Edinburgh Geometry Seminar 2011
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54:40
The surface of cuboids and Siegel modular threefolds - Damiano Testa (Warwick) motionmaker
Date: Thursday 2nd February 2012Speaker: Damiano Testa (Warwick).Title: The surface of cuboids and Siegel modular threefolds.Abstract: A perfect cuboid is a parallelepiped with rectangular faces all of whose edges, face diagonals and long diagonal have integer length. A question going back to Euler asks for the existence of a perfect cuboid. No perfect cuboid has been found, nor it is known that they do not exist. In this talk I will first compute the Picard group of the space of cuboids (joint with M. Stoll). Then, I will show that the space of cuboidsis a divisor in a Siegel modular threefold, thus allowing to translate the existence of a perfect cuboid to the existence of special torsion structures in abelian surfaces defined over number fields.http://www.maths.ed.ac.uk/cheltsov/seminar/
58:06
Deformations of G2-structures with torsion - Sergey Grigorian (Stony Brook)
Date: Thursday 26th January 2012Speaker: Sergey Grigorian (Stony Brook).Title: Deformations of G2-structures with torsion.Abstract: We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive a closed expression for the torsion of the deformed G2-structure. We then specialize to the case where the deformation liesin the seven-dimensional representation of G2 and is hence defined by a vector v. In this case, we explicitly derive the expressions for the different torsion components ofthe new G2-structure in terms of the old torsion components and derivatives of v. In particular this gives a set of differential equations for the vector v which have to be satisfied for a transition between G2-structures with particular torsions. For some specific torsion classes we then explore the solutions of these equations.http://www.maths.ed.ac.uk/cheltsov/seminar/
37:09
Constructions of generalized complex structures in dimension four - Rafael Torres (University of Oxford)
Date: Thursday 19th January 2012Speaker: Rafael Torres (University of Oxford).Title: Constructions of generalized complex structures in dimension four.Abstract: Recent constructions of exotic smooth structures on small 4-manifolds can be canonically used to expand our understanding of generalized complex structures. This talk will be an exposition of such an enterprise, whose produce include unbosoming unexpected phenomena on the number of type change loci of a generalized complex structure, and their construction on a myriad of 4-manifolds.http://www.maths.ed.ac.uk/cheltsov/seminar/
01:16:01
Topology from cosmology - Joan Simon (Edinburgh) motionmaker
Date: Thursday 1st December 2011Speaker: Joan Simon (Edinburgh).Title: Topology from cosmology.Abstract : I will informally discuss how the Euler number of Calabi-Yau 3-folds (CY) can be constrained due to cosmological inflationary measurements in certain string theory compactification set-ups. The emphasis of the talk will be on the links between seemingly different disciplines rather than in a precise theoretical presentation.http://www.maths.ed.ac.uk/cheltsov/seminar/
54:57
Weakly-Exceptional Quotient Singularities - Dmitrijs Sakovics (Edinburgh) - EDGE Seminar motionmaker
Date: Thursday 24th November 2011Speaker: Dmitrijs Sakovics (Edinburgh).Title: Weakly-Exceptional Quotient Singularities.Abstract: The classification of A-D-E singularities on surfaces and their relation to Platonic solids and regular polygons is a very well-known classical result. I will discuss one of the possible generalizations of this idea into higher dimensions and some recent results on the classification of such singularities.http://www.maths.ed.ac.uk/cheltsov/seminar/
57:43
Higher dimensional gauge theory and Fueter maps - Andriy Haydys (Bielefeld) - EDGE seminar motionmaker
Date: Thursday 10th November 2011Speaker: Andriy Haydys (Bielefeld).Title: Higher dimensional gauge theory and Fueter maps.Abstract: I will describe a generalization of the anti-self-duality equations for manifolds with exceptional holonomies due to Donaldson and Thomas. Under certain circumstances, higher dimensional instantons can degenerate to Fueter maps, which constitute a class of harmonic maps between hyperKaehler manifolds. I will describe some properties of Fueter maps and their relation to the compactified moduli space of higher dimensional instantons.http://www.maths.ed.ac.uk/cheltsov/seminar/
58:30
Window-shifts and Grassmannian twists - Ed Segal (Imperial) - EDGE seminar motionmaker
Date: Thursday 3rd November 2011Speaker: Ed Segal (Imperial).Title: Window-shifts and Grassmannian twistsAbstract: 'Window-shifting' is a new technique for constructing equivalences and symmetries of derived categories of sheaves. I'll warm up by discussing how we apply it in the case of the standard flop, and show that it produces the Siedel-Thomas spherical twist. I'll then go on to our main example which is a vector bundle over a Grassmannian, in this case we can produce an autoequivalence which seems to be completely new. I'll then describe a more geometric construction of this autoequivalence.This is joint work with Will Donovan.http://www.maths.ed.ac.uk/cheltsov/seminar/
51:44
Moduli spaces in graded ring theory - Sue Sierra (Edinburgh) - Edinburgh Geometry Seminar motionmaker
Date: Thursday 20th October.Speaker: Sue Sierra (Edinburgh).Title: Moduli spaces in graded ring theoryAbstract: Let R be a noetherian N-graded algebra, generated in degree 1,over the complex numbers. A point module is a cyclic R-module withHilbert series 1/(1-s). If R is strongly noetherian --- that is, it remains noetherian upon base extension --- then its point modules are parameterized by a projective scheme X, and this induces a canonical map from R to a twisted homogeneous coordinate ring on X. This technique was crucial in the analysis of noncommutative P^2's (regular algebras of dimension 3).We study a non-strongly noetherian case: the noncommutative Rees rings known as naive blowup algebras. We show there is a stack that represents point modules, and that a certain equivalence relation on point modules is corepresented by a projective scheme.We show that this geometry characterises naive blowup algebras. This is joint work with Tom Nevins.
54:22
Rationality of quotients by p-groups - Constantin Shramov (HSE/Steklov) - Edinburgh Geometry Seminar motionmaker
Date: Thursday 27th October.Speaker: Costya Shramov (Moscow).Title: Rationality of quotients by p-groups.Abstract: Let G be a p-group and V its complex representation.I'll survey the known results on rationality and stable rationality of V/G,in particular the obstructions to rationality.Web: http://www.maths.ed.ac.uk/cheltsov/seminar/This seminar is a part of Edinburgh-Glasgow-Aberdeen seminar.See http://www.maths.gla.ac.uk/~anc/highlandcow/
53:42
Laurent Polynomials in Mirror Symmetry - Victor Przyjalkowski - Edinburgh Geometry Seminar motionmaker
Date: Thursday 13th October 2011Speaker: Victor Przyjalkowski (Moscow, Vienna).Title: Laurent Polynomials in Mirror SymmetryAbstract: We discuss quantitative properties of Mirror Symmetry correspondencefor Fano varieties. Laurent polynomials naturally appear in this picture.They describe (the essential part of) dual Landau--Ginzburg models for Fanos.They are related to toric degenerations of the initial Fano varieties.Their relative compactifications are candidates for Landau--Ginzburgmodels from the Homological Mirror Symmetry point of view.We consider our main example --- Fano threefolds.We discuss why Landau--Ginzburg model (for given Fano variety)represented by Laurent polynomial is unique and why it is not unique.Web: http://www.maths.ed.ac.uk/cheltsov/seminar/
53:49
Noncommutative Minimal Models and applications to geometry - Michael Wemyss - Edinburgh Geometry Seminar motionmaker
Date: Thursday 6th October.Speaker: Michael Wemyss (Edinburgh).Title: Noncommutative Minimal Models and applications to geometry.Abstract: I will try and explain the ideas behindnoncommutative minimal models (=MMAs) and why they should notonly reprove parts of the MMP in dimension three, but also give usextra information that currently the geometry does not "see".This extra information (in the form of a quiver) should then allow usto run aspects of the MMP in a much easier way.The talk will mainly be example based,but as an application of the homological techniques,I will give (in the dimension three Gorenstein setting)a characterization of the Q-factorial propertyin terms of derived categories.This is joint with Iyama.Web: http://www.maths.ed.ac.uk/cheltsov/seminar/
54:21
Sergey Galkin - More symmetries of K3. motionmaker
Date: Thursday 29th September 2011Speaker: Sergey Galkin, IMPU (Tokyo, Japan).Title: More symmetries of K3.Abstract: I will discuss different natural extensions of the holomorphic symmetry group of K3 surface (using moduli and derived category) and will show new interesting examples.Web: http://www.maths.ed.ac.uk/cheltsov/seminar/