Unless noted otherwise, the geometry seminar is meeting Thursdays, 3pm in JCMB 6311.

It is organized by all faculty working in geometry, and currently coordinated by Arend Bayer.

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 16, JCMB 4312 Francesco Sala (Heriot-Watt) Framed sheaves on root stacks and gauge theory on ALE spacesThis talk is about a new approach to the study of supersymmetric gauge theories on ALE spaces of type $A_{k-1}$, for $k\geq 2$, by using the theory of framed sheaves on 2-dimensional root toric stacks. In particular, I will describe a ``stacky compactification" of a minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_{k}$ and moduli spaces of framed sheaves on it. These moduli spaces provide a new setting for the study of gauge theories on ALE spaces and can be used to define new geometric realizations of representations of affine/vertex algebras. In the last part of the talk I will focus on the case of rank one framed sheaves: in that case I will describe a geometric construction of a highest weight representation of $\widehat{\mathfrak{sl}}(k)$ at level one and will characterize some Carlsson-Okounkov type vertex operators on $\widehat{\mathfrak{sl}}(k)$ that have a gauge-theoretic meaning.
January 23, 4 pm, JCMB 5326 Diane Maclagan (Warwick) Tropical schemes, tropical cycles, and valuated matroidsThe tropicalization of a subvariety of a torus records the cycle of its compactification in an ambient toric variety. In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure. I will introduce these notions, and outline how to also recover the tropical cycle from this information. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon
January 30 Alastair King (Bath) Grassmannian cluster categories and dimers on a disc
February 6 Pierre Schapira (Paris VI) Microlocal Euler classes and Hochschild homology

This is a joint work with Masaki Kashiwara.
On a complex manifold $(X;O_X)$, the Hochschild homology is a powerful tool to construct characteristic classes of coherent modules and to get index theorems. Here, I will show how to adapt this formalism to a wide class of sheaves on a real manifold M by using the functor uhom of microlocalization. Hence, the analogue of the Hochschild homology lives now in T*M, the cotangent bundle. It is isomorphic to the inverse image by $\pi_M: T*M \to M$ of $\omega_M$, the topological dualizing complex on M. This construction applies in particular to constructible sheaves on real manifolds and D-modules on complex manifolds, or more generally to elliptic pairs.

February 13 Adam Boocher Closures of a linear spaceLet L be an affine linear space. Once we fix coordinates, it makes sense to discuss the closure of L inside a product of projective lines. In this talk I'll present joint work with Federico Ardila concerning the defining ideal of the closure. It turns out that the combinatorics of this ideal are completely determined by a matroid associated to L and we are able to explicitly compute its degree, universal Groebner basis, Betti numbers, and initial ideals. I'll include several examples along the way and discuss how this closure operation comes up naturally when one searches for ideals with "nice" behavior upon degeneration.
February 27 TBA TBA
March 6 Tara Holm (Cornell) The topology of toric origami manifoldsA folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will describe how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.
March 13 Paolo Stellari (Milano) Fourier-Mukai functors: derived vs dg categoriesFourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless a basic question is still open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? We discuss the recent advances in the subject and study the same question in the context of dg categories where the problem has been settled by B. Toen. In this talk we propose a simpler approach not based on the notion of model category. This is a joint work with A. Canonaco.
March 20, 4pm, JCMB 4312 Vladimir Baranovsky (UC Irvine) Deformation quantization of smooth Lagrangian subvarietyWe consider a smooth algebraic variety X with an algebraic symplectic form and a deformation O_h (with a formal parameter h) of the structure sheaf compatible with this form. For a smooth Lagrangian subvariety Y in X and a line bundle L on Y, we describe when L also admits a deformation to a module over O_h. Joint work with V. Ginzburg, D. Kaledin and J. Pecharich.
March 21 3 pm, Faculty Room North, David Hume Tower, George Square Hodge Seminar Etienne Ghys (Lyon) Some remarks on singularities of real analytic curves in the planeThe topological nature of singularities of planar complex analytic curves has been understood for many years. Indeed, Newton-Puiseux series allow a complete understanding of the link of the singularity. Amazingly, the analogous question in the real domain leads to interesting combinatorial developments. I will start my discussion with a clever remark of M. Kontsevich.
March 27, 4 pm, JCMB 5325 Nicolò Sibilla Ribbon graphs, skeleta and homological mirror symmetryIn this talk I will review recent work of mine, partially in collaboration with H. Ruddat, D. Treumann and E. Zaslow, which centers on various aspects of Kontsevich's Homological Mirror Symmetry in the large complex limit. The one-dimensional case will be emphasized, as a convenient testing ground for more general constructions.
April 3, JCMB 5325, 3pm-5pm Klaus Altmann Update on deformations of toric varieties

We give a summary about methods and known results in the deformation theory of affine toric varieties. The convex geometric counterpart of this is the decomposition of polyhedra into Minkowski summands.
However, the presence of singularities in codimension two or the combination of different multidegrees require new tools. This leads to the concepts of so-called thickenings of morphisms - meaning deformations "without base space".
In particular, we return to one of the original tasks when dealing with toric geometry: The understanding of algebraic invariants of toric objects in terms of their combinatorics.

Page last modified on Thursday 27 of March, 2014 14:48:13 UTC

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