William Veys' Abstracts

1. Arc spaces and motivic integration (3 lectures)
   
   The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in R, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications.
   
   The aim of these lectures is to provide a gentle introduction to these concepts. We will explain the basic notions and first results, and provide concrete examples. More precisely we will treat the (p-adic) number theoretic pre-history of the theory, spaces of jets and arcs on an algebraic variety, the Grothendieck ring of varieties, properties of generating series for jets and truncated arcs, motivic measure and motivic integration with values in the completed Grothendieck ring, some first applications, the motivic volume,motivic zeta functions and the monodromy conjecture.



2. Stringy invariants (2 lectures)
   
   Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological Mirror symmetry test for pairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori's Minimal Model Program.
   
   In these lectures we want to explain our generalizations. Batyrev's invariants can be constructed using either motivic integration or weak factorization. We first indicate where both approaches fail in order to define the 'same' invariants on general singular varieties. Then we construct our generalizations on all varieties without strictly log canonical singularities (these are a kind of 'border class' between mild and general singularities). In arbitrary dimension our stringy zeta functions are defined assuming the Minimal Model Program, more precisely using relative log minimal models.On surfaces we have two different generalized stringy invariants, defined unconditionally.