• Lorenzo Fantini: Non-archimedean geometry and singularities
    Valuation theory has played a fundamental role to resolve
    singularities at least since the work of Zariski in the late 1930s.
    In this course I will overview the role played by valuation spaces
    on singularity theory since then, focusing in particular on the
    theory of non-archimedean analytic spaces developed by Vladimir
    Berkovich fifty years later.
     
  • Shihoko Ishii: Arc spaces and singularities
    In 1968, Nash introduced the concept of arc spaces and posed what
    is now known as Nash's problem. This problem was solved in all
    dimensions in 2012, thanks to the contributions of many
    individuals. Nash's problem offers a fresh perspective on
    singularity theory, and the idea of arc spaces plays a significant
    role in its development.
    In my lecture, I will introduce the fundamental properties of arc
    spaces and demonstrate their applications within singularity
    theory. 

    References for the lectures
     
  • Hussein Mourtada: Introduction to resolution of singularities

    Abstract and references
     
  • Tamara Servi: O-minimality and singularities
    O-minimal geometry is situated between general differential
    topology and real algebraic geometry. The objects of study (which
    I will call "o-minimal structures") are categories of real
    sets and functions which have a "tame" topological behaviour:
    finite stratifications and a good dimension theory for sets in the
    category; almost everywhere regularity, factorization theorems,
    uniform asymptotics for parametric families of functions in the
    category. An example of o-minimal structure is given by the
    category S of globally subanalytic sets and functions, generated
    by real analytic functions restricted to compact boxes. Another
    example is given by the category S(exp), generated by S and the
    unrestricted real exponential function. There are also several
    examples of o-minimal structures which are generated by
    (interesting!) functions which are not analytic (such as the
    restriction to the positive reals of Euler's gamma function). The
    notion of o-minimal structure and some of the tools used in tame
    geometry come from model theory, a branch of mathematical
    logic. The main aim of this course is to give a self-contained,
    rather constructive and purely geometric proof of the o-minimality
    of S(exp).
    References
    • L. van den Dries, o-minimal structures and real analytic geometry,
      Current developments in mathematics, 1998 (Cambridge, MA) (Int. Press, Somerville, MA, 1999) 105–152.
    • J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmico-exponentielles,
      Ann. Inst. Fourier (Grenoble) 47 (1997) 859–884.
    • A. Parusiński, Lipschitz stratification of subanalytic sets,
      Ann. Sci. École Norm. Sup. (4) 27 (1994) 661–696.
    • L. van den Dries and P. Speissegger, O-minimal preparation theorems, 
      Model theory and applications, Quaderni di Matematica 11 (Aracne, Rome, 2002) 87–116.