Edinburgh Geometry Seminar 2012Playlist created by weebirationalist | 9 videos | See more playlists »
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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/