work with Karin Baur.

In this talk, based on joint work with Sebastian Opper and Pierre-Guy Plamondon, we will describe the construction of a geometric surface model for the bounded derived category of a gentle algebra. We will use the model to give a geometric description of silting objects and the singularity category of gentle algebras, the latter is joint work with Claire Amiot and Pierre-Guy Plamondon.

I will review the combinatorial description of partially wrapped Fukaya categories of punctured surfaces due to Haiden-Katzarkov-Kontsevich and show how it provides a geometric model for the derived category of any graded gentle algebra. As an application, I will discuss my joint work with A. Polishchuk, which uses the action of the mapping class group to give sufficient numerical criteria for two gentle algebras to be derived equivalent. Time permitting, I will also discuss homological mirror symmetry for punctured surfaces which links this story to another geometric model of gentle algebras related to the derived categories associated to nodal stacky curves.

This will be a Skype talk given by Mandy Chueng.

I will explain how geometric invariant theory gives rise to the numerical (and categorical) condition of theta-semistability for representations of quivers and describe its relationship to the classical slope semistability of Mumford and also to Schofield's construction of determinantal semi-invariants. I will also try to touch briefly on the conceptual evolution towards theta-torsion theories and the relationship with scattering diagrams (largely following Bridgeland).

Bridgeland proved that any triangulated category has a associated space of stability conditions which is a complex manifold. In general, such spaces of Bridgeland stability conditions are difficult to compute and relatively few examples are well understood. Discrete derived categories, as defined by Vossieck, form a class of triangulated categories which are sufficiently simple to make explicit computation possible, but also non-trivial enough to manifest interesting behaviour. For these examples, combinatorial techniques can be used understand the structure and, in particular, to prove the contractibility of the corresponding space of stability conditions. I will give an overview of this topic, introducing the key definitions. Finally, I will outline an approach to producing partial compactifications of the stability spaces, by considering generalised stability conditions. This is joint work with David Pauksztello, and David Ploog and Jon Woolf.

I will review the combinatorial description of partially wrapped Fukaya categories of punctured surfaces due to Haiden-Katzarkov-Kontsevich and show how it provides a geometric model for the derived category of any graded gentle algebra. As an application, I will discuss my joint work with A. Polishchuk, which uses the action of the mapping class group to give sufficient numerical criteria for two gentle algebras to be derived equivalent. Time permitting, I will also discuss homological mirror symmetry for punctured surfaces which links this story to another geometric model of gentle algebras related to the derived categories associated to nodal stacky curves.

This is a joint work with Pierre-Guy Plamondon and Sibylle Schroll. Combining a joint work with Grimeland, and recent results due to Kawazumi, Lekili and Polischuk, we manage to give a complete derived invariant to a subclass of gentle algebras called surface algebras. This invariant is easy to compute, and I will show how to do it on different examples.

Semistable subcategories were introduced in the context of Mumford's GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for path algebras of Dynkin and extended Dynkin quivers, the poset of semistable subcategories is isomorphic to the corresponding lattice of noncrossing partitions. We classify semistable subcategories for a family of algebras each of which is defined by the choice of a partial triangulation of the disk. Our description also shows that each such semistable subcategory is equivalent to a generalized noncrossing partition. This is joint work with Monica Garcia.

Alastair King applied geometric invariant theory to quiver representations (and more generally finite dimensional algebras) and gave useful criteria for stability of a quiver representations with respect to a given weight. Considering stability for representations of the triple flag quiver, one can prove many results for Littlewood-Richardson coefficients (the tensor product multiplicities for representations of the general linear group). In this talk, I will give present new applications. For generalized Kronecker quivers, Visu Makam and the I gave effective bounds for stability and generators of the invariant ring. This result has several applications to complexity theory such as non-commutative rational identity testing. For the $m$-subspace quiver, stability relates to Brascamp-Lieb inequalities that generalize the Hoelder inequality and other important inequalities. Recent work with Calin Chindris shows that there are natural generalizations to arbitrary quivers.

I will describe the relationship between two spaces associated to a quiver with potential. The first is a complex manifold parametrizing Bridgeland stability conditions on a triangulated category, and the second is a cluster variety with a natural Poisson structure. For quivers of type $A$, I will describe a local biholomorphism from the space of stability conditions to the cluster variety. The existence of this map follows from results of Sibuya in the classical theory of ordinary differential equations.

Associated to a quiver with potential are two distinguished (possibly non-Hausdorff) complex manifolds: the space of stability conditions and the $\mathcal{X}$-cluster variety. The work of Allegretti-Bridgeland, Gaiotto-Moore-Neitzke and Kontsevich-Soibelman show that there are various remarkable geometric relationships between these spaces. In particular, the walls of the second kind in the space of stability conditions are closely related to the walls of the scattering diagram of the $\mathcal{X}$-cluster variety. In this talk we will introduce $\mathcal{X}$-cluster varieties with coefficients. We use this notion to build a flat degeneration of every skew-symmetrizable specially completed X-cluster variety (in the sense of Fock-Goncharov) to the toric variety associated to a distinguished sub-fan (the so-called $\mathbf{g}$-fan or cluster complex) of its scattering diagram. We further show that our construction is cluster dual to Gross-Hacking-Keel-Kontsevich's toric degeneration of $\mathcal{A}$-cluster varieties. If time permits we will outline applications of our construction in the context of mirror symmetry. This is based on joint work with Lara Bossinger, Man-Wai Cheung, Bosco Frías-Medina and Timothy Magee.

Tau-tilting theory gives the Hasse quiver of functorially finite torsion classes, whose arrows are labelled by bricks. In this talk, I will explain results for all torsion classes, given in a joint work with Demonet, Reading, Reiten, Thomas.

Given an algebra $A$ and a set of automorphisms, one can define a Riemann-Hilbert (RH) problem, aimed to find meromorphic connections on the $\mathrm{Aut}(A)$-principl bundle over $\bC$ with prescribed generalised monodromy. It is of particular interest considering a family of `isomonodromic' RH problems on $A$, parametrised by a complex manifold $M$, as this induces interesting geometric structures on the space of parameters $M$. In the context of stability conditions, there is a Riemann-Hilbert problem naturally attached to a CY3 category endowed with a generalised Donaldson-Thomas theory counting semistable objects. It is defined on the torus algebra of character on the free abelian group generated by classes of simple stable objects and depends on the choice of a stability condition. The wall-crossing formulae from Donaldson-Thomas theory are interpreted as isomonodromy conditions. I will introduce this topic and discuss some recent developments, focusing on an example associated with some special quivers.

As a generalisation of Arnold's strange duality for unimodal singularities, Ebeling and Takahashi introduced a notion of strange duality for invertible polynomials, which shows a mirror symmetric phenomenon. For each of bimodal singularities, Ebeling produced a Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles by means of its defining equation, which is understood geometrically by Ebeling and Ploog. In my talk, we consider strange-dual pairs of bimodal singularities together with the projectivisations obtained by the one in Ebeling and Ploog's work, by which, we can construct families of K3 surfaces. We discuss whether or not the strange duality extends to dualities of polytopes and lattices for the families. As a consequence, we present that every strange-dual pair can extend to polytope duality, whilst with some exceptions, can extend to lattice duality, and a Hodge-theoretical reason for the lattice duality not being held.

We illustrate a new method to induce stability conditions on semiorthogonal decompositions and apply it to the Kuznetsov component of the derived category of cubic fourfolds. We use this to generalize results of Addington-Thomas about cubic fourfolds and to study the rich hyperkaehler geometry associated to these hypersurfaces. This is the content of joint works with Arend Bayer, Howard Nuer, Marti Lahoz, Emanuele Macri and Alex Perry.

For bounded derived categories of finite-dimensional algebras, due to the bijection of Koenig and Yang, silting objects correspond to t-structures whose hearts are equivalent to module categories of finite-dimensional algebras. Silting-discrete algebras are algebras which have only finitely many silting objects in any interval in the poset of silting objects. Examples of silting discrete algebras include hereditary algebras of finite representation type, derived-discrete algebras, symmetric algebras of finite representation type and many others. I will explain that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. its heart is equivalent to a module category of a finite-dimensional algebra. As a corollary, the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

In this talk we introduce the notion of an indexed chain of torsion classes in an abelian length category $\mathcal{A}$ and we show that any such chain induce a Harder-Narasimhan (like) filtration for every object in $\mathcal{A}$. Later, building on this result, we compare the properties of indexed chains of torsion classes with the stability functions introduced by Rudakov. Finally, following ideas of Bridgeland, we show that the space of all chains of torsion classes induced by the interval $[0,1]$ form a topological space having a wall and chamber structure and we characterise its chambers.

Given a quiver $Q$, it is a challenge to find `optimal' slopes making many representations stable with respect to this slope. In general there is no optimal one, in particular for tame quivers there are many choices (except for the Kronecker quiver, there is just one optimal). For Dynkin quivers we show there is always an optimal slope with the property: a representation is indecomposable precisely if it is stable.

Donaldson-Thomas theory has been developed to study moduli spaces in a 3-Calabi-Yau setting strongly related to superstring theory. It does not only apply to geometry but also to topology and to representation theory. The latter case is well-understood, and we start off by giving the definition and the main properties of the Cohomological Hall algebra for representations of quivers with potential. In the second part we sketch applications involving intersection cohomology of moduli spaces, quantum groups and Kac-Moody algebras. This is a report of joint work with Ben Davison and Markus Reineke.

Let $(Q,W)$ be a quiver with potential having finite-dimensional Jacobian algebra. I will construct from $(Q,W)$ a new algebra, which will be Gorenstein with 2-Calabi-Yau singularity category whenever it is Noetherian, this singularity category being conjecturally equivalent to Amiot's cluster category of $(Q,W)$. Both the Noetherianity and the equivalence are proved to hold when $Q$ is acyclic. The construction appears to be related to the Ginzburg dg-algebra of $(Q,W)$, and I will discuss both precise and conjectural connections to this dg-algebra. Time permitting, I will also discuss results in other Calabi-Yau dimensions.