Algebraic and Symplectic Geometry Seminar

I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on $R^4$, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group $U(1)$. I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to a structural result about vector spaces.

This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

Derived moduli stacks extend moduli stacks to give families over simplicial or dg rings. Lurie's representability theorem gives criteria for a functor to be representable by a derived geometric stack, and I will introduce a variant of it. This establishes representability for problems such as moduli of sheaves and moduli of polarised schemes.

A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring

satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative

differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology

is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes

without encountering gluing issues.

I show how to construct some Fano 3-folds that have the same Hilbert series but different Betti numbers, and so lie on different components of the Hilbert scheme. I would like to show where these fit in to a speculative (indeed fantastical) geography of Fano 3-folds, and how the projection methods I use may apply to other questions in the geography.

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topologica vertex formalism, for all toric Calabi-Yau's admitting the latter.

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topological vertex formalism, for all toric Calabi-Yau's admitting the latter.

I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of  Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.