Past Algebraic and Symplectic Geometry Seminar

17 June 2014
15:45
Lionel Darondeau
Abstract
The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Berczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Beside the original argument coming from equivariant geometry, we will explain our alternative proof of such a formula and we will particularly be interested in the interplay between the two approaches.
  • Algebraic and Symplectic Geometry Seminar
17 June 2014
14:00
Lionel Darondeau
Abstract
Hyperbolicity is the study of the geometry of holomorphic entire curves $f:\mathbb{C}\to X$, with values in a given complex manifold $X$. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^{n}\setminus X_{d}$ of projective hypersurfaces $X_{d}$ having sufficiently high degree $d\gg n$. Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.
  • Algebraic and Symplectic Geometry Seminar
3 June 2014
15:45
Gabriele Vezzosi
Abstract
We will give an idea of derived algebraic geometry and sketch a number of more or less recent directions, including derived symplectic geometry, derived Poisson structures, quantizations of moduli spaces, derived analytic geometry, derived logarithmic geometry and derived quadratic structures.
  • Algebraic and Symplectic Geometry Seminar
27 May 2014
15:45
Elizabeth Baldwin
Abstract
Auctioneers may wish to sell related but different indivisible goods in a single process. To develop such techniques, we study the geometry of how an agent's demanded bundle changes as prices change. This object is the convex-geometric object known as a `tropical hypersurface'. Moreover, simple geometric properties translate directly to economic properties, providing a new taxonomy for economic valuations. When considering multiple agents, we study the unions and intersections of the corresponding tropical hypersurfaces; in particular, properties of the intersection are deeply related to whether competitive equilibrium exists or fails. This leads us to new results and generalisations of existing results on equilibrium existence. The talk will provide an introductory tour to relevant economics to show the context of these applications of tropical geometry. This is joint work with Paul Klemperer.
  • Algebraic and Symplectic Geometry Seminar
27 May 2014
14:00
Abstract

Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will describe a new structure theory, motivated by Hamiltonian reduction (and in particular the Morse theory that results), for some categories (of D-modules) of interest to representation theorists. I will then explain how this implies a modified form of "hyperkahler Kirwan surjectivity" for the cohomology of certain Hamiltonian reductions. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

  • Algebraic and Symplectic Geometry Seminar
20 May 2014
14:00
Alessia Mandini
Abstract
After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold $(M, \omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in it. I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in real $3$-space. Under some genericity assumptions on the edge lengths, the polygon space is a symplectic manifold; in fact, it is a symplectic reduction of Grassmannian of 2-planes in complex $n$-space. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate for their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.
  • Algebraic and Symplectic Geometry Seminar
13 May 2014
15:30
Abstract
<p><span>Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X. &nbsp;Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a differential equation that controls the variation of Hodge structure). &nbsp;Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology. &nbsp;I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.</span></p>
  • Algebraic and Symplectic Geometry Seminar
13 May 2014
14:00
Abstract

Suppose that X and Y are Kahler manifolds or orbifolds which are related by a crepant resolution or flop F.  It is expected that the Gromov--Witten potentials of X and Y should be related by analytic continuation in Kahler parameters combined with a linear symplectomorphism between Givental's symplectic spaces for X and Y.  This linear symplectomorphism is expected to coincide, in a precise sense which I will explain, with the Fourier--Mukai transform on K-theory induced by F.  In this talk I will prove these conjectures, as well as their torus-equivariant generalizations, in the case where X and Y are toric.  
This is joint work with Hiroshi Iritani and Yunfeng Jian

  • Algebraic and Symplectic Geometry Seminar
29 April 2014
15:45
Cristina Manolache
Abstract
Counting curves with given topological properties in a variety is a very old question. Example questions are: How many conics pass through five points in a plane, how many lines are there on a Calabi-Yau 3-fold? There are by now several ways to count curves and the numbers coming from different curve counting theories may be different. We would then like to have methods to compare these numbers. I will present such a general method and show how it works in the case of stable maps and stable quasi-maps.
  • Algebraic and Symplectic Geometry Seminar

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