2018 Term 1
Week 1, 05/10/18
Aurelio Carlucci (University of Oxford)
Moduli spaces of stable pairs on the conifold and quiver representations
A quiver is a collection of arrows, that is an oriented graph. They appear in different branches of mathematics and theoretical physics, as a means to encode, in an efficient way, combinatorial information arising from algebraic or categorical data. In certain cases, they allow to describe moduli spaces appearing in algebraic geometry.
This talk is going to be an introduction on the latter aspect, with a perspective on my current research. We will start by considering classical examples, like the Kronecker quiver or the trefle quiver, and the corresponding moduli spaces, hinting at the applications to enumerative geometry.
We will move then to the topic of representations of the path algebra associated to a quiver, and the notion of stability introduced by King; there is indeed a remarkable translation, in the quiver case, of the more general notion of stability for an abelian category.
In particular, Nagao and Nakajima proved that some classes of moduli spaces of sheaves on the resolved conifold, appearing in enumerative geometry, can be interpreted as moduli spaces of representations of a certain framed quiver, allusively called conifold quiver. We will unravel the details of this association, showing how simple computations can lead to the characterisation of stability conditions for representations of the conifold quiver.
This leads to an explicit coordinate description of the moduli spaces of stable pairs on the resolved conifold: we confirm the results for the known cases, and we complement the information already obtained by a more geometric approach in a new case.
Week 2, 8/10/2018
Lorenzo De Biase (Cardiff University)
Generalised braid categorification
Ordinary braid group Brn is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continuous transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space Fln of complete flags in Cn. Generalised braids are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In this talk I will present some progress that have been made towards extending the result of Khovanov and Thomas to the categorification of the generalised braid category.
Week 3, 15/10/2018
Lucas das Dores (University of Liverpool)
Symmetric powers of varieties and rational equivalence of 0-cycles
For any quasi-projective variety X over a field k, its d-th symmetric power is the quotient of the its d-th Cartesian product by the action of the permutation group of d elements.
In particular, if the field is algebraically closed, the k-points of the d-th symmetric powers of the variety correspond to a zero-cycle of degree d on X. Moreover, rational equivalence of zero cycles is encoded by rational curves on symmetric powers of X.
In this talk, we review a classical result by Mumford stating that if X is a nonsingular projective surface with positive geometric genus, its group of 0-cycles of degree 0 modulo rational equivalence is not finite dimensional.
Week 4, 22/10/2018
Peter Spacek (University of Kent)
Mirror Symmetry for Homogeneous Spaces
In this talk we will take a look at mirror symmetry at the level of the small quantum cohomology, specialized to the case of homogeneous spaces. We will introduce Gromov-Witten invariants and how they are involved in the small quantum cohomology using complex projective space as an illustration. We will stay with projective space, but now consider it as a homogeneous space, and work out a method to construct a mirror pair for it: that is, a variety whose coordinate ring modulo relations defined by a rational function models the small quantum cohomology of projective space. Finally, if time allows, we will try to give some insights how the construction works for more general homogeneous spaces.