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AGTP 2012

9-14 July 2012

Organisers: Timothy Logvinenko and Miles Reid

AGTP: School on Algebraic Geometry and Theoretical Physics Mon 9th-Sat 14th July 2012, University of Warwick

Partly funded by an interdisciplinary grant of GBP9000 from University of Warwick Institute for Advanced Studies

Organised by Timothy Logvinenko and Miles Reid

Lecture courses by Paul Aspinwall (Duke), Mark Gross (San Diego), Amihay Hanany (Imperial), Alexander Kuznetsov(Steklov Institute) and Balázs Szendrői (Oxford)

Additional lectures by

Pawel Sosna (Universität Hamburg), John Calabrese(Oxford), Richard Thomas (Imperial), Paul Hacking (UMass/Amherst), Timothy Logvinenko (Warwick), Agnieszka Bodzenta-Skibińska (Warsaw), Matthew Ballard (Wien), Hokuto Uehara (Tokyo Metropolitan University), Miles Reid (University of Warwick),

----------- Mon Jul 9th -----------

10:00 - 11:00 Registration 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Paul Aspinwall(Duke) "D-Branes on Calabi-Yau Threefolds I" 12:30 - 14:00 Lunch at the Common Room 14:00 - 15:00 Mark Gross(San Diego) "Cluster varieties and mirror symmetry I" 15:00 - 15:30 Tea & Coffee at the Common Room 15:30 - 16:30 Pawel Sosna (Universität Hamburg) "On the derived category of the classical Godeaux surface" 17:00 - 18:00 John Calabrese(Oxford) "The Crepant Resolution Conjecture for Donaldson-Thomas Invariants"

18:00 - 19:00 Wine reception at the Common Room

------------ Tue Jul 10th ------------

10:00 - 11:00 Mark Gross(San Diego) "Cluster varieties and mirror symmetry II" 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Paul Aspinwall(Duke) "D-Branes on Calabi-Yau Threefolds II" 12:30 - 14:00 Lunch at the Common Room 14:00 - 15:00 Balázs Szendrői (Oxford) "Refined Donaldson-Thomas theory I" 15:00 - 15:30 Tea & Coffee at the Common Room 15:30 - 16:30 Richard Thomas (Imperial) "Cubic fourfolds and K3 surfaces" 17:00 - 18:00 Paul Hacking (UMass/Amherst) "Explicit mirror symmetry for surfaces"

19:00 - Conference Dinner at Radcliffe

------------ Wed Jul 11th ------------ 10:00 - 11:00 Mark Gross(San Diego) "Cluster varieties and mirror symmetry III" 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Paul Aspinwall(Duke) "D-Branes on Calabi-Yau Threefolds III" 12:30 - 14:00 Lunch at the Common Room 14:00 - 15:00 Balázs Szendrői (Oxford) "Refined Donaldson-Thomas theory II" 15:00 - 15:30 Tea & Coffee at the Common Room 15:30 - 16:30 Timothy Logvinenko (Warwick) "DG-categories for the working mathematician" 17:00 - 18:00 Problem session

------------ Thu Jul 12th ------------ 10:00 - 11:00 Alexander Kuznetsov (Steklov Institute) "Homological Projective Duality I" 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Amihay Hanany (Imperial): "Brane Tilings and Specular Duality I" 12:30 - 14:00 Lunch at the Common Room 14:00 - 15:00 Balázs Szendrői (Oxford) "Refined Donaldson-Thomas theory III" 15:00 - 15:30 Tea & Coffee at the Common Room 15:30 - 16:30 Agnieszka Bodzenta-Skibińska (Warsaw) "DG-categories of rational surfaces" 17:00 - 18:00 Problem session

------------ Fri Jul 13th ------------ 10:00 - 11:00 Alexander Kuznetsov(Steklov Institute) "Homological Projective Duality II" 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Amihay Hanany (Imperial): "Brane Tilings and Specular Duality II" 12:30 - 14:00 Lunch at the Common Room 14:00 - 15:00 Matthew Ballard (Wien) "Derived categories and variation of geometric invariant theory quotients" 15:00 - 15:30 Tea & Coffee at the Common Room 15:30 - 16:30 Amihay Hanany (Imperial): "Brane Tilings and Specular Duality III" 17:00 - 18:00 Hokuto Uehara (Tokyo Metropolitan University): "Exceptional collections on toric Fano threefolds and birational geometry"

------------ Sat Jul 14th ------------ 10:00 - 11:00 Alexander Kuznetsov(Steklov Institute) "Homological Projective Duality III" 11:00 - 11:30 Tea & Coffee at the Common Room 11:30 - 12:30 Miles Reid (University of Warwick) "Ice Cream and Orbifold Riemann-Roch" 12:30 - 14:30 Pub lunch in Kenilworth ====

Abstracts:

• Paul Aspinwall (Duke): D-Branes on Calabi-Yau Threefolds

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.

• Mark Gross (San Diego): Cluster varieties and mirror symmetry

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.

• Alexander Kuznetsov(Steklov Institute): Homological Projective Duality

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.

• Amihay Hanany (Imperial): Brane Tilings and Specular Duality

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.

• Balázs Szendrői (Oxford): Refined Donaldson-Thomas theory

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.

Talks:

• Matthew Ballard(Wien): Derived categories and variation of geometric invariant theory quotients

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).

• Agnieszka Bodzenta-Skibińska(Warsaw): DG-categories of rational surfaces

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.

• Paul Hacking (UMass/Amherst): Explicit mirror symmetry for surfaces

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.

• Timothy Logvinenko(Warwick): DG-categories for the working mathematician

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.

• Pawel Sosna(Universität Hamburg): On the derived category of the classical Godeaux surface

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.

• Richard Thomas(Imperial): Cubic fourfolds and K3 surfaces

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.

• Hokuto Uehara(Tokyo Metropolitan University): Exceptional collections on toric Fano threefolds and birational geometry

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.

Aerial photograph of Maths Houses

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Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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