Note time change: the geometry seminar is meeting Thursdays, 2.10-3pm in JCMB 6311 (except where noted otherwise).

It is organized by all faculty working in geometry, and currently coordinated by Arend Bayer. See the Seminars page for instructions on how to subscribe to the geometrytopology mailing list.

The seminar is named after William Edge (1904-1997), who is known for example for his work on finite geometry, and worked at University of Edinburgh for over 40 years (1932-1975).

Related seminars: Topology, MAXIMALS, EMPG, GLEN, COW.

Current Semester

Move your mousepointer on the title of a talk to see an abstract (if available). The schedule is also kept up to date in a google calendar, which you can find below.

September 18 No Seminar
September 25 Ivan Cheltsov (Edinburgh) What are the worst singular points of plane curves of given degree?
October 2 Martin Kalck (Edinburgh) Relative singularity categories(Relative) singularity categories are triangulated categories associated with (non-commutative resolutions of) singular varieties. I will explain these notions and their mutual relations focusing on the simplest examples - the singularities of type A_1, e.g. k[x|/x^2. For these examples, everything can be understood in a rather elementary way. In particular, familiarity with triangulated categories will NOT be necessary to follow the talk. In the end, I will mention what we know for ADE-singularities in general. This is based on joint work with Dong Yang.
October 9 Brent Pym (Oxford) Quantum deformations of projective three-spaceIn noncommutative projective geometry, quantum versions of projective space are often described in terms of their homogeneous coordinate rings, which are noncommutative analogues of polynomial rings. The algebras corresponding to quantum projective planes were classified in geometric terms by Artin, Tate and Van den Bergh in a celebrated 1990 paper. The related problem for projective three-space has received considerable attention, but the full classification remains elusive. I will describe some recent progress on this problem, in which deformation quantization is combined with Cerveau and Lins Neto's classification of foliations on projective space to give a classification of the flat deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.
October 16 Balazs Szendroi (Oxford) Motivic Donaldson-Thomas series of deformed Calabi-Yau geometriesMotivic DT theory gives a refined count of objects in 3-CY categories, for example sheaves on Calabi-Yau threefolds. At least in the local quiver setting, it is easy to ask about motivic counts of objects in categories defined by deformed Calabi-Yau spaces, such as quantum three-space or the affine cone over the Jordan plane. We compute the answer in some cases, and conjecture it in some others, based on an intriguing and sofar not always precise formula involving the motivic count of simple objects only. (Joint work with Andrew Morrison and Brent Pym)
October 23 Evgeny Smirnov (HSE Moscow) Spherical double flag varieties

Classical Schubert calculus deals with orbits of a Borel subgroup in GL(V) acting on a Grassmann variety Gr(k,V) of k-planes in a finite-dimensional vector space V. These orbits (Schubert cells) and their closures (Schubert varieties) are very well studied both from the combinatorial and the geometric points of view.

One can go one step farther, considering the direct product of two Grassmannians Gr(k,V)x Gr(l,V) and the Borel subgroup in GL(V) acting diagonally on this variety. In this case, the number of orbits still remains finite, but their combinatorics and geometry of their closures become much more involved. However, something still can be said about them. I will explain how to index the closures of a Borel subgroup in Gr(k,V)xGr(l,V) combinatorially and construct their desingularizations, which are similar to Bott-Samelson desingularizations for ordinary Schubert varieties. I will also mention the analogues of these results for direct products of partial flag varieties for reductive groups of type different from A_n.

October 30 Chunyi Li (Edinburgh) Topics on the Hilbert scheme of points on projective planeThe Hilbert scheme of n points in P2 parameterizes all 0-dimensional subschemes on P2 with length n. It is a smooth version of n-th symmetric product of P2. I will introduce some of its 'neighborhoods' in the area of birational geometry and deformation theory.
November 6, JCMB 5215 Gavin Brown (Loughborough) Elliptic Gorenstein projection following Coughlan

The archetype elliptic Gorenstein singularity on a 3-fold
is the vertex of the affine cone over a K3 surface.
In favourable circumstances, within a family of 3-folds
of general type, one may be able to degenerate to
a 3-fold with an elliptic Gorenstein singularity, and then
project away from this point to a 3-fold of general type
in a smaller ambient projective space.
Used in reverse, this resembles Fano's famous strategy for
constructing Fano 3-folds whose anticanonical embedding is
in high codimension: construct the image of some projection
and then undo the projection. We try the same with canonical
3-folds (embedded by K_X = O(+1), rather than O(-1) for Fanos)
and projection from elliptic Gorenstein points.
As ever, the more subtle phenomena begin to arise
in codimension 4, when we try to write X in wP^7.
This is joint with Stephen Coughlan (Hannover).

November 13 Susan J. Sierra (Edinburgh) Moduli of point representations, the enveloping algebra of the Virasoro algebra, and noetherianityI will explain the geometric methods underlying my 2013 proof (with Walton) that the universal enveloping algebra of the Virasoro algebra is not noetherian. This was a problem that had been open for more than 20 years and had resisted many attempts at an algebraic proof.
November 20 Georg Oberdieck (Zurich) Curve counting in K3 x P1 and the Hilbert scheme of points of a K3 surfaceLet S be a K3 surface. Recent results suggest a correspondence between the Gromov-Witten and Donaldson-Thomas theory of S x P1 and the genus 0 GW theory of the Hilbert scheme of points of S. I will explain these results and how they lead to precise conjectures about the correspondence. As an application, I will discuss a conjecture for the full GW theory of the Calabi Yau threefold S x E, where E is an elliptic curve. This is joint work with R. Pandharipande.
November 27 No seminar
December 11, 3pm, JCMB 6206 Cristina Manolache (Imperial) Mirror symmetry without localizationMirror Symmetry predicts a surprising relationship between the number of curves in a target space X and certain hypergeometric functions. Existing proofs rely on complicated localisation computations. I will describe a new, more conceptual proof.
January 15 Michael Wemyss (Edinburgh) Aspects of the Homological MMP

I will outline the main ideas of arXiv:1411.7189, which uses noncommutative deformations and universal properties to jump between minimal models in the MMP in a satisfyingly algorithmic fashion. As part of this, a flop is constructed not by changing GIT, but instead by changing the algebra keeping GIT fixed, and flops are detected by whether certain contraction algebras are finite dimensional. Carrying this extra information allows us to continue to flop, and thus continue the MMP, without having to calculate everything at each stage.

Proving things in this canonical categorical manner allows us then to say things about GIT. In fact the HomMMP computes the full wall and chamber structure, and also gives a method for determining which walls produce flops and which do not. If there is time, I will explain that it also can be used to prove that flop functors braid in dimension three, however the combinatorics are not the expected one, and higher length braid relations naturally appear.

February 5 Mario García Fernández (ICMAT) TBA
February 12 Alice Rizzardo (Edinburgh) TBA

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