Forthcoming events in this series
Cohomology of Hilbert schemes of plane curve singularities and the triply graded Khovanov-Rozansky homology of their links
Abstract
I describe a conjecture equating the two items appearing in the title.
Mirror symmetry and mixed Hodge structures I
Abstract
I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will discuss computable Hodge theoretic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.
Derived Categories of Cubic 4-Folds
Abstract
If $X$ is a Fano variety with canonical bundle $O(-k)$, its derived category
has a semi-orthogonal decomposition (I will say what that means)
\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]
where the subcategory $A$ is the "interesting piece" of $D(X)$. In the previous talk we saw that $A$ can have very rich geometry. In this talk we will see a less well-understood example of this: when $X$ is a smooth cubic in $P^5$, $A$ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $X$ is rational if and only if $A$ is geometric, relate it to Hassett's earlier work on the Hodge theory of $X$, and mention an autoequivalence of $D(Hilb^2(K3))$ that I came across while studying the problem.
Complete Intersections of Quadrics
Abstract
There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used
as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.
(HoRSe seminar) Localized virtual cycles, and applications to GW and DT invariants II
Abstract
We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).
(HoRSe seminar) Localized virtual cycles, and applications to GW and DT invariants I
Abstract
We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).
Wall-crossing and invariants of higher rank stable pairs
Abstract
rank stable pairs (which we call frozen triples) given by the data $(F,\phi)$ where $F$ is a pure coherent sheaf with one dimensional support over $X$ and $\phi:{\mathcal O}^r\rightarrow F$ is a map. We compute the Donaldson-Thomas type invariants associated to the frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. This work is a sequel to arXiv:1011.6342, where we gave a deformation theoretic construction of a higher rank enumerative theory of stable pairs over a Calabi-Yau threefold, and we computed similar invariants using Graber-Pandharipande virtual localization technique.
Cobordisms of sutured manifolds
Abstract
Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.
Gravitational instantons from rational elliptic surfaces
Abstract
Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.
(HoRSe seminar) On the calculus underlying Donaldson-Thomas theory II
Abstract
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli
space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
(HoRSe seminar) On the calculus underlying Donaldson-Thomas theory I
Abstract
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
15:45
Counting invariants for the ADE McKay quivers
Motivic Donaldson-Thomas invariants and 3-manifolds
Abstract
I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.
Topological quantum field theory structure on symplectic cohomology
Abstract
Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the
homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in
symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed
Hamiltonian orbits and Reeb chords.
Finite time singularities for Lagrangian mean curvature flow
Abstract
I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold.
(HoRSe seminar) Spherical objects on K3 surfaces II
Abstract
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
(HoRSe seminar) Spherical objects on K3 surfaces I
Abstract
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
(HoRSe seminar) ADHM Sheaves, Wallcrossing, and Cohomology of the Hitchin Moduli Space II
Abstract
The second talk will present conjectural motivic generalizations
of ADHM sheaf invariants as well as their wallcrossing formulas.
It will be shown that these conjectures yield recursive formulas
for Poincare and Hodge polynomials of moduli spaces of Hitchin
pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.
(HoRSe seminar) ADHM Sheaves, Wallcrossing, and Cohomology of the Hitchin Moduli Space I
Abstract
The first talk will present a construction of equivariant
virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.