• Enrica Floris: Singularities in the MMP
    We will survey the different definitions of singularities of pairs
    (terminal, canonical, klt, lc) and give many examples illustrating the
    subtleties of the various definitions. We will prove some geometric
    properties and explain how the singularities play a role in vanishing
    theorems.
     
  • Anne Moreau: Vertex algebras and singularities
    In this series of lectures, we will explain how the geometry of
    certain Poisson varieties, their singularities and their arc spaces,
    can be used to investigate some interesting problems on vertex
    algebras. Among the topics, we will explore remarkable connections
    between the characters of some vertex algebras (in the framework of
    W-algebras) and the Hilbert series of arc spaces associated with
    certain Poisson varieties.
    Reference for the lectures 
     
  • Mircea Mustaţă: Mixed Hodge modules and singularities
    Lecture I: V-filtrations and b-functions 
    Lecture II: The Hodge filtration on local cohomology
    Lecture III: The minimal exponent, the Hodge filtration, and the Du Bois complex

    Abstract: D-modules on a smooth complex algebraic variety X are
    sheaves of modules over the sheaf of differential operators on X.
    Saito's theory of Hodge modules provides a functorial framework for
    doing Hodge theory, in which the basic objects are D-modules endowed
    with a canonical good filtration (the Hodge filtration). In this
    lecture series, we will discuss the connection between the Hodge
    filtration on the localization O_X[1/f] and the singularities of the
    hypersurface defined by f.

    Lecture I: A key notion in the theory of Hodge modules is that of
    V-filtration, a concept introduced by Malgrange and Kashiwara. Its
    main feature is that it provides the D-module counterpart for the
    notions of nearby and vanishing cycles, and, as such, it provides
    invariants of singularities that are interesting in their own
    right. In the first lecture, we will discuss the V-filtration and its
    connection with the notion of b-function (or Bernstein-Sato
    polynomial).

    Lecture II: After a brief introduction to Hodge modules, we will
    discuss the Hodge filtration on the localization O_X[1/f] and its
    connection with the V-filtration and the b-function associated to f.
    If time permits, we will discuss the more general case of Hodge
    filtrations on local cohomology sheaves.

    Lecture III: We will review the Du Bois complex of complex algebraic
    varieties and then discuss its connection with Hodge modules and, in
    particular, to the minimal exponent of hypersurface singularities.

    Prerequisites: Some familiarity with D-module theory would be useful,
    at least to provide context and motivation. For example, the overview
    in Chapter I of the notes available at
    https://websites.umich.edu/~mmustata/NotesTokyo.pdf 
    should be enough for this purpose.