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58:15
Compact moduli spaces for slope-semistable sheaves - Daniel Greb motionmaker
Date: Thursday 14th March 2013 Speaker: Daniel Greb (Bochum) Title: Compact moduli spaces for slope-semistable sheaves Abstract: While the variation of moduli spaces of H-slope/Gieseker-semistable sheaves on surfaces under change of the ample polarisation H is well-understood, research on the corresponding question in the case of higher-dimensional base manifolds revealed a number of pathologies. After presenting these, I will discuss recent joint work with Matei Toma (Nancy), which resolves some of these pathologies by looking at curves instead of divisors. This naturally leads to the question whether higher-dimensional analogues of the Donaldson-Uhlenbeck compactification exists, and I will discuss our construction of such a compactification. http://www.maths.ed.ac.uk/cheltsov/seminar/
53:13
Noether-Lefschetz theory for hypersurfaces in toric 3-folds - Ugo Bruzzo motionmaker
Date: Thursday 14th March 2013 Speaker: Ugo Bruzzo (SISSA) Title: Noether-Lefschetz theory for hypersurfaces in toric 3-folds Abstract: We prove a sufficient criterion for a very general hypersurface in simplicial toric 3-fold to have the same Picard number as the ambient variety. We also give some results about the estimate of the codimension of the locus where the Picard number is bigger. http://www.maths.ed.ac.uk/cheltsov/seminar/
58:03
The cavalcade of Poncelet's theorem - Emma Praviato motionmaker
Date: Tuesday 12th March 2013 Speaker: Emma Praviato (Boston) Title: The cavalcade of Poncelet's theorem Abstract: This talk will review results of Poncelet type that involve special functions and Hamiltonian systems. http://www.maths.ed.ac.uk/cheltsov/seminar/
01:05:09
Cluster varieties and stability conditions - Tom Bridgeland motionmaker
Date: Thursday 28th February 2013 Speaker: Tom Bridgeland (Oxford) Title: Cluster varieties and stability conditions Abstract: I will begin in a very elementary way by explaining the operation of mutation of quivers, and using it to define two complex manifolds, one of them being the cluster variety, and the other being a baby version of the space of stability conditions. I will then describe what these spaces look like for a class of quivers related to triangulations of surfaces, and point out the connection with moduli spaces of Higgs bundles and the Hitchin map. I will finish by discussing some ideas of physicists Gaiotto, Moore and Neitzke which suggest that these examples should fit into a more general framework. http://www.maths.ed.ac.uk/cheltsov/seminar/
01:00:33
An extension of Kodaira vanishing in arbitrary codimension - Victor Lozovanu motionmaker
Date: Thursday 7th February 2013 Speaker: Victor Lozovanu (Paris 6) Title: An extension of Kodaira vanishing in arbitrary codimension Abstract: One of the most celebrated theorems in complex algebraic geometry is Kodaira vanishing, together with its extension due to Kawamata and Viehweg. In this talk I will discuss about a joint work with Greg G. Smith, where we generalize Kodaira vanishing to arbitrary codimension. I will present a few applications to this work; by giving answers to questions of projective normality and bounding the multigraded regularity. I will also discuss some future projects that derive nicely from this work. http://www.maths.ed.ac.uk/cheltsov/seminar/
59:39
Spherical DG-functors - Timothy Logvinenko motionmaker
Date: Thursday 29th November 2012Speaker: Timothy Logvinenko (Warwick)Title: Spherical DG-functorsAbstract: Seidel-Thomas twists are autoequivalences of the derived category D(X) of an algebraic variety X. They are the mirror symmetry analogues of Dehn twists along Lagrangian spheres on a symplectic manifold. Given an object E in D(X) with numerical properties of such a sphere, Seidel and Thomas defined the spherical twist of D(E) along E, proved it to be an autoequivalence and gave braiding criteria for several such twists.It was long understood that all of the above should generalise to the notion of the twist along a spherical functor into D(X). In full generality this was long obstructed by some well-known imperfections of working with triangulated categories. In this talk, I present joint work with Rina Anno, where we fix this by working with the standard DG-enhancement of D(X). We define the notion of a spherical DG-functor and give the braiding criteria for twists along such functors.http://www.maths.ed.ac.uk/cheltsov/seminar/
01:08:02
Non-commutative resolutions and Grothendieck groups - Charles Vial motionmaker
Date: Thursday 25th October 2012Speaker: Charles Vial (Cambridge)Title: Non-commutative resolutions and Grothendieck groupsAbstract: This is joint work with Hailong Dao, Osamu Iyama and Ryo Takahashi. A finitely generated module M over a commutative noetherian ring R is said to give a non-commutative resolution (NCR) of R if M is faithful and End_R(M) has finite global dimension. The aim of this talk is to discuss the relevance of such a definition and to give necessary conditions for the existence of NCRs. These conditions focus on the Grothendieck group of the category of finitely generated modules over R and its subcategories. This group is related, via Riemann-Roch, to the group of so-called algebraic cycles. I will explain how methods from the theory of algebraic cycles can be used in that setting and I will show that a standard graded Cohen-Macaulay algebra R over a field of zero characteristic with only rational singularities outside the irrelevant ideal has a NCR only if R has rational singularities.http://www.maths.ed.ac.uk/cheltsov/seminar/
59:46
Towards algebro-geometric understanding of K-stability of Fano varieties - Yuji Odaka motionmaker
Date: Tuesday 27th November 2012Speaker: Yuji Odaka (Imperial)Title: Towards algebro-geometric understanding of K-stability of Fano varieties.Abstract: The K-stability of Fano manifold was introduced by Tian in 90s and reformulated, generalized in more algebraic way by Donaldson. It is a (conjectural) counterpart of existence of Kahler-Einstein metrics which gives a "unique" way to regard Fano variety as Kahler (thus Riemannian) manifold, which the speaker has been expected to give application to construction of projective moduli variety of Fano varieties and even more general polarized varieties ("K-moduli") as well.However, for a given Fano manifold, it is actually hard to see it is K-stable or not in general, even for simple examples. This is the main focus of the talk. Reviewing the developments so far, I would like to discuss for future developments.http://www.maths.ed.ac.uk/cheltsov/seminar/
55:15
Mori Dream Spaces as fine moduli of quiver representations - Alastair Craw motionmaker
Date: Thursday 1st November 2012Speaker: Alastair Craw (Glasgow)Title: Mori Dream Spaces as fine moduli of quiver representationsAbstract: Mori Dream Spaces provide a relatively large class of examples in algebraic geometry that nevertheless exhibit many beautiful properties, the most obvious being that all operations of Mori Theory can be carried out by variation of GIT quotients. In particular, every Mori Dream Space can be constructed as a GIT quotient in a natural (though not canonical) way. In this talk I will introduce alternative GIT constructions using noncommutative algebra. As an application, we refine a result of Bergman-Proudfoot by showing that del Pezzo surfaces can be reconstructed from a tilting bundle.http://www.maths.ed.ac.uk/cheltsov/seminar/
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/
53:49
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/
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/