András Sándor: Deformations of a nice class of non-isolated toric singularities through mirror symmetry.
I will talk about an application of the exciting correspondance between the deformations of non-isolated Gorenstein toric singularities and Laurent polynomials through mirror symmetry. We will study a nice class of Gorenstein toric 3-folds defined by certain triangles and we want construct formal deformations. For this, we will pass through the mirror and dive into the world of Laurent polynomials and their mutations. This leads us to smoothings and Fano fibres too.
Davide Accadia: Completed cycles k-Leaky Hurwitz Numbers
Double Hurwitz numbers count the number of branched coverings from a genus g curve to P^1 with two fixed branching points and ramification profiles. The tropicalization procedure highlights a parallel with counting covering graphs with a fixed flow or current
through the graph. It is possible to define a $k$-leaky version of these graphs, so that every node "leaks" a constant amount of the flow. Double Hurwitz numbers can be written in the form of Vacuum Expectation Values in the Fock space, and so can Leaky Hurwitz numbers. Fock space formalism has made it possible to adapt proofs for many structural results, namely: piecewise polynomiality, wall crossing formulae, and cut and join equation. This talk is based on joint work with D. Lewański
and M. Karev.
Okke van Garderen: Quiver moduli, knots and Hall algebras
The knot-quiver correspondence is a conjectural relation between knot-invariants and Betti numbers of quiver moduli spaces. In this talk I will give some evidence that this relation can be categorified, by showing that skein relations from the knot-side can be expressed using the cohomological Hall algebra of the quiver.
Francesco Zucconi: Supported deformations and birational geometry
If the general Kodaira-Spencer classes of a fibration vanish, then the general fibers are isomorphic.
In the talk, we present a birational analogue of the above result assuming that the general fiber is smooth of general type and the Kodaira-Spencer classes are supported on a horizontal divisor D.
In this case, we can construct a horizontal meromorphic vector field with poles only on D.
If D restricts fiberwise to a non-movable divisor and if D is invariant for the two foliations naturally encoded in the hypothesis then the vector field can be locally integrated outside D and this forces the fibers to be birational.
Søren Gammelgaard: Quiver schemes and reducedness
Let Q be a quiver, and A a quiver algebra (i.e., a quotient of the path algebra of Q). We say that a quiver scheme is a moduli scheme of A-modules. Such quiver schemes are versatile tools in algebraic geometry, and we will focus on the class of Nakajima quiver schemes. In much recent work, these have been used to construct various moduli spaces attached to the Kleinian singularity C^2/G, such as the Hilbert schemes Hilb^n(C^2/G), the equivariant Hilbert schemes nG-Hilb(C^2), and various "equivariant Quot schemes". However, a persistent issue is that Nakajima quiver schemes are not known to be reduced in general. Using a variant of the Le Bruyn-Procesi theorem, we will sketch our proof that Nakajima quiver schemes whose underlying reduced scheme is a point, are themselves isomorphic to a point. We will also give an example of a closely related quiver scheme that is not reduced. This is joint work with Lukas Bertsch.
10:15: Okke van Garderen (SISSA)
11:30 : Davide Accadia (Università di Trieste)
12:30: Lunch
14:30: Francesco Zucconi (Università di Udine)
15:40: Andras Sandor (University of Ljubljana)
16:50: Søren Gammelgaard (Università di Ferrara)