Past Algebraic Geometry Seminar

E.g., 2019-09-18
E.g., 2019-09-18
E.g., 2019-09-18
25 June 2019
15:30
Kobi Kremnizer
Abstract

In this talk I will describe how to make sense of the function $(1+t)^x$ over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz. 

  • Algebraic Geometry Seminar
18 June 2019
15:30
Marco Gualtieri
Abstract

After reviewing our recent description of generalized Kahler structures in terms of holomorphic symplectic Morita equivalence, I will describe how this can be used for explicit constructions of toric generalized Kahler metrics.  Then I will describe how these ideas, combined with concepts from geometric quantization, provide a new approach to noncommutative algebraic geometry.

  • Algebraic Geometry Seminar
11 June 2019
15:30
Alastair Craw
Abstract

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

  • Algebraic Geometry Seminar
28 May 2019
15:30
Dominic Bunnett
Abstract

The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory.  I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces in certain weighted projective spaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.

  • Algebraic Geometry Seminar
21 May 2019
15:30
Adam Gyenge
Abstract

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

  • Algebraic Geometry Seminar
14 May 2019
15:30
Masaki Kashiwara
Abstract

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

  • Algebraic Geometry Seminar
7 May 2019
15:30
Fatemeh Mohammadi
Abstract

Many toric degenerations and integrable systems of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can also be seen to arise from the cones of the tropicalisation of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians Gr(k,n) and prove a necessary condition for such an initial degeneration to be toric. I will present several combinatorial conjectures and computational challenges around this problem.  This is based on joint works with Kristin Shaw and with Oliver Clarke.

  • Algebraic Geometry Seminar
26 February 2019
15:30
Sven Meinhardt
Abstract

I will explain the notion of a singular ring and sketch how singular rings provide field and vertex algebras introduced by Borcherds and Kac. All of these notions make sense in general symmetric monoidal categories and behave nicely with respect to symmetric lax monoidal functors. I will provide a complete classification of singular rings if the tensor product is a cartesian product. This applies in particular to categories of topological spaces or (algebraic) stacks equipped with the usual cartesian product. Moduli spaces provide a rich source of examples of singular rings. By combining these ideas, we obtain vertex and field algebras for each reasonable moduli space and each choice of an orientable homology theory. This generalizes a recent construction of vertex algebras by Dominic Joyce.

  • Algebraic Geometry Seminar

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