Thursday, November 1 |
07:30 - 09:00 |
Breakfast (Restaurant at your assigned hotel) |
09:30 - 10:15 |
Anna Barbieri: A Riemann-Hilbert problem from stability conditions ↓ Given an algebra A and a set of automorphisms, one can define a Riemann-Hilbert (RH) problem, aimed to find meromorphic connections on the 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. (Conference Room San Felipe) |
10:30 - 11:00 |
Coffee Break (Conference Room San Felipe) |
11:00 - 11:45 |
Makiko Mase: Duality of families of K3 surfaces and bimodal singularities ↓ 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. (Conference Room San Felipe) |
12:00 - 12:45 |
Paolo Stellari: Cubic fourfolds, noncommutative K3 surfaces and stability conditions ↓ 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. (Conference Room San Felipe) |
13:00 - 15:00 |
Lunch (Restaurant Hotel Hacienda Los Laureles) |
15:00 - 15:45 |
Alexandra Zvonareva: Contractibility of the stability manifold for silting-discrete algebras ↓ 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. (Conference Room San Felipe) |
16:00 - 16:45 |
Hipolito Treffinger: Algebraic Harder-Narasimhan filtrations ↓ In this talk we introduce the notion of an indexed chain of torsion classes in an abelian length category A and we show that any such chain induce a Harder-Narasimhan (like) filtration for every object in 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. (Conference Room San Felipe) |
16:45 - 17:15 |
Coffee Break (Conference Room San Felipe) |
17:30 - 18:15 |
Lutz Hille: Stable representations for Dynkin quivers ↓ 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. (Conference Room San Felipe) |
19:00 - 21:00 |
Dinner hosted by Dr. José Antonio de la Peña (Restaurante Catedral located at: Calle de Manuel García Vigil #105, Centro, Oaxaca.) |