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.

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) Stability data, irregular connections and tropical curvesI will give an overview of recent joint work with S. Filippini and J. Stoppa, in which we construct isomonodromic families of irregular meromorphic connections on P1, with generalized monodromy in the automorphisms of a class of infinite-dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R → 0 “conformal limit” we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). In the R → ∞ “large complex structure limit” the families relate to tropical curves in the plane and tropical/GW invariants. The connections we construct are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto–Moore–Neitzke.
February 12 Alice Rizzardo (Edinburgh) An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves
February 26 Sergey Arkhipov (Aarhus) Quasi-coherent Hecke categories and affine braid group actions

We propose a geometric setting leading to categorical braid group actions. First we consider the quasi-coherent Hecke category QCHecke(G,B) for a reductive group G with a Borel subgroup B. We show that a monoidal action of QCHecke(G,B) on a triangulated category gives rise to a categorification of degenerate Hecke algebra representation known as Demazure Descent Data. Next we replace the group G by the derived group scheme LG of topological loops with values in G and consider QCHecke(LG,LB). A monoidal action of the category QCHecke(LG,LB) gives rise to a categorical action of the affine Braid group.
Trying to avoid heavy derived algebraic geometry methods, we present an example of the construction above in classical algebro-geometric terms. For a G-variety X, we construct a Braid group action on a category of equivariant matrix factorizations on the product of T∗X and the Grothendieck variety for the Lie algebra of G. The potential for matrix factorizations is provided by the moment map.

March 5 No EDGE seminar (GLEN in Glasgow)
March 12 Michael Groechenig (Imperial) Infinite-dimensional vector bundles and reciprocity
March 19 No seminar
March 26 (two talks!) Joseph Karmazyn (Edinburgh) Moduli, McKay, and Minimal Models

Moduli spaces are often used to realise derived equivalences in algebraic geometry. I will recall the examples of the derived equivalence of 3-fold flops in the minimal model program and the derived SL2 McKay correspondence.

These derived equivalences can be translated to noncommutative algebra where they have been extended to include more general settings. I will discuss how a moduli interpretation can be extended to include these derived equivalences with noncommutative algebras.

Rebecca Tramel (Edinburgh) Bridgeland stability on surfaces with curves of negative self-intersection

I consider X a smooth projective complex surface containing a curve C whose self-intersection is negative. In 2002, Bridgeland defined a notion of stability for the objects in Db(X), which generalized the notion of slope stability for vector bundles on curves. The space of such stability conditions is a complex manifold, Stab(X). If we fix a numerical class, then we can decompose Stab(X) into open chambers where the moduli space of stable objects of this class remains constant, and codimension one walls where this moduli space may change.
I consider objects in Db(X) whose numerical class is that of the skyscraper sheaf of a point. In the geometric chamber of Stab(X), the moduli space of stable objects of this class is X itself. In 2012 Toda showed that if C has self-intersection -1, there is a wall to the geometric chamber along which the points of C become semistable. I generalize this result to find such a wall when C has arbitrary negative self-intersection. I then describe the moduli space of stable objects after wall-crossing.

April 23 Tyler Kelly (Cambridge) Equivalences of (Stacky) Calabi-Yaus in Toric VarietiesGiven Calabi-Yau complete intersections in a fixed toric variety, there are possibly various constructions to compute its mirror. Sometimes these mirrors are isomorphic but sometimes not. Mirror symmetry predicts a relationship amongst these so-called double mirrors. In this talk, we will show that the stacky versions of these varieties are derived equivalent. In the proving of this theorem, we get some applications which involve polarisations of K3 surfaces, special degenerate families of CY hypersurfaces in toric varieties, and a generalization of the BHK mirror constructions to families.
May 28, JCMB 6206 Daniel Halpern-Leistner (Columbia) Equivariant Hodge theoryRecent results have revealed a mysterious foundational phenomenon: some quotient stacks for algebraic groups G acting on non-proper schemes X still behave as if they are proper schemes. I will report on one instance of this yoga: one can consider the non-commutative Hodge-to-de-Rham sequence, from Hochschild homology to periodic cyclic homology, for the derived category of coherent sheaves. This spectral sequence degenerates on the first page for smooth and proper schemes, and it turns out that this degeneration also occurs for many "cohomologically proper" quotient stacks. With a little work, this leads to a canonical weight 0 Hodge structure on the Atiyah-Segal equivariant K-theory of the complex analytification of X. The associated graded of the Hodge filtration is the space of functions on the "derived loop space" of the stack.

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