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I will review the A-model and B-model for string theory on Calabi-Yau threefolds and how this is related to homological mirror symmetry. These ideas lead to the notion of stability conditions. The structure of a superpotential and A-infinity algebras will also be discussed.
I will talk about connections between the theory of cluster varieties and mirror symmetry, and in particular with the work of myself, Siebert, Hacking, Keel and Kontsevich. I will explain how our work gives insights into some of the standard conjectures in the theory of cluster varieties, in particular concerning the existence of canonical bases. These turn out to correspond to theta functions as constructed in the above work.
Homological Projective Duality is a relation between a pair of (noncommutative) algebraic varieties (with some additional data) which on one hand generalizes classical projective duality, and on the other hand, captures homological properties of linear sections of the dual varieties. I will explain the general statement of HPD, discuss the proof, and show as many examples as possible.
Brane Tilings are a set of supersymmetric gauge theories in 3+1 dimensions which have the special property of having a moduli space of vacua which is a 3 dimensional Calabi Yau singularity. Brane tilings received recent attention in both mathematics and in physics. These lectures will cover the physics aspects of brane tilings and will introduce a new duality, called specular duality. To characterize this duality we will discuss the combined mesonic and baryonic moduli space, called the master space, and show how this moduli space behaves for a host of examples.
I will start by reviewing some of the ideas around Donaldson-Thomas theory, the theory of counting sheaves on Calabi-Yau threefolds. Then I will discuss some geometric and physical motivations for the existence of a q-deformation or refinement of this theory. Then I will actually construct this theory in a model case, and discuss some sample computations.
I will begin by reviewing some useful stratifications associated to one-parameter subgroups that naturally appear in Mumford's GIT and reviewing how the stratifications behave under varying the linearization defining the GIT quotient. When a single stratum is "flipped" by passing through a wall in the space of linearizations, the derived categories of the GIT quotients on each side of the wall are related by a semi-orthogonal decomposition. The complementary admissible subcategory admits a semi-orthogonal decomposition each of whose components is equivalent to the derived category of a GIT quotient of the fixed locus of the associated one-parameter subgroup. The direction of the semi-orthogonal decomposition and the number of copies of the derived category of the fixed locus depends on parameter analogous to a Morse index which measures the difference in the "size" of the linearized canonical bundles of the strata. Throughout the talk I will try to illustrate connections with well-established results and emphasize examples old and new. All results in the talk are based on joint work, arXiv:1203.6643, with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).
DG-enhancement of a triangulated category allows to assign to every full exceptional collection a finite DG-quiver such that the triangulated category is equivalent to the category of twisted complexes over this quiver. I will describe how to calculate this DG-quiver for exceptional collections with vanishing k-th Ext groups for k > 0. In particular I will discuss DG-quivers of full exceptional collections of line bundles on smooth rational surfaces. I will finish with presenting examples of non-commutative deformations of these surfaces.
The crepant resolution conjecture for DT invariants relates the DT generating series of a CY3 orbifold and of a resolution of its coarse moduli space. I will discuss an approach to the conjecture which uses Bridgeland's perverse coherent sheaves. This ties in quite nicely with how the DT generating series vary under flops and, more generally, birational morphisms.
A long-standing question in the theory of derived categories of coherent sheaves is to establish the existence of derived equivalences corresponding to flops. We will discuss in particular the Grassmannian flop, and show how appropriate equivalences can be quickly established in this case using a non-unique "grade restriction window" inspired by the work of Herbst-Hori-Page on gauged linear sigma models. (This can be viewed as extension of their ideas to non-abelian gauge groups.) We also discuss autoequivalences produced by appropriate composition of equivalences associated to different windows, and give a description of these autoequivalences as twists of certain spherical functors. This is joint work with Ed Segal.
We describe an explicit construction of the mirror to a non-compact Calabi--Yau surface. The mirror is the spectrum of an algebra with a distinguished basis analogous to theta functions on abelian varieties. The algebra structure is constructed using counts of holomorphic curves, similar to the construction of the quantum cohomology ring for compact symplectic manifolds (although unlike that case the algebra has positive Krull dimension). This is joint work with Mark Gross and Sean Keel.
For a variety X acted on by a reductive group, one can consider the derived category of equivariant coherent sheaves on X, or the derived category of a GIT quotient of X. In this talk I will describe a relationship between these two categories: among other things the category of the GIT quotient can be embedded as a full subcategory of the equivariant category. I will explain how it is a common generalization of several classical results: Serre's description of the category of modules on a projective variety, localization and surjectivity theorems for the equivariant cohomology of a variety with a Hamiltonian group action, and the 'quantization commutes with reduction' theorem. I will present some applications of the theory: constructing derived equivalences for some non-abelian VGIT's and stratified Mukai flops, and I will offer an explanation for the appearance of spherical twists as auto equivalences.
I discuss why would anyone sensible want to DG-enhance their triangulated category and what should they expect to gain from it.
I explain twisted complexes and their convolutions, the notion of a pre-triangulated DG-category, the homotopy category of DG-categories as a natural habitat for DG-enhancements, the representability result of Toen and the uniqueness results of Lunts and Orlov. I finish by mentioning some recent developments in this area.
We give an orbifold Riemann–Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of terms each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold terms are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi–Yau 3-folds.
The classical Godeaux surface X is the quotient of the Fermat quintic surface by the action of Z/5Z. I will start by reviewing some aspects of the geometry of this surface. I will then show how this information can be used to prove that the bounded derived category of coherent sheaves on X admits an exceptional collection consisting of 11 line bundles. This is joint work with Christian Böhning and Hans-Christian Graf von Bothmer.
I will start by reviewing, with examples, some of the amazing similarities between cubic 4-folds and K3 surfaces.
Hassett: the (interesting bit of the) Hodge diamond of a cubic 4-fold looks remarkably like that of a K3 surface; conjecturally it is exactly that of a K3 surface iff the cubic 4-fold is rational.
Kuznetsov: the (interesting bit of the) derived category of a cubic 4-fold looks remarkably like that of a K3 surface; conjecturally it is exactly that of a K3 surface iff the cubic 4-fold is rational.
Then I will discuss joint work with Nick Addington. A cubic 4-fold is Hassett if it is Kuznetsov, and the converse is true over at least a Zariski open subset of (each irreducible component of) the moduli space of Hassett cubic 4-folds. If there's time I will review our attempts to close this statement up.
Bernardi and Tirabassi demonstrated that if one assumes a conjecture of Bondal, which states that the Frobenius push-forward of the structure sheaf O_X generates the derived category D^b(X) of a smooth projective toric variety X, then one can show that any smooth toric Fano 3-fold admits a full strong exceptional collection consisting of line bundles.
We prove Bondal's conjecture for smooth toric Fano 3-folds, and improve Bernardi and Tirabassi's result and proof using some birational geometry.