• Lukas Braun:   Some results on (non-)finitely generated quasiaffine algebras.
    In this talk, I will report about work in progress on questions
    related to (non-)finite generation of quasiaffine algebras. This in
    particular relates to certain Cox rings, reductive invariant rings of
    quasiaffine algebras, and singularities of such rings. Results
    include abstract statements as well as computational methods.

     
  • Ana Belen de Belén de Felipe:  Resolution of reduced curve singularities via one toric morphism.
    The existence of resolution of singularities in positive
    characteristic remains among the major unsolved problems in algebraic
    geometry. In this talk, I will focus on reduced plane curve germs,
    presenting some examples to illustrate the following conjecture stated
    by Teissier (proved for plane branches by Goldin and Teissier, and for
    reduced curve singularities by de Felipe, González-Pérez and
    Mourtada): Let k be an algebraically closed field and let 
    An(k) be the n-dimensional affine space. Given an algebraic
    variety X in An(k), there exist m and an embedding of
    An(k) in Am(k) such that the singularities of X⊂Am(k)
    can be resolved by a toric modification of Am(k). 
     
  • Pierrette Cassou-Noguès: Sur le nombre de Milnor d'une singularité de courbe plane en caractéristique arbitraire. 
     
  • Christopher Chiu: On singularities of the arc space of an algebraic variety.
    The arc space X∞ of an algebraic variety X is typically a
    non-Noetherian scheme of infinite dimensions. Yet its local geometry
    still carries deep information on the singularities of X. We will
    first review the well-established connection between invariants of
    divisorial valuations on X, and invariants of associated maximal
    divisorial sets of X∞ and their generic points. The main result of
    this talk says that one of the latter invariants, the embedding
    codimension, is generically constant over maximal divisorial sets. The
    proof relies on an extension of the theorem of Drinfeld, Grinberg and
    Kazhdan to non-rational arcs, which might be of independent interest.

     
  • Georges Comte:  Bézout's bounds for rational and lacunary complex algebraic plane curves
    I will explain which Bézout's bounds one can obtain in the complex
    case for rational plane curves and lacunary algébraic curves. More
    precisely, I will give lower and upper fewnomial bounds on the number
    of intersection points in a ball of the complex plane, between a
    rational curve P(C) and a lacunary algebraic curve Q=0. These
    bounds depend only on the initial terms of P and on the support of Q.
    This is a joint work with Sébatien Tavenas. 
     
  • Eva Elduque:  On the algebro-geometric realization of group homomorphisms.
    Given a smooth complex quasi-projective curve with an extra orbifold
    structure (which we interpret as a finite set of points with
    multiplicities), we can define its orbifold fundamental group and
    refer to the groups defined in this way as "curve orbifold
    groups". The class of curve orbifold groups includes interesting
    families of groups, like finitely generated free products of cyclic
    groups and triangle groups. In this talk, we will address the
    following problem: given a smooth quasi-projective variety U such that
    its fundamental group surjects onto an infinite curve orbifold group,
    when is this epimorphism induced by an algebraic morphism from U to a
    curve C at the level of (orbifold) fundamental groups? We will also
    see some geometric consequences of this result, as well as discuss
    interesting examples in the case when U is the complement of a plane
    curve in CP². Joint work with José Ignacio Cogolludo-Agustín. 
     
  • Anne Frühbis-Krüger: Simple singularities beyond complete intersections
    Known lists of simple singularities have often been a source and
    testing ground for conjectures on properties of singularities. But
    those lists do not contain non-isolated singularities up to now and
    that has a good reason: there are no simple non-isolated hypersurfaces
    or complete intersections. Hence the first question is the one of
    existence of a simple non-isolated singularity. The class of
    determinantal singularities provides finitely determined non-isolated
    singularities and will be in the center of this talk.
     
  • Ursula Ludwig:  A Morse theoretical complex computing intersection homology
    In this talk we present for a compact pseudomanifold and a given
    perversity in the sense of Goresky and MacPherson a Morse theoretical
    cochain complex, which computes the intersection cohomology of the
    space. The complex generalises the famous Morse-Thom-Smale complex
    associated to a smooth Morse function on a smooth compact manifold to
    pseudomanifolds equipped with socalled radial Morse functions. 
     
  • Laurentiu Maxim:  Higher singularities of local complete intersections
    Higher analogues of rational and Du Bois singularities were introduced
    recently through Hodge theoretic methods, and applied in the context
    of deformation theory, birational geometry, etc. In this talk, I will
    give a brief overview of these singularities, and explain how they can
    be studied through the lens of characteristic classes in the case of
    local complete intersections. (Joint work with Bradley Dirks and
    Sebastian Olano). 
     
  • Irma Pallarès Torres: Rational homology manifolds via cubical hyperresolutions.
    In this talk, we will introduce a characterization of rational
    homology manifolds in terms of cubical hyperresolutions and discuss
    some applications in the theory of characteristic classes of singular
    spaces. 
     
  • Aftab Patel:  The Geometry of Locally Bounded Rational Functions.
    This talk concerns locally bounded rational functions defined on
    non-singular real algebraic varieties — that is, rational functions
    that are bounded in a neighborhood of every point in their
    domain. These functions have already been studied in the past from an
    algebraic point of view. However, in the context of this talk they
    will be examined from geometric perspective, commencing with their
    characterisation. There are two equivalent approaches to
    characterising a locally bounded rational function. The first is via
    the curve selection lemma. The second uses the resolution of
    singularities. The most interesting observation here is that this
    class of functions coincides exactly with the class of functions that
    can be transformed into regular functions with values in the ground
    field (we consider a real closed field here). Since these functions
    are, in general, multi-valued, a careful analysis is required to
    construct a suitable notion of their zero-sets. This talk will
    conclude with an exposition of some other desirable properties of
    these functions including a version of the Łojasiewicz inequality,
    which will lead naturally to the reformulation of the zero sets of
    these functions as subsets of arc-spaces. This, in turn, allows one to
    obtain, in the case of two dimensions, the usual correspondence
    between algebra and geometry that exists for other classes of rational
    functions over real closed fields such as regulous functions. This
    talk is based on joint work with Victor Delage and Goulwen Fichou.

     
  • Tomasz Pełka:  Symplectic geometry "at radius zero" and applications.
    Consider a degeneration of Kähler manifolds over a punctured disk
    whose central fiber is snc. The A'Campo space (constructed
    jointly with J. F. de Bobadilla) extends it to a symplectic fibration
    over an annulus, whose restriction to the inner ("radius zero") circle
    has a simple combinatorial description. In fact, it is decomposed to
    pieces which are isotropic torus bundles over the strata of the
    central fiber, and the monodromy acts as translation in these tori
    (this description agrees with the topological construction of
    N. A'Campo from the 70s, hence the name). In my talk I will survey
    this construction and highlight its two applications. First, the
    combinatorial description of the monodromy allows to compute its Floer
    homology using McLean's spectral sequence, and thus infer a positive
    answer to the Zariski multiplicity conjecture for μ-constant families
    of isolated hypersurface singularities. Second, over the 0-dimensional
    strata we get Lagrangian tori which have very similar properties as
    those expected by the Kontsevich-Soibelman conjecture for maximal
    Calabi-Yau degenerations. 
     
  • Enrico Savi:  The Nash-Tognoli theorem over Q & the Q-algebraicity problem for isolated singularities
    Abstract and references 
     
  • Bernd Schober:  Frieze patterns and combinatorics of curve singularities
    Conway-Coxeter frieze patterns are arrays of integers satisfying a
    determinantal condition, the so-called diamond rule. Recently, these
    combinatorial objects have been of considerable interest in
    representation theory since they encode cluster combinatorics of type A. 
    In this talk, we will discuss a new connection between
    Conway-Coxeter frieze patterns and complex plane curve singularities:
    via the beautiful relation between friezes and triangulations of
    polygons one can relate each frieze to the so-called lotus of a curve
    singularity, which was introduced by Popescu-Pampu. In particular,
    this allows to interpret certain entries in the frieze in terms of
    invariants of the curve singularity. 
    This is joint work with Eleonore Faber.