Etats de la Recherche : 12 - 15 June 2017
Derived Algebraic Geometry and Interactions

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Résumés / Abstracts

  • Tony Pantev - Gabriele Vezzosi : Introduction to derived algebraic geometry

    These lectures will introduce the main tools that will be used in the other lectures, namely derived stacks and differential calculus over them.

    Plan of the lectures:

    • Lecture 1. Higher stacks and higher derived stacks. Artin derived stacks. Examples.
    • Lecture 2. Differential calculus on Artin derived stacks.
    • Lecture 3. Symplectic and lagrangian structures on Artin derived stacks.


  • Marco Robalo : Introduction to infinity-categories


  • David Ben-Zvi : Derived Algebraic Geometry of Topological Field Theories

    We will explore some of the algebraic structures underlying topological field theories in three and four dimensions. By examining the local, line and surface operators in the theory (the values on spheres of various dimensions) and their multiplication (little disc operads) we will see emerging the main features of Seiberg-Witten/Nekrasov-Shatashvili geometry of moduli spaces of vacua: Poisson varieties carrying canonical quantizations, their resolutions and deformations, classical and quantum integrable systems and their duality. We will illustrate these structures through their realization in the geometric Langlands program.


  • Damien Calaque : Formal moduli problems and formal derived stacks

    In this lecture series we will present Lurie's approach to formal moduli problems and their description via tangent dg Lie algebras. We will also explain how to extend Lurie's result to formal moduli problems under a given derived affine scheme Spec(A), and their relation with the formal derived stacks appearing in the theory of formal localization.

    Plan of the lectures:

    • Lecture 1: formal moduli problems after Lurie (following [Lu,To] and [He]).
    • Lecture 2 : "under Spec(A) formal moduli problems", A-Lie algebroids and graded mixed cdgas.
    • Lecture 3 : relation with the affine formal derived stacks of [CPTVV].


    References:

    • [CPTVV] D. Calaque, T. Pantev, B. Toën, M. Vaquié, G. Vezzosi, Shifted Poisson Structures and Deformation Quantization, arXiv:1506.03699 (Section 2)
    • [He] Benjamin Hennion, Tangent Lie algebra of derived Artin stacks, arXiv:1312.3167
    • [Lu] Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems, http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf (Sections 1, 2 & 5)
    • [To] Bertrand Toën, Problèmes de modules formels, https://perso.math.univ-toulouse.fr/btoen/files/2012/04/Bourbaki-Toen-2016-final1.pdf


  • Dominic Joyce : Derived Differential Geometry

    Derived Differential Geometry is the study of derived smooth manifolds and orbifolds, where “derived” is in the sense of Derived Algebraic Geometry. There are by now several different models for (higher) categories of derived manifolds and derived orbifolds: the “derived manifolds” of Spivak and Borisov-Noel, my “d-manifolds” and “d-orbifolds”, and my “m-Kuranishi spaces” and “Kuranishi spaces”. For derived manifolds without boundary, these are known to be all roughly equivalent, at least at the level of homotopy categories.
    Actually, a prototype version of derived orbifolds has been used for many years in the work of Fukaya, Oh, Ohta and Ono as their “Kuranishi spaces”, a geometric structure on moduli spaces of J-holomorphic curves in symplectic geometry, but it was not understood until recently that these are part of the world of derived geometry.
    Derived manifolds are a (higher) category of geometric spaces which include ordinary smooth manifolds, but also many more singular objects. One reason derived manifolds and derived orbifolds are important is that many moduli spaces in differential geometry, and in complex algebraic geometry, can be given the structure of derived manifolds and derived orbifolds. For example, any moduli space of solutions of a nonlinear elliptic partial differential equation on a compact manifold is a derived manifold.
    Also, compact, oriented derived manifolds and derived orbifolds have “virtual classes”, generalizing the fact that a compact, oriented _n_-dimensional manifold _X_ has a fundamental class [_X_] in its top-dimensional homology group. This means that derived manifolds and orbifolds have applications in enumerative invariant problems (e.g. Donaldson invariants, Gromov-Witten invariants, Seiberg-Witten invariants, Donaldson-Thomas invariants, ...), and generalizations such as Floer homology theories and Fukaya categories.


  • Etienne Mann - Marco Robalo : Recent Progresses in Gromov-Witten invariants and derived algebraic geometry

    Plan of the lectures:

    • Lecture 1: Etienne Mann
      In this talk, we will state our main result. We will explain how derived algebraic geometry comes into play in Gromov-Witten theory.
    • Lecture 2: Marco Robalo
      In this talk, we will give some details of the proof, namely what is the notion of an infini-operad and how from a general result of Toën, one can get back our result.
    • Lecture 3: Etienne Mann
      We will recall the construction of Berhend-Fantechi and Lee about virtual sheaf, and explain how we can compare the derived sheaf with this virtual sheaf.


  • Pavel Safronov : Shifted Poisson geometry

    The goal of these lectures is to describe the theory of Poisson and coisotropic structures on derived algebraic stacks. Many examples of such structures come from Poisson-Lie groups and their generalizations. The development of the theory is motivated by mathematical physics and higher deformation quantization.

    Plan of the lectures:

    • Lecture 1. P_n-algebras and their deformations.
    • Lecture 2. Shifted coisotropic structures and a comparison with shifted symplectic geometry.
    • Lecture 3. Examples.