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.
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.

Wednesday, May 21, 11.30-12.30, JCMB 5205 Sam Payne (Yale) Nonarchimedean methods for multiplication maps

Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the homogeneous coordinate ring of a projective variety. I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry.

This work is parallel in many ways to the limit linear series of Eisenbud and Harris. One key difference is that we focus on degenerations in which the special fiber is not of compact type. In this context, the tropical Riemann-Roch theory of Baker and Norine and Baker’s specialization lemma are starting points for sometimes intricate calculations in component groups of Neron models and on skeletons of Berkovich curves.

Tuesday, May 27, 2pm-3pm, JCMB 5215 Jason Lo (MPI Bonn) Fourier-Mukai transforms on elliptic fibrationsIn their 2002 JAG paper, Bridgeland-Maciocia laid out a few ideas for computing Fourier-Mukai transforms for sheaves on elliptic threefolds, which allowed them to construct an isomorphism between two moduli spaces of stable sheaves. I will explain how their ideas can be developed into a toolchest for computing Fourier-Mukai transforms on elliptic fibrations. On elliptic fibrations of any dimension, this toolchest can be used to construct an open immersion from a moduli of polynomial stable complexes to a moduli of stable sheaves, as well as a 1-1 correspondence between line bundles and spectral sheaves.
Tuesday, May 27, 3pm-4pm, JCMB 5215 Ziyu Zhang (Bath) Birational geometry of singular moduli spaces of O'Grady typeWe consider 10 dimensional singular moduli spaces of semistable objects on a projective K3 surface with respect to generic Bridgeland stability conditions. We will show that they admit symplectic resolutions, which are all deformation equivalent to the exceptional 10 dimensional holomorphic symplectic manifold constructed by O'Grady. Moreover, many properties of smooth holomorphic symplectic manifolds still hold for these singular moduli spaces. By generalizing work of Bayer and Macri, we can relate wall crossing on the stability manifold of the K3 surface to birational transformations of these singular moduli spaces. This is a joint work with C.Meachan.
Thursday, June 12, 2pm, JCMB 6311 Nathan Ilten (Berkeley) Vanishing of Higher Cotangent Cohomology and Applications
Monday, June 16, 3pm, JCMB 6206 Dave Anderson (IMPA) Schubert calculus and the Satake correspondenceRecent work of Laksov, Thorup, Gatto, and Santiago has given a perspective in which Schubert calculus on Grassmannians can be realized via certain operators on exterior algebras. This point of view also applies to equivariant and quantum versions of Schubert calculus. In this mostly expository talk, I will explain how their work fits into a more general framework via the geometric Satake correspondence.
Wednesday, June 18, 3pm, JCMB 5215 Pablo Solis (Berkeley) Degenerations of the Moduli space of G bundles on a curveThis talk is about three things: infinite dimensional Lie groups, representation theory and moduli problems. The group in question is called the loop group; it is the space of maps from a circle into a topological group. The loop group has a class of representations -positive energy representations- that generalize the highest weight representations of semisimple groups over the complex numbers. These representations allow one to construct an infinite dimensional space that one can relate to the finite dimensional moduli problem of parametrizing G bundles on a compact Riemann surface. I'll explain this setup and show how it can be used to compactify this moduli space of G bundles when the Riemann surface is allowed to develop singularities. Over the last 30 years other compactifications have been presented by Caparoso, Gieseker, Pandharipande, Seshadri and Nagaraj but this is the first construction that provides a compactification not just for vector bundles but for principal G bundles for an arbitrary simple group over the complex numbers.
Tuesday, July 29, 2pm Emanuele Macri (Ohio State) Stability conditions on abelian threefolds

I will present a new proof of a result by Maciocia and Piyaratne on the existence of Bridgeland stability conditions on abelian threefolds.
As in their work, we prove a Bogomolov-type inequality involving Chern classes of certain stable objects in the derived category. Our approach uses the multiplication maps, and has the advantage of applying to abelian threefolds with arbitrary Picard number.
As a corollary, we can show the existence of Bridgeland stability conditions on all the orbifolds quotients of abelian threefolds, including Calabi-Yau threefolds of abelian-type, and those constructed by Borcea-Voisin.
This is based on joint work in progress with Arend Bayer and Paolo Stellari.

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